Riemann-Hilbert方法在与高阶谱问题相联系的可积方程中的应用

发布时间：2018-03-06 11:17

本文选题：可积方程　切入点：高阶谱问题　出处：《郑州大学》2017年博士论文　论文类型：学位论文

【摘要】：本文的研究对象是与高阶谱问题相联系的可积方程,研究内容分为两大部分:利用基于Riemann Hilbert问题的Fokas统一变换方法研究可积方程在半直线上的初边值问题,以及利用基于Riemann Hilbert问题的Deift Zhou非线性最速下降法研究可积方程初值问题解的长时间渐近行为.在本文的第二章中,我们研究耦合非线性Schr¨odinger方程的初边值问题.首先,利用初始和所有边值数据分别定义谱函数(6))和(6)),并借助Fredholm积分方程构造相关的Riemann Hilbert问题,其中跳跃矩阵是由(6))和(6))确定的.然后,我们证明方程的解可以由Riemann Hilbert问题的解给出,且得到的解与初值和边值数据吻合.对于适定问题,一部分边值是未知的,然而对于一类特殊的边界条件,即线性化边界条件,我们可以利用(6))表示(6)).对于非线性化边界条件,利用扰动展开法给出未知边界值的一个有效表示.在第三章中,我们研究耦合修正Korteweg de Vries方程的初边值问题.与第二章不同的是,本章我们采取“打包”的思想,将方程的高阶Lax对写成2×2分块矩阵的形式,并利用谱问题的特征函数的组合来构造相关的Riemann Hilbert问题.此外,引入Sherman Morrison公式,巧妙地推出了Riemann Hilbert问题的跳跃矩阵和相应的留数条件.在本文的第四章中,我们利用矩阵分块将高阶谱问题低阶化,构造一个2×2的分块矩阵Riemann Hilbert问题.首先,引入矩阵函数并作原Riemann Hilbert问题的一个等价变换,然而满足矩阵Riemann Hilbert问题的函数的不可解性给我们带来了挑战,幸运的是,文中利用的渐近逼近克服了这一困难.然后,利用非线性最速下降法,将Riemann Hilbert问题经过一系列变换后,我们得到一个抛物柱面方程.最终,借助标准抛物柱面函数的渐近性研究了耦合非线性Schr¨odinger方程在衰减初值条件下解的长时间渐近行为.与第四章采用的方法类似,在第五章中,我们从Sasa Satsuma方程的Lax对出发,构造相应的Riemann Hilbert问题,并借助非线性最速下降法以及标准抛物柱面函数,求出了该方程的渐近主部.与第四章不同的是,本章涉及到两个驻相点,相应Riemann Hilbert问题的跳跃曲线更复杂.
[Abstract]:The object of this paper is the integrable equation associated with higher order spectral problems. The research contents are divided into two parts: the Fokas unified transformation method based on Riemann Hilbert problem is used to study the initial-boundary value problem of integrable equations on half-line. And the Deift Zhou nonlinear steepest descent method based on Riemann Hilbert problem is used to study the long time asymptotic behavior of the solution of the initial value problem of integrable equation. In the second chapter of this paper, we study the initial-boundary value problem of the coupled nonlinear Schr odinger equation. Using the initial and all boundary value data to define the spectral function and the Hilbert problem respectively, and using the Fredholm integral equation to construct the related Riemann Hilbert problem, in which the jump matrix is determined by the two methods. Then, we prove that the solution of the equation can be obtained by the solution of the Riemann Hilbert problem. Some of the boundary values are unknown for the fitness problem, but for a class of special boundary conditions, that is, linearized boundary conditions, we can express the boundary conditions by using Yi-6o). For the nonlinear boundary conditions, An efficient representation of the unknown boundary value is given by using the perturbation expansion method. In chapter 3, we study the initial-boundary value problem of the coupled modified Korteweg de Vries equation. Different from the second chapter, we take the idea of "packing" in this chapter. The higher order Lax pairs of the equation are written in the form of 2 脳 2 block matrices, and the associated Riemann Hilbert problem is constructed by the combination of the eigenfunctions of the spectral problems. In addition, the Sherman Morrison formula is introduced. In chapter 4th of this paper, we deduce the jump matrix and the corresponding residue condition of the Riemann Hilbert problem. In Chapter 4th, we use the matrix block to order the higher order spectrum problem and construct a 2 脳 2 block matrix Riemann Hilbert problem. This paper introduces the matrix function and makes an equivalent transformation of the original Riemann Hilbert problem. However, the irsolvability of the function satisfying the matrix Riemann Hilbert problem brings us a challenge. Fortunately, the asymptotic approximation used in this paper overcomes this difficulty. By using the nonlinear steepest descent method, we obtain a parabolic cylindrical equation after a series of transformations for the Riemann Hilbert problem. By means of the asymptotic behavior of standard parabolic cylindrical functions, the long time asymptotic behavior of the coupled nonlinear Schr odinger equation under the condition of decay initial value is studied. Similar to the method used in Chapter 4th, in Chapter 5th, we proceed from the Lax pair of the Sasa Satsuma equation. The corresponding Riemann Hilbert problem is constructed, and the asymptotic principal part of the equation is obtained by means of the nonlinear steepest descent method and the standard parabolic cylindrical function. Different from Chapter 4th, this chapter deals with two stationary phase points. The jump curve of the corresponding Riemann Hilbert problem is more complicated.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.5

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