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

IF 2.7 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2025-01-29 DOI:10.1016/j.physd.2025.134543

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

{"title":"Cauchy matrix scheme and the semi-discrete Toda and sine-Gordon systems","authors":"Tong Shen , Chunxia Li , Xinyuan Zhang , Songlin Zhao , Zhen Zhou","doi":"10.1016/j.physd.2025.134543","DOIUrl":null,"url":null,"abstract":"<div><div>This paper aims to explore the relations between the Sylvester equation and semi-discrete integrable systems. Starting from the Sylvester equation <span><math><mrow><mi>K</mi><mi>M</mi><mo>+</mo><mi>M</mi><mi>L</mi><mo>=</mo><mi>r</mi><msup><mrow><mi>s</mi></mrow><mrow><mo>⊤</mo></mrow></msup></mrow></math></span>, 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 <span><math><mi>K</mi></math></span> and <span><math><mi>L</mi></math></span>. As examples, soliton solutions and multi-pole solutions are analyzed and illustrated. The connection of the <span><math><mi>τ</mi></math></span> function of the semi-discrete Toda equation with Cauchy matrix approach is clarified. Under the constraint <span><math><mrow><mi>K</mi><mo>=</mo><mi>L</mi></mrow></math></span>, 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.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"473 ","pages":"Article 134543"},"PeriodicalIF":2.7000,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physica D: Nonlinear Phenomena","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167278925000223","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

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.

查看原文本刊更多论文

求助全文

约1分钟内获得全文求助全文

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.