diff --git "a/RNFAT4oBgHgl3EQf0x4X/content/tmp_files/load_file.txt" "b/RNFAT4oBgHgl3EQf0x4X/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/RNFAT4oBgHgl3EQf0x4X/content/tmp_files/load_file.txt" @@ -0,0 +1,954 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf,len=953 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='08705v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='SI] 20 Jan 2023 Multi-soliton solutions of the sine-Gordon equation with elliptic-function background Daisuke A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Takahashi1,2∗ 1Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 2Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan E-mail: takahashi@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='chuo-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='jp Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The multi-soliton solution of the sine-Gordon equation in the presence of elliptic- function background is derived by the inverse scattering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The key tool in our formulation is the Lax pair written by 4×4 matrix differential operators given by Takhtadzhyan and Faddeev in 1974, which enables us to use the conventional form of the integral representation of the Jost solutions and Krichever’s theory of commuting differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' As a by-product we also provide generalized orthogonality and completeness relations for eigenfunctions associated with indefinite inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The multi-soliton solution is expressed by a determinant of theta functions and the shift of the background lattice due to solitons is also determined using addition formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' One kink and one breather solutions are presented by animated gifs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Introduction The sine-Gordon (SG) equation has a wide variety of applications in physics and other natural sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' For example, it appears as a continuum limit of the Frenkel-Kontorowa model describing the commensurate-incommensurate transition (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [1], chapter 6 and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [2], chapter 10), the effective model for the chiral magnet and soliton propagation on it [3, 4], and kinks in the long Josephson junction of superconductors [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The model where the partial derivative is changed from hyperbolic to elliptic type has also been solved and applied to vortices and dislocations in spatially two-dimensional systems [8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The rogue-wave solutions with unstable background has been recently investigated [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' From the view of the theory of classical integrable systems, the SG equation is one of the earliest equations formulated via the zero-curvature expression, known as the Ablowitz-Kaup-Newell-Segur (AKNS) formalism [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The analogical integrable models with higher group symmetries have been also investigated (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [14] and references found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [15], part II, chapter I, §8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In some of the above-mentioned works, the multi-soliton excited states not only for uniform background but also for oscillating non-uniform background attract considerable physical attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' While the multi-soliton solutions of an integrable equation in the presence of the elliptic-function, or more general quasiperiodic Riemann theta function background, can be in principle obtained by taking a special limit of the finite-zone quasiperiodic solutions [16], the procedure is often complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Furthermore, in physical applications, the eigenfunctions of the Lax pair, which is necessary in construction of soliton solutions, often ∗ The current primary affiliation is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 2 play additional roles in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', calculation of dispersion relations of linearized waves and linear stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, constructing the soliton solutions with non-uniform background not from a degenerate case of quasiperiodic solutions but by adding solitons to the stationary background using eigenfunctions of the Lax pair via various conventional methods provides helpful by-products in investigation of physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Motivated by these circumstances, in this paper, we derive the multi-soliton solution of the SG equation in the presence of elliptic-function background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Though this problem itself has been solved in many references including the above-mentioned ones, we believe that the following features (i)-(iii) will bring some new light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (i) We use the Lax formalism based on 4 × 4 matrix differential operator by Takhtadzhyan and Faddeev [17], which is a variant of the Lax pair written by an integral operator [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' While the famous treatment of the SG equation in soliton theory is the zero-curvature expression using 2 × 2 matrices which depend on the spectral parameter [12, 13], the use of this method has the following advantages: (a) the theory of commuting differential operators [19] can be applied and (b) the common integral representation of Jost solutions can be used without modification so we do not need trial and error to find suitable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (ii) The final expression of the multi-soliton solution, which is obtained by formulating the inverse scattering method (ISM) and solving the Gelfand-Levitan-Marchenko (GLM) equation, is compactly summarized as a determinant of theta functions, and its asymptotic form is also derived using the addition formulas in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (iii) The orthogonal and completeness relations of eigenfunctions arising from the indefinite inner product is discussed in detail, which is necessary in formulation of the ISM and determination of the possible emerging patterns of discrete eigenvalues corresponding to solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This treatment is regarded as a generalization of the corresponding finite- dimensional linear algebra [21] to continuous space and differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 2, we write down the Lax pair of the SG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 3, we point out that the Lax operators are “σ-self-adjoint”, and introduce the associated indefinite inner product which defines the generalized orthogonal relations to eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 4, we derive simultaneous eigenfunctions of the Lax pair for the stationary soliton lattice solution, and identify the corresponding algebraic curve, based on the theory of commuting differential operators [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The eigenvalues and eigenfunctions are parametrized by uniformization variable on torus, and their symmetries are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 5, we define left and right Jost solutions and the scattering matrix for the system where the background potential asymptotically tends to the stationary soliton lattice at spatial infinities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 6, we introduce the integral representation of the Jost solution, and present the GLM equation which determines the kernel function of the integral representation from the scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 7, we solve the GLM equation for reflectionless case and determine the multi-soliton solution, which is expressed by determinant of theta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The phase shift of the background lattice is also found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 8, by solving the time-dependent Lax equation, we determine the time-evolution of the scattering matrix, and using this, we derive the time- dependent multi-soliton solution for the SG equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 9, we provide the constraint that the discrete eigenvalues must satisfy in order for the resultant solution to become real and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 10, we address the gif animations of the soliton solutions generated by Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The method of visualization and used numerical values are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 11 is devoted to summary and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The appendices discuss several technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Appendix A, the integration formula necessary to obtain the eigenfunction is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Appendix B, we show the detailed derivation of the eigenfunctions given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Appendix C, we derive the completeness relation for eigenfunctions of the soliton lattice potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' There, we also discuss several technical important points on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', the zero- norm eigenfunction defined by indefinite inner product and the necessity of the expression 3 by meromorphic integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Appendix D, we provide the detailed derivation of the GLM equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Appendix E, we calculate the determinant of theta functions appearing in the asymptotic form of the soliton solution using addition formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Lax pair We write an n × n identity and zero matrix as In and On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let σ1, σ2, and σ3 be the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The Lax pair and the Lax equation for the SG equation is given by [17] 4i∂ ˆL ∂t = [ ˆL, ˆB], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) ˆL = −4i∂x �σ3 O2 � + � iσ1w e−iφσ2/2 e−iφσ2/2 O2 � , w = φx + φt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) ˆB = 4i∂x �I2 −I2 � − 2 � σ3e−iφσ2/2 e−iφσ2/2σ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) then reduces to the SG equation φtt − φxx + sin φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Here we slightly changed prefactors of operators and choice of the SU(2) basis in the way (σ1, σ2, σ3) → (σ1, σ3, −σ2) from the original work Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This makes no essential difference in formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Symmetry of the Lax operator and orthogonality of eigenfunctions Henceforth we simply write σ := σ3 ⊕ σ3 = I2 ⊗ σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, ˆL and ˆB are both “σ-self- adjoint”[21], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', σ ˆL†σ = ˆL, σ ˆB†σ = ˆB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) Therefore the orthogonal and completeness relations of eigenfunctions of these operators are defined through the indefinite inner product, called “σ-inner product” in [21] : (f1, f2)σ := � dxf † 1 σf2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The correct identification of the completeness relation is important in derivation of the GLM equation (Appendix C and Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Following the established procedure of the ISM, we first consider the eigenvalue problem of ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Since the highest-order coefficient matrix of ˆL is given by σ3 ⊕ O2, which obviously has rank 2, there exist two linearly independent eigenfunctions for a given eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' They can be written in the form f = � g λ−1e−iφσ2/2g � with g a two-component column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If we rewrite the equation with respect to g using the light-cone coordinate x′ = x+t 2 , t′ = x−t 2 , the famous 2 × 2 AKNS form [12] is reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let f1 = � g1 1 λ e−iφσ2/2g1 � and f2 = � g2 1 λ e−iφσ2/2g2 � be eigenfunctions of ˆL with eigenvalues λ1 and λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, we can show 4i � f † 1 (I2 ⊕ O2)f2 � x = (λ∗ 1 − λ2)f † 1 σf2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) Therefore, if λ∗ 1 � λ2, then we have � dxf † 1 σf2 = 0, which is the above-mentioned orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' On the other hand, when λ∗ 1 = λ2, we obtain f † 1 (I2 ⊕ O2)f2 = g† 1g2 = (x-independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We can also show that if λ1 = λ2, det(g1, g2) does not depend on x, which 4 can be more easily shown using the AKNS form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In addition to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1), ˆL further has the following symmetry: (I2 ⊗ σ2) ˆL(I2 ⊗ σ2) = ˆL∗, (σ3 ⊗ σ2) ˆL(σ3 ⊗ σ2) = − ˆL, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) which immediately means ˆLf = λf ↔ ˆL[(I2 ⊗ σ2)f ∗] = λ∗(I2 ⊗ σ2)f ∗ ↔ ˆL[(σ3 ⊗ σ2)f] = −λ(σ3 ⊗ σ2)f ↔ ˆL[(σ3 ⊗ I2)f ∗] = −λ∗(σ3 ⊗ I2)f ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Thus the eigenvalues λ, λ∗, −λ, −λ∗ always appear simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' ˆL also has a little unfamiliar symmetry: ˆL f = λf ↔ ˜L[(σ2 ⊗ σ3)f] = −λ−1(σ2 ⊗ σ3)f, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) ˜L := an operator such that w in ˆL is replaced by ˜w = 2φx − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) We thus find the identity ˆL−1 = −(σ2 ⊗ σ3) ˜L(σ2 ⊗ σ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) It is unusual that the inverse of a matrix differential operator can be written down explicitly, and furthermore, its expression does not include an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Such situation seems to occur when the highest-order coefficient matrix is not full-rank, which is −4iσ3 ⊕ O2 for the present ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In particular, for the stationary solution of the SG equation φt = 0, we find w = ˜w = φx and hence ˆL = ˜L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In this case ˆL−1 = −(σ2 ⊗ σ3) ˆL(σ2 ⊗ σ3) holds and the eigenvalues λ and −λ−1 appear in pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We, however, again emphasize that this relation does not hold for general w � φx, and therefore when we consider the scattering matrix S in the ISM in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 5, we must not impose this symmetry to ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Soliton lattice solution, Riemann surface, and eigenfunctions Let us consider the eigenfunction for the stationary solution, which we henceforth write φ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Since the Lax pair commutes [ ˆL, ˆB] = 0 for the stationary solution, we consider the simultaneous eigenfunction: ˆLf0 = λf0, ˆBf0 = ωf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) We first determine φ0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The first integral for the stationary SG equation is φ2 0x + 2 cos φ0 = 4 m − 2 ↔ ( 1 2φ0x)2 1 − m sin2 φ0−π 2 = 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) Integrating this equation once again yields the soliton lattice solution as φ0(x) = π + 2 am � x √m ���m � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) φ′ 0(x) = 2 √m dn � x √m ���m � , cos φ0(x) 2 = − sn � x √m ���m � , sin φ0(x) 2 = cn � x √m ���m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Here and henceforth, we use the notations of elliptic functions in the Abramowitz-Stegun book [22] unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If 0 < m < 1, it is the rotating solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If m = 1, it 5 represents the stationary one-kink solution, and if m > 1, it is an oscillating solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In this paper we only consider the rotating background 0 < m < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Generally, the pair of commuting differential operators satisfies an algebraic relation P( ˆL, ˆB) = 0 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Strictly speaking, Krichever’s original paper [19] only consider the case where the highest-order coefficient matrices of the matrix differential operators are invertible, but similar nature can be seen even if this assumption is not satisfied and an algebraic curve can be defined in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In the present case, P(λ, ω) = 0 gives a genus-one curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', an elliptic curve: λ2ω2 = λ4 + 2 � 2 m − 1 � λ2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) It can be parametrized by elliptic functions: λ(z) = −i √m sn(iz|m) cn(iz|m) dn(iz|m) = i √m sn(iz|m) sn(iz − K|m), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) ω(z) = sn2(iz|m) − sn2(iz − K|m) λ(z) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) where K = K(m) and K′ = K(1 − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The eigenfunction f0(x, z) for the above parametrized eigenvalue λ = λ(z) and the corresponding crystal momentum k(z) is k(z) = Z(2iz + iK′|m) + Z(2iz − iK′|m) 4i √m , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) f0(x, z) = eik(z)xΘ4(0) 2Θ4( x √m) \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Θ1( x √m − iz)/Θ4(iz) Θ2( x √m − iz)/Θ3(iz) −iΘ4( x √m − iz)/Θ1(iz) −iΘ3( x √m − iz)/Θ2(iz) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) Here, we define the scaled theta functions by Θi(u|m) := [ϑi( πu 2K , q)]Abramowitz-Stegun with the nome q = e−πK′/K, and the second argument m is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using this scaled theta function, the Jacobi zeta function is expressed as Z(u|m) = d du ln Θ4(u|m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' λ(z), ω(z), k(z) satisfy the following (quasi-)periodicity, parity, and the complex conjugation relations: λ(z) = (−1)l+nλ(z + nK′ + ilK)(−1)n = −λ(−z) = λ(z∗)∗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10) ω(z) = (−1)nω(z + nK′ + ilK) = −ω(−z) = ω(z∗)∗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11) k(z) = k(z + nK′ + ilK) + nπ 2K √m = −k(−z) = k(z∗), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12) for n, l ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The eigenfunction f0(x, z) also has the symmetries f0(x, z + nK′ + ilK) = il � (σl 3σn 2) ⊗ (σl 2σn 3) � f0(x, z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13) f0(x, −z∗) = (σ3 ⊗ I2)f0(x, z)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14) The two linearly independent solution for a given eigenvalue λ = λ(z) are f0(x, z) and f0(x, −z − iK), because λ(z) = λ(−z − iK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' To cover all eigenfunctions, the region we must consider is the rectangle with vertices z = ±K′ ± iK, which is the fundamental period parallelogram of λ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The bounded 6 eigenfunctions with k(z) ∈ R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', the scattering eigenstates, lie on the lines z ∈ R + iK 2 Z, and there are four independent lines in one fundamental period parallelogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Among them, R, R−iK gives monotonically increasing k(z) and real λ(z), while R± iK 2 gives monotonically decreasing k(z) and pure imaginary λ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The region satisfying Im k(z) > 0 is given by −K < Im z < − K 2 , 0 < Im z < K 2 , and the positions of zeros of a(z) corresponding to bound eigenstates are chosen from this region in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The completeness relation satisfied for scattering eigenstates is δ(x − y)I4 = � C1+C2 dz 2π f0(x, z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15) Here, C1 is the rectangular contour passing through the vertices −K′ → K′ → K′ + iK 2 → −K′ + iK 2 → −K′ and C2 is the one −K′ − iK → K′ − iK → K′ − iK 2 → −K′ − iK 2 → −K′ − iK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Note that the integrations along vertical lines cancel out due to the periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, the actual path with non-zero contribution is � C1+C2 = � K′ −K′ − � K′+ iK 2 −K′+ iK 2 + � K′−iK −K′−iK − � K′− iK 2 −K′− iK 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15) is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Scattering matrix Henceforth, we consider the scattering problem under the condition that the background potential tends to the stationary soliton lattice at spatial infinities x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let the potential φ(x) satisfy the boundary condition φ(x) → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 φ0(x) (x → −∞), φ0(x − x0) (x → +∞), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) where x0 represents the phase shift of the background lattice caused by solitons and ripple waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We define the left and right Jost solutions by the asymptotic form f−(x, z) → f0(x, z) (x → −∞), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) f+(x, z) → f0(x − x0, z) (x → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) The scattering matrix S (z) is then defined by the linear transformation between these two: � f+(x, z) f+(x, −z − iK) � = � f−(x, z) f−(x, −z − iK) � S (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) The uniqueness of the definition of the Jost solution by its asymptotic form and the properties of eigenfunctions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14) imply f±(x, z) = (−iσ3 ⊗ σ2)l f±(x, z + 2nK′ + ilK) = (σ3 ⊗ I2)f±(x, −z∗)∗, n, l ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) Note that while f0(x, z) has an additional symmetry with respect to translation of z by (2n + 1)K′ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13)), the corresponding symmetry does not exist for Jost solutions f±(x, z), because this additional symmetry is coming from special nature of φ0(x) satisfying ˆL = ˜L in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5), we have S (z) = σl 3S (z + 2nK′ + ilK)σl 3 = S (−z∗)∗ = σ1S (−z − iK)σ1, n, l ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) Equating the integration constant obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) at x → ±∞, we find S (z)−1 = S (z∗)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) 7 We can also show det S = 1 from the constancy of the Wronskian det(g1, g2) (see the sentence after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, S and its inverse matrix has the forms S (z) = �a(z) −b(z∗)∗ b(z) a(z∗)∗ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) S (z)−1 = �a(z∗)∗ b(z∗)∗ −b(z) a(z) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) with matrix elements satisfying a(z)a(z∗)∗ + b(z)b(z∗)∗ = 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10) a(z) = a(−z∗)∗ = a(z − iK), b(z) = b(−z∗)∗ = −b(z − iK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11) From this, the zeros of a(z) always appear simultaneously at four points z, −z∗, z−iK, −z∗−iK, and the orders of these zeros are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Integral representation of Jost solution and Gelfand-Levitan-Marchenko equation Let us introduce the integral representation for the left Jost solution f−(x, z) = f0(x, z) + � x −∞ dyΓ(x, y)f0(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) The kernel function Γ(x, y) is a 4 × 4 matrix and assumed to decrease exponentially in the limits x, y → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In order for this integral to converge, the integrand must decrease y → −∞, and this condition is satisfied at least for eigenfunctions with Im k ≤ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', −K/2 ≤ Im z ≤ 0, K/2 ≤ Im z ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4), we immediately conclude that Γ(x, y) = (σ3 ⊗ σ2)Γ(x, y)(σ3 ⊗ σ2) = (σ3 ⊗ I2)Γ(x, y)∗(σ3 ⊗ I2) = (I2 ⊗ σ2)Γ(x, y)∗(I2 ⊗ σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) Let us now write ˆL0(x) = −4iσ3 ⊕ O2 + U0(x), U0(x) = � iσ1w0 e−iφ0σ2/2 e−iφ0σ2/2 O2 � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) ˆL(x) = −4iσ3 ⊕ O2 + U(x), U0(x) = � iσ1w e−iφσ2/2 e−iφσ2/2 O2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Following the same derivation as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [23], section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8, (In the present case, the first-order coefficient matrix of ˆL is not full-rank, but it makes no difference to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=') U(x) − U0(x) = 4i�σ3 ⊕ O2, Γ(x, x)�, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) ˆL(x)Γ(x, y) = Γ(x, y) ˆL0(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5), w = w0 + 2Γ12, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) eiφ/2 = eiφ0/2 + 4(iΓ13 + Γ14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' � ↔ cos φ 2 = cos φ0 2 + 4iΓ13, sin φ 2 = sin φ0 2 − 4iΓ14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) where we briefly write the matrix element of Γ(x, x) as Γi j = [Γ(x, x)]i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Note that Γ12, iΓ13, and iΓ14 are real-valued functions due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using these relations, we can re-construct 8 the potential from the kernel Γ, which is determined from scattering data by solving the GLM equation shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The integral kernel Γ(x, y) is uniquely determined from the following scattering data: (i) the values of the reflection coefficient for scattering states r(z) = b(z)/a(z), where scattering states mean the bounded eigenfunction s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' k(z) ∈ R, appearing on lines z ∈ R + iK 2 Z, and (ii) the list of the zeros of a(z), which we write z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' , zn, and the normalization factor c2 j := b(z j) i˙a(z j), where zj’s are to be chosen from the region Im k(zj) > 0, and due to the property (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11), the zeros simultaneously appear at z, −z∗, z − iK, −z∗ − iK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The values of c2 j also have some constraint, whose detail will be described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Here, we only treat the case where all zeros of a(z) are first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' For uniform background, the solitons corresponding to higher- order zeros are discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The case of unstable uniform background is considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The GLM equation that determines the kernel Γ(x, y) from the above-mentioned scattering data is given by Γ(x, y) + Ω(x, y) + � x −∞ dwΓ(x, w)Ω(w, y) = 0, y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) Here, Ω = Ωbd + Ωsc with Ωbd(x, y) := W(x)W(y)T(I2 ⊗ σ1), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10) W(x) := −iσ3 ⊗ σ2 (f0(x, −z1)c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', f0(x, −zn)cn) , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11) Ωsc(x, y) := \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � K′ −K′ − � K′+ iK 2 −K′+ iK 2 + � K′+iK −K′−iK − � K′− iK 2 −K′− iK 2 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb dz 2π f0(x, −z − iK)r(z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12) Ωbd represents the contribution from bound states, and Ωsc is that from scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The derivation of the GLM equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12) is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Reflectionless potentials Let us solve the GLM equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) for the reflectionless case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Ωsc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The ansatz for the kernel Γ(x, y) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let us introduce the notation H(x) = (h1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', hn(x)), where each hi(x) is four-component column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We then assume Γ(x, y) = H(x)W(y)T(I2 ⊗ σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) Substituting it to the GLM equation, we obtain the equation for H(x): H(x) + W(x) + H(x)G(x) = 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) where we define the n × n Gram matrix by G(x) = � x −∞ dyW(y)T(I2 ⊗ σ1)W(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2), H = −W(In+G)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let Gl j(x) be the (l, j)-component of G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) and the symmetries of λ(z) and f0(x, z), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14), we find Gl j(x) = clc j 4 f0(x, −zl)T(σ2 ⊕ O2)f0(x, −zj) λ(zl) − λ(zj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) 9 It can be simplified using the Weierstrass three-term formula in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [20], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The result is Gl j(x) = clc je−i[k(zl)+k(z j)]x Θ2Θ4Θ3( x √m + i(zl + zj)) Θ3Θ2(i(zl + zj))Θ4( x √m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) Thus, Γ(x, y) = −W(x)[In + G(x)]−1W(y)T(I2 ⊗ σ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) Let us write W as an array of row vectors: W = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed W1 W2 W3 W4 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8, where each Wi is a 1 × n row vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then we get the expressions Γ13 = −W1(In + G)−1WT 4 , Γ14 = −W1(In + G)−1WT 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using the relation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) and the Woodbury-type identity 1 + ⃗a†A−1⃗b = det(A+⃗b⃗a†) det A , cos φ 2 = cos φ0 2 + 4iΓ13 = � cos φ0 2 � det(I + P) det(I + G), Pi j = Gi j − 4i cos φ0 2 f03(x, −zi)f02(x, −zj), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) sin φ 2 = sin φ0 2 − 4iΓ14 = � sin φ0 2 � det(I + Q) det(I + G), Qi j = Gi j − 4i sin φ0 2 f04(x, −zi)f02(x, −zj), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) where f0 j(x, z) represents the j-th component of f0(x, z) defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' These can be simplified by using Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [20], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Rewriting C j = 1 c2 j √m, and introducing new matrices G, P, Q through the relations Gi j = m1/4cic je−i[k(zi)+k(z j)]xGi j, Pi j = m1/4cic je−i[k(zi)+k(z j)]xPi j, Qi j = m1/4cic je−i[k(zi)+k(z j)]xQi j, we obtain the final expressions cos φ 2 = � cos φ0 2 � det(E + P) det(E + G), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) sin φ 2 = � sin φ0 2 � det(E + Q) det(E + G), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10) where the components of n × n matrices E, G, P, and Q are given by Ei j = δi jC je2ik(z j)x, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11) Gi j = Θ4Θ3( x √m + i(zi + zj)) Θ2(i(zi + zj))Θ4( x √m) , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12) Pi j = �Θ4(izi) Θ1(izi) � \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed −Θ4Θ2( x √m + i(zi + zj)) Θ2(i(zi + zj))Θ1( x √m) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 �Θ2(izj) Θ3(izj) � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13) Qi j = �Θ4(izi) Θ2(izi) � \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Θ4Θ1( x √m + i(zi + zj)) Θ2(i(zi + zj))Θ2( x √m) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 �Θ1(izj) Θ3(izj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14) Since the wavenumbers used to make bound states all satisfy Im k(zj) > 0, E → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 ∞ (x → −∞), 0 (x → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15) 10 Therefore, the asymptotic form of φ is given by cos φ 2 → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 cos φ0 2 (x → −∞), cos φ0 2 det PG−1 (x → +∞), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='16) sin φ 2 → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 sin φ0 2 (x → −∞), sin φ0 2 det QG−1 (x → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='17) The determinant appearing in x → +∞ is calculated in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The resultant asymptotic form is φ(x) → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 φ(x) (x → −∞), φ(x − x0) (x → +∞), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='18) x0 := −2i √m n � j=1 zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='19) x0 has the meaning of the phase shift of the background lattice caused by solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The realness of x0 follows from the fact that the zeros of a(z) always simultaneously emerge at four points z, z − iK, −z∗, and −z∗ − iK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Time evolution Finally, let us determine the time dependence of scattering matrix if the system obeys the Lax equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let f be a time-dependent eigenfunction of ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, the Lax equation implies ˆLf = λf, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) 4ift = − ˆBf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) We now define the time-dependent eigenfunction by the initial condition ˜f+(t = 0, x, z) = f+(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Though ˆB is now a time-dependent operator for finite x, it is time-independent at x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, the asymptotic forms of this ˜f+(t, x, z) is given by ( ˜f+(t, x, z), ˜f+(t, x, −z − iK)) → \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 (f0(x, z), f0(x, −z − iK))eiω(z)tσ3/4S (z) (x → −∞), (f0(x − x0, z), f0(x − x0, −z − iK))eiω(z)tσ3/4 (x → +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) Here, we have used the relation ω(−z − iK) = −ω(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, if we define the time- dependent right Jost solution by the asymptotic form f+(t, x, z) → f0(x − x0, z) for x → +∞, which is different from ˜f+(t, x, z), then the relation ( ˜f+(t, x, z), ˜f+(t, x, −z − iK)) = (f+(t, x, z), f+(t, x, −z − iK))eiω(z)t/4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then the time evolution of the scattering matrix is given by S (t, z) = eiω(z)tσ3/4S (z)e−iω(z)tσ3/4, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) or for each component, a(t, z) = a(z), b(t, z) = e−iω(z)t/2b(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) Since C j is defined by C j = 1 c2 j √m = i˙a(z j) √mb(z j), its time evolution is written as C j(t) = C jeiω(z j)t/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) 11 Therefore, we can define the time-dependent Ei j(t) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11), we replace C j by C j(t) = C jeiω(z j)t/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Ei j(t) = δi jC je2ik(z j)x+iω(z j)t/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) Then the time-dependent solution of the SG equation is given by replacing E with E(t) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10): cos φ 2 = � cos φ0 2 � det(E(t) + P) det(E(t) + G), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) sin φ 2 = � sin φ0 2 � det(E(t) + Q) det(E(t) + G), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) where the definition of G, P, and Q remains unchanged (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12)-(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Limitation to eigenvalues zj’s and normalization coefficients C j’s In order for the function φ(x, t) to be real and bounded, zj’s, which are the zeros of a(z) corresponding to the discrete eigenvalues for bound states, and the normalization coefficients C j’s must satisfy several conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1 Here we describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' First, depending on its value, zj’s must satisfy the following (i) or (ii): (i) zj, zj − iK, −z∗ j, −z∗ j − iK appear simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Due to this symmetry, one of these four zj can be chosen from 0 ≤ Re zj ≤ K′, 0 < Im zj < K 2 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (ii) If, as a special case, zj = irj or zj = irj + K′ with rj ∈ (0, K 2 ), then just two zeros zj and zj − iK appear simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The case (i) corresponds to the breather solution and the case (ii) the traveling one kink solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The total number of zj’s are always even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If there are breathers and even number of kinks, the number is a multiple of four, while if there exists odd number of kinks, the number is of the form 4n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The velocity of the soliton is given by v j = − Im ω(z j) 4 Im k(z j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In particular, for one kink solution, it reduces to v j = − ω(z j) 4k(z j), which is negative if zj = irj and positive if zj = irj + K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Next let us consider the condition for C j’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11), the derivative of a(z) satisfies ˙a(z) = −˙a(−z∗)∗ = ˙a(z − iK), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) and thus if we define C(z) := i˙a(z) √mb(z), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) then it has the symmetry C(z) = C(−z∗)∗ = −C(z − iK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) This symmetry is the same as that of b(z) given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, for the breather solution where four zeros appear, if we temporarily write these four as zj, zj+1 = zj−iK, zj+2 = −z∗ j, zj+3 = −z∗ j − iK, then we should choose the corresponding coefficients as C j+1 = −C j, C j+2 = C∗ j, C j+3 = −C∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' For the kink solution, where two zeros zj and zj+1 = zj − iK appear, C j+1 = −C j are both real and must have the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Whether the solution becomes kink or anti-kink, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', the phase rotation becomes counterclockwise or clockwise, is determined by which one is chosen positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 1 If we consider application to scientific problems where the complex-valued and/or divergent solutions have some appropriate physical interpretations, the restriction stated here can be loosened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 12 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' A snapshot of one kink solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The number of zeros is n = 2 and the parameters are m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='95, z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='02iK + K′, z2 = z1 − iK, C1 = −1, C2 = −C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The snapshot at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Animation of soliton solutions Here, we present a few examples of the soliton solutions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9), by gif animations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We visualize the solution through the 3D plot (x, cos φ(x, t), sin φ(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' An example of snapshot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' See attached files, testx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='gif with x=1,2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Below, we provide the parameters for each solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We write K = K(m) and K′ = K(1 − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (i) test1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='gif: one kink solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The number of zeros is n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The parameters are m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='99, z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='02iK + K′, z2 = z1 − iK, C1 = −1, C2 = −C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The time range is −10 ≤ t ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (ii) test2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='gif: one anti-kink solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' C1 = 1, C2 = −C1, and other parameters are the same as (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (iii) test3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='gif: one breather solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The number of zeros is n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The parameters are m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='99, z1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1iK + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5K′, z2 = z1 − iK, z3 = −z∗ 1, z4 = −z∗ 2, C1 = −1, C2 = −C1, C3 = C∗ 1, C4 = C∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The time range is −10 ≤ t ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Solutions with more number of solitons will be generated using the Mathematica file in google drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Summary and discussion In this paper, we have derived the multi-soliton solutions of the SG equation with elliptic- function background by the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The result is expressed by a determinant of theta functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The shift of the background lattice caused by solitons (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='18) with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='19) has also been found using the addition formula of theta functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' One key tool in our work is the Lax pair represented by 4×4 matrix differential operators (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3), originally introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This most conventional but seemingly outdated approach makes it possible to use the common form of the integral representation of the Jost solution (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) 2 In this file, since the default build-in Jacobi elliptic functions are a little slow, the functions defined by the ratio of theta functions are used alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5 10 -213 without modification and the formulation of the ISM is simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Also, the completeness relations in an indefinite inner product space, which is necessary in the formulation of the ISM, has been discussed in detail (Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Application to various physical phenomena including the dynamics of defects with periodic background, extension to unstable oscillating background (m > 1) and studying multi rogue waves, and the solitons associated with higher- order zeros of a(z), are left as possible future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Before concluding, we provide several technical remarks and perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In this work, we can introduce the concept of orthogonality between eigenfunctions based on the indefinite inner product, since the Lax operators have the symmetry (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If the Lax pair has no such symmetry and any inner product cannot be defined, the completeness relation will be constructed from the set of left and right eigenfunctions, an analog of left and right eigenvectors for finite dimensional matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Such basis is sometimes called a bi-orthogonal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Whether we can always reduce the classical integrable systems written by zero-curvature condition (or a compatibility condition) and/or the Lax pair including an integral operator to the Lax formalism with differential operators seems to be unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (The inverse operation is easy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' if one has an integrable equation written by the Lax form using matrix differential operators, one can immediately rewrite it in a zero-curvature expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=') If we can do it, the integral representation of the Jost solution will be widely applicable and multi-soliton solutions will be easily obtained by the dressing method [26], where we can even omit the full formulation of the ISM if we do not have an interest in general initial-value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We also note that, as emphasized in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 3 and 4, the highest-order coefficient matrix of the Lax operator ˆL is not full-rank and ˆL−1 can be written down explicitly without using an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Though we do not show detail here, a similar property also emerges in the derivative nonlinear Schr¨odinger equation, which is usually treated by the Kaup-Newell form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' That is, if we rewrite it to the Lax form, the highest-order coefficient matrix is not full-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Even in these systems, the algebraic curve from the pair of commuting operators can be appropriately defined (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5)), though, strictly speaking, these cases are not included in the seminal work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Exploring these examples more comprehensively and exhaustively might bring a slight extension to the theory of commuting differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Acknowledgments This work was supported by MEXT-Supported Program Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' S1511006 and JSPS KAKENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' JP19H05821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Integral often emerging in calculation of eigenfunctions for elliptic potentials Let us define scd(z|m) := sn(z|m) cn(z|m) dn(z|m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' When an elliptic-function potential is given in some hierarchy of integrable systems, the calculation of its eigenfunctions often reduces to the following integral: � dxα[scd(αx) + scd β] sn2(αx) − sn2 β = αZ(β)x + ln Θ1(αx − β) Θ4(β)Θ4(αx), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) even if the corresponding Riemann surface has higher genus g > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This formula can be derived by using [27], appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If we interpret this formula using Weierstrass’s functions, it reduces to the formula given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [28], II, Kap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 6, §3, and is called the standard form 14 of the elliptic integral of the third kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using it, the solution to the first-order differential equation fx f = γ + α[scd αx + scd β] sn2 αx − sn2 β (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) is given by f(x) = Ceikx Θ1(αx − β) Θ4(β)Θ4(αx), k = −i(γ + αZ(β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) If m ∈ [0, 1], α > 0, and k is real, then f becomes a twisted periodic function f � x + 4Kl α � = eik 4Kl α f(x), l ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Therefore, if k can be physically interpreted to be crystal momentum of the Bloch function, it is defined up to mod απ 2K and the corresponding Brillouin zone is given by [− απ 4K , απ 4K ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Eigenfunctions for soliton lattice In this appendix, we derive the simultaneous eigenfunction f0(x, z) with its crystal momentum k(z) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9) for the time-independent Lax pair ˆL and ˆB with the soliton lattice potential (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3), when the eigenvalues are parametrized by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The method is in fact the special case of the one given by Krichever [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let us write f0 = � g 1 λ e−iφσ2g � , where g is a two-component vector g = � g1 g2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Eliminating g′ 1 and g′ 2 using the eigenequation of ˆL in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1), and substituting them to that of ˆB, one obtains two linear relations with respect to g1 and g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The determinant of this coefficient matrix must vanish, which gives an elliptic curve (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Using it, one can eliminate either of g1 and g2 and obtain the first-order differential equation only containing one function: g′ 1 g1 = −i(2λ − ω) 4 + i(−1 + λ4 − 2λ3ω + λ2ω2) 2λ(−1 + λ2 − λω + 2 cos2 φ0 2 ) + −φ0x cos φ0 2 sin φ0 2 −1 + λ2 − λω + 2 cos2 φ0 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) g′ 2 g2 = i(2λ + ω) 4 + −i(−1 + λ4 + 2λ3ω + λ2ω2) 2λ(−1 + λ2 + λω + 2 cos2 φ0 2 ) + −φ0x cos φ0 2 sin φ0 2 −1 + λ2 + λω + 2 cos2 φ0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) and the parametrizations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7), the above equations reduce to g′ 1 g1 = −i(2λ − ω) 4 + 1 √m scd( x √m) + scd(iz) sn2( x √m) − sn2(iz) , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) g′ 2 g2 = i(2λ + ω) 4 + 1 √m scd( x √m) + scd(iz − K) sn2( x √m) − sn2(iz �� K) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Therefore, using the formula in Appendix A, we find g1 = C1eik1x Θ1( x √m − iz) Θ4(iz)Θ4( x √m), k1 = ω − 2λ 4 − i√mZ(iz), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) g2 = C2eik2x Θ2( x √m − iz) Θ3(iz)Θ4( x √m), k2 = ω + 2λ 4 − i√mZ(iz − K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) 15 Due to the Bloch (Floquet) theorem, g1 and g2 must share the same crystal momentum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', k1 and k2 must be related by k1 ≡ k2 mod π 2 √mK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In fact, using the periodicity of the Jacobi elliptic and zeta functions, we can check k1 = k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, henceforth we simply write k1 = k2 = k(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The ratio C1/C2 must also be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Rewriting g1/g2 using the Jacobi elliptic function, and checking its consistency with the linear relation between g1 and g2 obtained from the eigenequation of ˆB, we find C1/C2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The crystal momentum k(z) can be further simplified as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The derivative k′(z) can be simplified using the double-angle formula of the cs function: k′(z) = 1 √m � − cs2(2iz) − E K � = 1 2 √m � dn2(2iz + iK′) + dn2(2iz − iK′) − 2E K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) Since Z′ = dn2 − E K , integrating this expression soon gives the Jacobi zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Further- more, the constant of integration can be fixed using some specific value at any point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', k1( K′ 2 ) = − π 4K √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Thus we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The third and fourth components of the eigenfunction, g3 = λ−1 � − sn( x √m)g1 − cn( x √m)g2 � and g4 = λ−1 � cn( x √m)g1 − sn( x √m)g2 � , can be simplified using another expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6), λ(z) = −i Θ1(iz)Θ2(iz) Θ3(iz)Θ4(iz), and the addition formula of theta functions, in particular, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [20], Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5b) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The final expression is then given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Completeness relation In this appendix, we derive the completeness relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15) for eigenfunctions of the soliton lattice potential φ0(x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3 The first important point is that ˆL is not self-adjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', ˆL† � ˆL and instead satisfies ˆL† = σ ˆLσ with σ = I2 ⊗ σ3 as stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' For such operator, the eigenfunction satisfies the orthogonal relation with respect to the indefinite inner product (f, g)σ = � dxf †σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The normalization of eigenfunction is also made based on (f, f)σ and all eigenfunctions are classified into the ones possessing positive, negative, and zero norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The completeness relation for the case of finite-dimensional linear algebra is shown in section 3 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [21], and 3 In order to eliminate confusion, we should explain the difference of the usage of the terminology “eigenvalues an eigenfunctions” between classical integrable systems and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In theoretical physics, particularly in quantum mechanics, when one finds a solution of an eigenequation of a given differential operator ˆL f = λf , the operand function f is called an eigenfunction only when it is a bounded function and only in this case λ is included to the set of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The eigenfunctions possessing everywhere-bounded plane-wave type behavior are called a scattering state and constitute the set of continuous eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', a band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The eigenfunction with localized profile and finite norm is called a bound state and make a discrete eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The completeness relation, which is in bra-ket notation of quantum mechanics often expressed as 1 = � n |n⟩ ⟨n|, is constructed by gathering all these scattering and bound eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In classical integrable systems, the exponentially divergent solution of an eigenequation, which is usually not counted as an eigenfunction, plays an important role in generating multi-soliton solutions by various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' An elementary example is the divergent solution of the Schr¨odinger operator −fxx = −κ2 f with eigenvalue λ = −κ2 < 0 and f = e±κx, which is indeed used to construct the multi-soliton solution of the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Thus, identification of all solutions of the eigenequation for any complex λ is important, and therefore, in the context where no confusion occurs, these divergent solutions f and corresponding values λ are also sometimes called eigenfunctions and eigenvalues, without distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This manuscript also adopts this loose use of terminology in several sections, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' To preserve the logical unambiguity, the “genuine” eigenfunctions with no divergent behaviors are explicitly referred to as scattering and bound eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The completeness relation includes only these “genuine” eigenfunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The solution of eigenequation for arbitrary complex eigenvalue is also often called the Baker- Akhiezer function, whose original meaning is the single-valued function defined on a Riemann surface and possessing finitely many essential singular points but meromorphic excepting these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 16 we now need to consider its analog in continuous space for a differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (A specific example for a differential operator in continuous space is found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [29, 30], though not fully general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=') In mathematical physics, a space equipped with indefinite inner product is called the Krein space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Eigenfunctions with nonzero norm can be almost analogously treated to the case of self-adjoint operators whose eigenvalues are real and eigenfunctions are normalized by positive-definite norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' On the other hand, we must carefully consider eigenfunctions with zero norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In particular, the eigenfunction with complex eigenvalue always has zero norm due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In this case, in order to prepare a σ-orthogonal basis, we should construct positive- and negative-norm functions by linear combination of a pair of zero-norm eigenfunctions with mutually complex conjugate eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The second important point in writing down the completeness relation supposed to be applied in the ISM is that the integrand must be expressed by meromorphic function with respect to spectral variable λ or its parametrization variable z, because, in the derivation of the GLM equation in Appendix D, we want to use the residue theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, we must eliminate z∗ from the integrand using the complex conjugation relations described in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The key relation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' A little calculation using the addition formula of theta functions shows f0(x, z∗)†σf0(x, z) = 4 � i=1 (−1)i−1gi(x, z∗)∗gi(x, z) = −1 √m � dn2( x √m) + cs2(2iz) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) This quantity is meromorphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', only including z due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' The spatial average of this quantity is, using dn2 = E K and recalling Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7), f0(x, z∗)†σf0(x, z) = 1 √m � − E K − cs2(2iz) � = k′(z), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) which is used to normalize scattering eigenstates and rewrite the k-integral to z-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' As shown below, the spatial average of the norm density f0(x, z)†σf0(x, z) for scattering states all reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Thirdly, if we have a σ-orthogonal basis of the Bloch-type functions parametrized by crystal momentum k and band index α, and all of them have non-vanishing norm and are normalized so that the spatial average becomes φ±(k, α)†σφ±(k, α) = ±1, where ± represents the sign of norm, the completeness relation is given by � α � dk 2π(φ+(k, α)φ+(k, α)† − φ−(k, α)φ−(k, α)†)σ = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' This fact can be proved by considering the finite-length system where the scattering states can be explicitly normalizable and their spectrum is discretized, and then taking the infinite-length limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Here we omit the detail of this argument, since it is tedious but straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' For readers’ reference, we note that a similar discussion in another physical problem for discretization of scattering states of a self-adjoint operator and its in��nite limit can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [31], section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Keeping in mind the above-mentioned things, let us now write down the completeness relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Since the potentials (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) are periodic, there are only scattering states by the Bloch theorem and no bound state exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Hence, the completeness relation is written only by scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' As stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 4, up to periodicity, there are four distinct lines in z-plane where crystal momentum becomes real k(z) ∈ R, representing the scattering states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Here we discuss the scattering states on each line in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (i) z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We can check λ(z) ∈ R and f0(x, z) has positive norm, so f0(x, z∗) = f0(x, z) and hence 17 f0(x, z∗)†σf0(x, z) = f0(x, z)†σf0(x, z) = k′(z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, the contribution of these states to completeness relation is � dk 2π f0(x,z)f0(y,z)†σ k′(z) = � K′ −K′ dz 2π f0(x, z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (ii) z ∈ R − iK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We can check λ(z) ∈ R and f0(x, z) has negative norm, and f0(x, z∗) = −f0(x, z) using periodicity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Thus, f0(x, z)†σf0(x, z) = −f0(x, z∗)†σf0(x, z) = −k′(z) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, the contribution to these states to completeness relation is � dk f0(x,z)f0(y,z)†σ −k′(z) = � K′+iK −K′−iK dz 2π f0(x, z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (iii) z ∈ R ± iK 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' We can check λ(z) ∈ iR and hence f0(x, z) has zero norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore we must make nonzero-norm functions by linear combination of a pair of eigenfunctions with mutually complex conjugate eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Since λ(z∗) = λ(z)∗ and k(z∗) = k(z)∗, this pair share the same wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Now let us assume that f1 and f2 are zero-norm eigenfunctions of the Bloch type with complex eigenvalues λ and λ∗ and share the same and real-valued wavenumber k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' By definition the spatial average of norm density is zero: f † i σfi = 0, i = 1, 2, but f † 1 σf2 � 0 unless they belong to nontrivial Jordan blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, if an overall prefactors of f1 and f2 are adjusted to satisfy the relation f † 1 σf2 = f † 2 σf1 > 0, then f± = f1± f2 √ 2 are positive- and negative-norm functions and σ-orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Thus these two can be used as a σ-orthogonal basis, and the contribution of these states to completeness relation is written as � dk 2π f+ f † +σ− f− f † −σ f † 2 σf1 = � dk 2π (f1 f † 2 + f2 f † 1 )σ f † 2 σf1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Now let us take f1 = f0(x, z), z ∈ R + iK 2 and f2 = −f0(x, z∗), z∗ ∈ R − iK 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2), f † 2 σf1 = −k′(z) > 0 is real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (Note: k(z) is a decreasing function on z ∈ R ± iK 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=') Therefore, the contribution of these states to completeness relation is given by � dk 2π (− f0(x,z)f0(y,z∗)†− f0(x,z∗)f0(y,z)†)σ −k′(z) = − � K′+ iK 2 −K′+ iK 2 dz 2π(f0(x, z)f0(y, z∗)† + f0(x, z∗)f0(y, z)†)σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Changing the dummy variable z → z∗, the latter term can be rewritten as − � K′− iK 2 −K′− iK 2 dz 2π f0(x, z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Summarizing all contributions from (i)-(iii), the completeness relation is given by δ(x − y)I4 = � K′ −K′ dz 2π f0(x, z)f0(y, z∗)†σ + � K′+iK −K′−iK dz 2π f0(x, z)f0(y, z∗)†σ − � K′+ iK 2 −K′+ iK 2 dz 2π f0(x, z)f0(y, z∗)†σ − � K′− iK 2 −K′− iK 2 dz 2π f0(x, z)f0(y, z∗)†σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) Adding vertical paths which cancel out by periodicity, we obtain the desired form δ(x − y)I4 = � C1+C2 dz 2π f(x, z)f(y, z∗)†σ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) where the definitions of the contours C1 and C2 are already described in the sentences after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 18 Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Derivation of the GLM equation In this appendix we drive the GLM equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' By the relation between right and left Jost solutions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) and the integral representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1), f+(x, z) 1 a(z) − f0(x, z) = � x −∞ dyΓ(x, y)f0(y, z) + � f0(x, −z − iK) + � x −∞ dyΓ(x, y)f0(y, −z − iK) � b(z) a(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) Multiplying f(w,z∗)†σ 2π , w ≤ x to both sides of the above equation from right, we integrate the expression by z along the contour C1 + C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' First, let us evaluate the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Defining Ωsc(x, w) := � C1+C2 dz 2π f0(x, −z − iK)b(z) a(z) f0(w, z∗)†σ, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) and using the completeness relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='15), we find R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' = Γ(x, w) + Ωsc(x, w) + � x −∞ dyΓ(x, y)Ωsc(y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) In the contour integral of Ωsc, actually the integrals for scattering states only contribute, and hence it can be rewritten as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Next, let us evaluate the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In the SG equation, generally a(z) can have higher-order zeros [24], but here we only consider the case where all zeros are of the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Let zeros of a(z) be z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' As already mentioned before, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11), the zeros of a(z) simultaneously emerge at z, −z∗, z − iK, −z∗ − iK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Therefore, not all zj’s can be freely chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Here, however, we formally assign independent labels to all zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, the left-hand side can be calculated using the residue theorem: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' = n � j=1 if+(x, zj)f0(w, z∗ j)† ˙a(zj) = n � j=1 if−(x, −zj − iK)b(zj)f0(w, z∗ j)† ˙a(zj) , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) where the relation f+(x, zj) = b(zj)f−(x, −zj − iK) arising from the condition a(zj) = 0 has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' If we define Ωbd(x, w) := n � j=1 f0(x, −zj − iK)c2 j f0(w, z∗ j)†σ, c2 j := b(zj) i˙a(zj), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) then L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' = −Ωbd(x, w) − � x −∞ dyΓ(x, y)Ωbd(y, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) To summarize, writing Ω = Ωsc + Ωbd, we obtain the final result Γ(x, w) + Ω(x, w) + � x −∞ dyΓ(x, y)Ω(y, w) = 0, w ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) If we simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='14), it is rewritten as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='10) with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 19 Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Calculation of determinant Here we calculate the asymptotic form of the multi-soliton solution for x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In particular, we provide the shift of the lattice x0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' First, we write P = S 1 ˜PS 2 and Q = S 3 ˜QS 4, where S 1 = diag �Θ4(iz1) Θ1(iz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Θ4(izn) Θ1(izn) � , S 2 = diag �Θ2(iz1) Θ3(iz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Θ2(izn) Θ3(izn) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='1) S 3 = diag �Θ4(iz1) Θ2(iz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Θ4(izn) Θ2(izn) � , S 4 = diag �Θ1(iz1) Θ3(iz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', Θ1(izn) Θ3(izn) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='2) and ˜Pi j = −Θ4Θ2( x √m + i(zi + zj)) Θ2(i(zi + zj))Θ1( x √m) , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='3) ˜Qi j = Θ4Θ1( x √m + i(zi + zj)) Θ2(i(zi + zj))Θ2( x √m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='4) Then, using the formula in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' [27], appendix D, we obtain det ˜P det G = (−1)n Θ4( x √m)Θ1( x √m + nK + 2i �n j=1 zj) Θ1( x √m)Θ4( x √m + nK + 2i �n j=1 zj), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='5) det ˜Q det G = (−1)n Θ4( x √m)Θ2( x √m + nK + 2i �n j=1 zj) Θ2( x √m)Θ4( x √m + nK + 2i �n j=1 zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='6) This relation itself is generally valid even when all zj’s are independent and n is not even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' 9, for the real and bounded solution of the SG equation, n is even and zj and zj − iK simultaneously appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' In this case, we label the zeros, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', as zn/2+ j = zj − iK, j = n 2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' Then, det S 1 = �n/2 j=1 Θ4(iz j)Θ3(iz j) Θ1(iz j)Θ2(iz j), det S 2 = (−1)n/2 det S −1 1 , det S 4 = �n/2 j=1 Θ1(iz j)Θ2(iz j) Θ3(iz j)Θ4(iz j), det S 3 = (−1)n/2 det S −1 4 , and hence det P det G = Θ4( x √m)Θ1( x √m + 2i �n j=1 zj) Θ1( x √m)Θ4( x √m + 2i �n j=1 zj) = cos φ0(x−x0) 2 cos φ0(x) 2 , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content='7) det Q det G = Θ4( x √m)Θ2( x √m + 2i �n j=1 zj) Θ2( x √m)Θ4( x √m + 2i �n j=1 zj) = sin φ0(x−x0) 2 sin φ0(x) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFAT4oBgHgl3EQf0x4X/content/2301.08705v1.pdf'} +page_content=' (E.' metadata={'source': 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