Cauchy matrix scheme and the semi-discrete Toda and sine-Gordon systems

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED Physica D: Nonlinear Phenomena Pub Date : 2025-03-01 Epub Date: 2025-01-29 DOI:10.1016/j.physd.2025.134543

Tong Shen , Chunxia Li , Xinyuan Zhang , Songlin Zhao , Zhen Zhou

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Abstract

This paper aims to explore the relations between the Sylvester equation and semi-discrete integrable systems. Starting from the Sylvester equation

K M + M L = r s^{⊤}

, the semi-discrete Toda equation, the modified semi-discrete Toda equation and their discrete Miura transformation are constructed by Cauchy matrix approach in a systematic way. Lax pair is derived for the modified semi-discrete Toda equation. Explicit solutions are presented for the semi-discrete Toda equation and classified according to the canonical forms of the constant matrices

K

and

L

. As examples, soliton solutions and multi-pole solutions are analyzed and illustrated. The connection of the

τ

function of the semi-discrete Toda equation with Cauchy matrix approach is clarified. Under the constraint

K = L

, the semi-discrete sine-Gordon equation, the modified semi-discrete sine-Gordon equation and their discrete Miura transformation are obtained, which can also be treated as the two-periodic reductions of the corresponding results of the semi-discrete Toda system. Integrable properties including the bilinear form, Lax pair and various types of solutions are investigated for the semi-discrete sine-Gordon equation. In particular, kink solutions and breathers of the semi-discrete sine-Gordon equation are analyzed and their dynamical behaviors are illustrated.

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Cauchy矩阵格式与半离散Toda系统和sin - gordon系统

本文旨在探讨Sylvester方程与半离散可积系统之间的关系。从Sylvester方程KM+ML=rs出发，采用柯西矩阵法系统地构造了半离散Toda方程、修正半离散Toda方程及其离散Miura变换。导出了改进的半离散Toda方程的Lax对。给出了半离散Toda方程的显式解，并根据常数矩阵K和l的标准形式进行了分类。作为例子，分析并说明了孤子解和多极解。阐明了半离散Toda方程的τ函数与柯西矩阵法的联系。在K=L约束下，得到了半离散正弦-戈登方程、修正的半离散正弦-戈登方程及其离散Miura变换，也可视为半离散Toda系统相应结果的两周期约简。研究了半离散正弦-戈登方程的双线性形式、Lax对和各种解的可积性。特别地，分析了半离散正弦-戈登方程的扭结解和呼吸解，并说明了它们的动力学行为。

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来源期刊

Physica D: Nonlinear Phenomena 物理-物理：数学物理

CiteScore

7.30

自引率

7.50%

发文量

213

审稿时长

65 days

期刊介绍： Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.