The Sylvester equation and the elliptic Korteweg-de Vries system

Abstract

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 $\mathbf{k}\mathbf{M}+\mathbf{M}\mathbf{k}=\mathbf{r}{\mathbf{c}}^T-g{\mathbf{K}}^{-1}\mathbf{r}{\mathbf{c}}^T{\mathbf{K}}^{-1}$ where $\mathbf{k}$ and $\mathbf{K}$ are commutative matrices and obey the matrix relation ${\mathbf{k}}^2=\mathbf{K}+3e_1\mathbf{I}+g{\mathbf{K}}^{-1}$. The obtained elliptic equations, both discrete and continuous, are formulated by the scalar function $S^{(i,j)}$ which is defined using $(\mathbf{k},\mathbf{K},\mathbf{M},\mathbf{r},\mathbf{c})$ and constitute an infinite size symmetric matrix. Lax pairs for both the discrete and continuous system are derived. The explicit solution $\mathbf{M}$ of the Sylvester equation and generalized solutions of the obtained elliptic equations are presented according to the canonical forms of matrix $\mathbf{k}$.

27 pages

References in corpus (5)

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• Notes on solutions in Wronskian form to soliton equations: KdV-type
• Explicit solutions to the Korteweg-de Vries equation on the half line
• An Inverse Scattering Transform for the Lattice Potential KdV Equation
• On an elliptic extension of the Kadomtsev-Petviashvili equation