[Paper Review] The Sylvester equation and the elliptic Korteweg-de Vries system

Ying‐ying Sun, Da‐jun Zhang+1 more|arXiv (Cornell University)|Jul 20, 2015

Nonlinear Waves and Solitons23 references3 citations

TL;DR

This paper introduces a novel solution framework for the elliptic Korteweg-de Vries (KdV) system using a Sylvester-type matrix equation with a Cauchy matrix structure. By deriving explicit solutions via canonical forms of the matrix $m{k}$, the authors re-derive both discrete (elpKdV) and continuous (epKdV) elliptic KdV systems, establish their Lax pairs, and show that scalar functions $S^{(i,j)}$ form an infinite symmetric matrix that generates the nonlinear equations through recurrence relations and continuum limits.

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

Motivation & Objective

• To generalize soliton solutions of the elliptic KdV system using a Sylvester-type matrix equation instead of the standard direct linearisation approach.
• To re-derive the discrete (elpKdV) and continuous (epKdV) elliptic KdV systems through a Cauchy matrix formulation based on matrix relations tied to an elliptic curve.
• To construct explicit solutions for the matrix $m{M}$ in the Sylvester equation using diagonal and Jordan block forms of $m{k}$, enabling soliton solutions.
• To establish Lax pairs for both the discrete and continuous systems using the derived matrix structures and scalar functions $S^{(i,j)}$.
• To demonstrate that the scalar function $S^{(i,j)}$, defined from $(m{k}, m{K}, m{M}, m{r}, m{c})$, forms an infinite symmetric matrix $m{S}$ that generates the nonlinear equations via recurrence and continuum limits.

Proposed method

• The study starts from the Sylvester equation $\bm{k}\bm{M} + \bm{M}\bm{k} = \bm{r}\bm{c}^T - g\bm{K}^{-1}\bm{r}\bm{c}^T\bm{K}^{-1}$, where $\bm{k}$ and $\bm{K}$ are commuting matrices satisfying $\bm{k}^2 = \bm{K} + 3e_1\bm{I} + g\bm{K}^{-1}$, linking the system to an elliptic curve.
• The scalar function $S^{(i,j)}$ is defined via the matrix elements of $\bm{M}$, forming an infinite symmetric matrix $\bm{S}$ that encodes the nonlinear dynamics.
• Explicit solutions for $\bm{M}$ are constructed for three canonical cases: diagonal $\bm{k}$, Jordan block $\bm{k}$, and their combinations, using the matrix $\bm{k}$'s spectral decomposition.
• Dispersion relations for the elpKdV and epKdV systems are derived from shift and derivative conditions: $ (a\bm{I} - \bm{k})\widetilde{\bm{r}} = (a\bm{I} + \bm{k})\bm{r} $, and $ \bm{r}_x = \bm{k}\bm{r}, \bm{r}_t = 4\bm{k}^3\bm{r} $, respectively.
• The Lax pairs for both systems are derived by analyzing the compatibility of the matrix evolution equations and the scalar function $S^{(i,j)}$.
• Continuum limits of the elpKdV system are analyzed to recover the epKdV system, showing consistency between discrete and continuous formulations.

Experimental results

Research questions

• RQ1 How can the elliptic KdV system be re-derived using a Sylvester-type matrix equation and a Cauchy matrix approach?
• RQ2 What are the explicit solutions for the matrix $\bm{M}$ in the Sylvester equation when $\bm{k}$ is diagonal, a Jordan block, or a combination of both?
• RQ3 How do the scalar functions $S^{(i,j)}$, derived from $\bm{M}$, generate the nonlinear elpKdV and epKdV equations through recurrence and continuum limits?
• RQ4 What is the role of the matrix $\bm{k}$'s canonical form in constructing soliton solutions and deriving Lax pairs for the elliptic KdV systems?
• RQ5 How are the discrete and continuous elliptic KdV systems related through the continuum limit of the Sylvester equation framework?

Key findings

• The Sylvester equation $\bm{k}\bm{M} + \bm{M}\bm{k} = \bm{r}\bm{c}^T - g\bm{K}^{-1}\bm{r}\bm{c}^T\bm{K}^{-1}$ with $\bm{k}^2 = \bm{K} + 3e_1\bm{I} + g\bm{K}^{-1}$ provides a new algebraic framework to derive the elliptic KdV systems.
• Explicit solutions for $\bm{M}$ are constructed for diagonal $\bm{k}$, Jordan block $\bm{k}$, and their combinations, using the canonical forms of $\bm{k}$ and the associated $\bm{r}, \bm{c}$ vectors.
• The scalar function $S^{(i,j)}$, defined from $\bm{M}$, forms an infinite symmetric matrix $\bm{S}$ that satisfies recurrence relations on $\mathbb{Z} \times \mathbb{Z}$, which are essential for deriving the discrete elpKdV system.
• The Lax pairs for both the elpKdV and epKdV systems are derived from the matrix evolution equations and the scalar function $S^{(i,j)}$, confirming integrability.
• The continuous epKdV system is obtained as a continuum limit of the elpKdV system, with the dispersion relations $\bm{r}_x = \bm{k}\bm{r}, \bm{r}_t = 4\bm{k}^3\bm{r}$ and $\bm{c}^T_x = \bm{c}^T\bm{k}, \bm{c}^T_t = \bm{c}^T\bm{k}^3$ governing the continuous dynamics.
• Solutions for $\bm{r}$ and $\bm{c}$ are explicitly given in terms of $\rho_i = \left(\frac{a+k_i}{a-k_i}\right)^n\left(\frac{b+k_i}{b-k_i}\right)^m\rho_i^0$ (for diagonal $\bm{k}$) and $\rho_1 = e^{k_1x + 4k_1^3t + \xi_1^{(0)}}$ (for continuous case), enabling full soliton solution construction.