Riemann-Hilbert方法在孤子方程求解中的应用

发布时间：2018-05-17 13:00

本文选题：孤子方程 + 矩阵谱问题　；参考：《郑州大学》2016年博士论文

【摘要】：孤子现象的出现是非线性效应和色散效应巧妙平衡的结果.孤子理论,特别是寻找具有重要物理背景的孤子方程的孤子解,已经吸引了许多研究者,这是由于孤子理论在许多领域中具有重要的作用,例如水波理论,非线性光学,等离子体物理,流体力学等.我们还知道获得孤子方程的孤子解的方法包括反散射变换,Darboux变换,B¨acklund变换,Hirota双线性方法,Wronskian技巧等.本文的主要目的是推广应用Riemann-Hilbert方法导出五个孤子方程的多孤子解,这些方程包括耦合m Kd V方程,广义Sasa-Satsuma方程,耦合SasaSatsuma方程,耦合Gerdjikov-Ivanov方程以及长短波方程.这五个孤子方程均和高阶的谱问题相联系,其中耦合Sasa-Satsuma方程的Lax对涉及5×5的矩阵谱问题,其余四个方程相联系于3×3的矩阵谱问题.通过对孤子方程的Lax对进行谱分析,我们构造了相关的分区域解析矩阵函数,进而得到了这些方程的Riemann-Hilbert问题.然后,通过求解对应于无反射情形下的特殊Riemann-Hilbert问题,我们分别系统导出了五个孤子方程的多孤子解公式.特别地,为描述相关孤子解的特点,我们给出了一些孤子解的有趣的三维图形.此外,在每章的最后,我们均将获得的多孤子解公式表示为了行列式之比,这种紧凑而整洁的形式非常便于进行符号计算.
[Abstract]:The soliton phenomenon is the result of the clever balance of nonlinear effect and dispersion effect. Soliton theory, especially in finding soliton solutions of soliton equations with important physical background, has attracted many researchers. This is because soliton theory plays an important role in many fields, such as water wave theory, nonlinear optics, and so on. Plasma physics, hydrodynamics, etc. We also know that the methods of obtaining soliton solutions for soliton equations include the inverse scattering transformation Darboux transformation and acklund transformation Hirota bilinear method and Wronskian technique. The main purpose of this paper is to generalize the multi-soliton solutions of five soliton equations by using the Riemann-Hilbert method. These equations include the coupled m Kd V equation, the generalized Sasa-Satsuma equation, the coupled SasaSatsuma equation, the coupled Gerdjikov-Ivanov equation and the long-short wave equation. The five soliton equations are all related to higher order spectral problems, where the Lax pair of coupled Sasa-Satsuma equations involves a 5 脳 5 matrix spectral problem, and the other four equations are associated with 3 脳 3 matrix spectral problems. Based on the spectral analysis of the Lax pair of soliton equations, we construct the related domain analytic matrix functions, and then obtain the Riemann-Hilbert problem of these equations. Then, by solving the special Riemann-Hilbert problem corresponding to the non-reflection case, we systematically derive the multiple soliton solutions of the five soliton equations. In particular, in order to describe the characteristics of the associated soliton solutions, we give some interesting three-dimensional graphs of the soliton solutions. In addition, at the end of each chapter, the formula of multi-soliton solution obtained is expressed as the ratio of determinants. This compact and neat form is very convenient for symbolic calculation.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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