离散可积系统在求解非线性晶格方程中的应用研究  

Application of Discrete Integrable System in Solving Nonlinear Lattice Equation

作　　者：朱永芳 ZHU Yongfang(Anhui Sanlian University Foundation Department,Hefei,Anhui 230601,China)

机构地区：[1]安徽三联学院基础部,安徽合肥230601

出　　处：《九江学院学报（自然科学版）》2023年第1期98-102,共5页Journal of Jiujiang University：Natural Science Edition

基　　金：安徽三联学院校级科研项目(编号KJZD2022011);安徽省高校自然科学研究项目(编号KJ2021A1175)的研究成果之一。

摘　　要：为了探究离散可积系统的可积性,找寻一种将其应用至非线性晶格方程求解中的有效途径,文章研究利用离散可积系统获取了Toda晶格方程的一个精确解。主要研究内容为:求解离散微分差分方程族的可积性及其Bargmann约束下的双非线性化,得到了有限维完全可积的Hamilton系统.使用高阶Bargmann约束求解方程的Lax对和伴随Lax对,将方程双非线性化为一个可积辛映射和一个有限维Liouville可积的Hamilton系统.研究提供了一种求解Toda晶格方程精确解的思路,展现了双非线性化方法在孤立子理论研究领域的重要性。In order to explore the integrability of discrete integrable systems and find an effective way to solve nonlinear lattice equations,an exact solution of Toda lattice equation was obtained by using the discrete integrable system.The main research contents were as follows:to solve the integrability of discrete differential difference equations and its Bi nonlinearity under bargmann constraint,a finite dimensional fully integrable Hamiltonian system was obtained.Using higher order bargmann constraints to solve the lax pair and adjoint lax pair of the equation,the equation was bilinear transformed into an integrable symplectic map and a finite dimensional Liouville integrable Hamiltonian system.The research provided an idea to solve the exact solution of Toda lattice equation,and showed the importance of the double nonlinear method in soliton theory.

关 键 词：离散可积系统 非线性晶格方程 孤立子 Toda晶格方程 无限维Hamilton系统 

分 类 号：O175[理学—数学]