[Paper Review] Solutions to the ABS lattice equations via generalized Cauchy matrix approach

Da‐jun Zhang, Song‐lin Zhao|arXiv (Cornell University)|Aug 18, 2012

Nonlinear Waves and Solitons18 references3 citations

TL;DR

This paper introduces a generalized Cauchy matrix approach for solving ABS lattice equations by starting from an undetermined matrix equation set for the plain wave factor vector $\bm{r}$ and dressed Cauchy matrix $\bm{M}$, rather than assuming them a priori. By deriving shift relations for scalar functions $S^{(i,j)}$, the method yields broader solution families than standard soliton solutions, including generalized multi-soliton and non-solitonic solutions for key equations in the ABS list such as lpKdV, lpmKdV, lSKdV, and H1–H3.

ABSTRACT

The usual Cauchy matrix approach starts from a known plain wave factor vector $r$ and known dressed Cauchy matrix $M$. In this paper we start from a matrix equation set with undetermined $r$ and $M$. From the starting equation set we can build shift relations for some defined scalar functions and then derive lattice equations. The starting matrix equation set admits more choices for $r$ and $M$ and in the paper we give explicit formulae for all possible $r$ and $M$. As applications, we get more solutions than usual multi-soliton solutions for many lattice equations including the lattice potential KdV equation, the lattice potential modified KdV equation, the lattice Schwarzian KdV equation, NQC equation and some lattice equations in ABS list.

Motivation & Objective

• To extend the standard Cauchy matrix approach by treating the plain wave factor $\bm{r}$ and dressed Cauchy matrix $\bm{M}$ as unknowns in a matrix equation set.
• To derive lattice equations and their solutions from a generalized starting framework that admits more solution choices than the canonical approach.
• To systematically construct explicit forms of $\bm{r}$ and $\bm{M}$ for a generic canonical matrix $\bm{\Gamma}$, enabling broader solution families.
• To demonstrate that the generalized approach yields solutions more general than standard multi-soliton solutions, including non-solitonic and limit cases.
• To apply the method to derive solutions for key equations in the ABS list, including lpKdV, lpmKdV, lSKdV, NQC, and H1–H3 equations.

Proposed method

• Start from a matrix equation set with unknown $\bm{r}$ and $\bm{M}$, where the first two equations determine $\bm{r}$ and the third defines $\bm{M}$ via a dressed Cauchy matrix form $\bm{M} = \bm{F}\bm{G}\bm{H}$.
• Define scalar functions $S^{(i,j)} = {}^{t}\hskip-2pt\bm{c} \bm{K}^j (\bm{I} + \bm{M})^{-1} \bm{K}^i \bm{r}$ and derive their shift relations using the matrix equation set.
• Prove the symmetric property $S^{(i,j)} = S^{(j,i)}$ using algebraic techniques and the structure of $\bm{K}$, which is essential for generating closed recurrence relations.
• Use the canonical form $\bm{\Gamma}$ of $\bm{K}$ to classify solutions based on eigenvalue structure, simplifying the analysis and solution derivation.
• Express solutions for lattice equations via $S^{(i,j)}$ and $S(a,b)$ functions, with explicit formulae for $\bm{r}$ and $\bm{M}$ in terms of parameters like $p, q, \rho^0_i, c_j$.
• Apply the framework to derive solutions for multiple equations in the ABS list, including lpKdV, lpmKdV, lSKdV, NQC, Q3–Q1, H3–H1, using known solution structures from prior work.

Experimental results

Research questions

• RQ1 Can a generalized Cauchy matrix approach be formulated that treats $\bm{r}$ and $\bm{M}$ as unknowns rather than given inputs?
• RQ2 What is the structure of the solution space for $\bm{r}$ and $\bm{M}$ when derived from a consistent matrix equation set?
• RQ3 How do the derived shift relations for $S^{(i,j)}$ lead to discrete lattice equations such as lpKdV and H1?
• RQ4 What types of solutions (e.g., solitonic, non-solitonic, limit cases) emerge from this generalized framework compared to the standard Cauchy matrix method?
• RQ5 Can this approach generate solutions for all equations in the ABS list, including H1–H3 and Q3–Q1, with explicit parameterization?

Key findings

• The generalized Cauchy matrix approach starts from a matrix equation set with undetermined $\bm{r}$ and $\bm{M}$, leading to broader solution families than the standard method.
• Explicit formulae are derived for all possible $\bm{r}$ and $\bm{M}$ corresponding to a generic canonical matrix $\bm{\Gamma}$, based on its eigenvalue structure.
• The symmetric property $S^{(i,j)} = S^{(j,i)}$ is rigorously proven, which is essential for generating closed recurrence relations that yield lattice equations.
• Solutions for the lpKdV equation are given by $u = \zeta - S^{(0,0)}$, with $\zeta = pn + qm + \zeta_0$, and $S^{(0,0)}$ defined via $\bm{M}$.
• For the H1 equation, the solution is $u = A(\zeta - S^{(0,0)}) + (-1)^{n+m}B(\zeta + c_0 - 2S^{(0,0)})$, where $A^2 - B^2 = 1$.
• The method does not yield rational solutions because $\bm{\Gamma}$ must be invertible, excluding zero eigenvalues, but allows non-solitonic solutions via Jordan block structures.