非线性模型的怪波解、孤子解及可积性

发布时间：2018-03-16 18:03

  本文选题：孤立子　切入点：可积系统　出处：《华东师范大学》2016年博士论文　论文类型：学位论文

【摘要】：本文基于符号计算,分别利用推广的Darboux变换、经典的Darboux变换、双Darboux变换和对称性理论,研究了非线性数学物理中若干非线性模型的怪波解、孤子解及可积性.在符号计算软件Maple平台上开发了Darboux变换与怪波求解的自动推演程序包.主要内容及创新点包括：第一章,绪论部分.主要介绍了怪波、孤子与可积系统、Darboux变换及符号计算的背景与研究现状,并阐明了本文的主要研究结果.第二章,研究了2+1维非线性模型的经典的Darboux变换与孤子解.构造了2+1维CDGKS方程和2+1维nKdV方程的经典的Darboux变换,给出了两个方程的N-孤子解的一般表达式,分析了亮、暗及扭结孤子的动力学行为.第三章,研究了与2×2谱问题联系的非线性模型的推广的Darboux变换与怪波解.构造了AB系统和Kundu-Eckhaus (KE)方程的推广的Darboux变换,给出了两个方程N-阶怪波解的统一表达式.对于AB系统,发现了“四尖峰”型怪波.对于KE方程,通过数值计算,发现高次非线性项和Raman散射项只影响怪波的空间分布,而对怪波出现的时间和振幅没有影响.第四章,讨论了与3×3谱问题联系的非线性模型的推广的Darboux变换与怪波解.构造了耦合Hirota方程、Manakov系统和三波共振(TWR)方程的推广的Darboux变换,给出了三个不同类型方程的N-阶怪波解的统一表达式.对于耦合Hirota方程,发现了向量形式的高阶怪波、高阶怪波与多个暗-亮孤子以及高阶怪波与多个呼吸子相互作用的三种非线性波结构.对于Manakov系统,讨论了高阶怪波与其它非线性波的相互作用性质.对于TWR方程,研究了组合型高阶怪波的分类问题,并基于数值模拟的方法,对高阶怪波解的稳定性进行了分析.第五章,讨论了变系数与离散非线性模型的可积性与孤子解.获得了2+1维变系数Gardner方程的Lax对和共轭Lax对,构造了方程的双Darboux变换,得到了N-孤子解的一般表达式,并分析了亮、暗孤子和共振亮、暗孤子的动力学行为.研究了四位势Blaszak-Marciniak晶格方程的Lie点对称、广义对称、守恒律和孤子解,从对称的角度证明了模型的可积性.第六章,自动推演程序包的开发.在Maple平台上开发了Darboux变换与怪波求解程序包DTRWSI,并以多个实例验证了程序包的实用性和有效性.第七章,总结与展望部分.对全文进行了总结,并就下一步工作做了展望.
[Abstract]:This paper based on symbolic computation, Darboux transform were used to promote the classical Darboux transform, Darboux transform and dual symmetry theory, study several nonlinear models in nonlinear mathematical physics strange wave solutions, soliton solutions and integrability. Maple software platform developed Darboux transform and wave solution of the automatic deduction procedure of blame wrapped in symbols. The main content and innovations include: the first chapter is the introduction part. This paper mainly introduced the strange wave, soliton and integrable system, the background and research status of Darboux transform and symbolic computation, and expounds the main research results of this paper. The second chapter, Darboux transformation and soliton of 2+1 dimensional nonlinear model of the classic the solution is constructed. Darboux transform 2+1 dimensional CDGKS equation and the 2+1 dimensional nKdV equation classic, the general expression of N- two soliton equations are given, analysis of the bright, dark and kink soliton dynamical behaviors . the third chapter studied with 2 * 2 spectral problem of contact nonlinear model of the generalized Darboux transform and strange wave solutions. To construct the AB system and Kundu-Eckhaus (KE) equation of the generalized Darboux transform, gives two equations of order N- wave solutions. Strange expressions for the AB system, found "four spike type strange wave. For the KE equation, through the numerical calculation, found that higher order nonlinear terms and Raman scattering only affects distribution of strange wave space, but have no influence on the strange wave amplitude and time. The fourth chapter discusses the nonlinear model with 3 * 3 spectral problem linked to the promotion of the Darboux transform and the blame wave solutions constructed coupled Hirota equation, Manakov system and three wave resonance (TWR) equation of the generalized Darboux transform, gives the N- order of three different types of equations of strange wave solutions. A unified expression for the coupled Hirota equation, found high order vector form strange wave, High order strange wave and multiple dark bright soliton and high-order strange waves with multiple breathers interaction of three kinds of nonlinear wave structure. For Manakov system, discusses the high order strange wave and other nonlinear wave interaction properties. For the TWR equation, the classification problem of combination type high order strange wave then, based on the method of numerical simulation, the stability of high order wave solutions of strange analysis. The fifth chapter discusses the variable coefficient nonlinear model and discrete Integrability and soliton solutions. The 2+1 dimensional Gardner equation with variable coefficients of Lax and Lax on the conjugate double Darboux transform, structural equation, general the expression of N- soliton solution is obtained, and the analysis of the bright, bright and dark soliton resonance, dynamics of dark solitons. On four potential Blaszak-Marciniak lattice equation Lie point symmetry, generalized symmetry, conservation and soliton solutions from the symmetric angle proves that the model can be Product. The sixth chapter, the development of the automatic deduction. The package is developed on the platform of Maple Darboux transform and strange wave solver package DTRWSI, and a plurality of examples to verify the practicality and validity of the package. In the seventh chapter, conclusion and outlook. The summary of the thesis, and prospected the next step of work.

【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29
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本文编号：1621044