2015/07/20 by Ying-ying Sun, Ying‐ying Sun, Da-jun Zhang +6 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Boundary value problem #Cauchy matrix #Combinatorics #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Korteweg–de Vries equation #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Physics #Polynomial and algebraic computation #Quantum mechanics #Scalar (mathematics) #nlin.SI
paper · pdf · doi:10.48550/arxiv.1507.05476
published in White Rose Research Online (University of Leeds, The University of Sheffield, University of York) (White Rose University Consortium) · 27 pages
arxiv created 2015/07/20 · openalex publication_date 2015/07/20 · arxiv updated 2015/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of ~\boldsymbolk \boldsymbolM+ \boldsymbolM \boldsymbolk = \boldsymbolr \boldsymbolcT-g\boldsymbolK-1 \boldsymbolr \boldsymbolcT \boldsymbolK-1 where \boldsymbolk and \boldsymbolK are commutative matrices and obey the matrix relation \boldsymbolk2=\boldsymbolK+3e1\boldsymbolI+g\boldsymbolK-1. The obtained elliptic equations, both discrete and continuous, are formulated by the scalar function S(i,j) which is defined using (\boldsymbolk,\boldsymbolK, \boldsymbolM, \boldsymbolr,\boldsymbolc) and constitute an infinite size symmetric matrix. Lax pairs for both the discrete and continuous system are derived. The explicit solution \boldsymbolM of the Sylvester equation and generalized solutions of the obtained elliptic equations are presented according to the canonical forms of matrix \boldsymbolk.