Boussinesq方程的复解和两个全离散方程的守恒律

发布时间：2018-04-26 16:02

本文选题：Boussinesq方程 + PT-对称　；参考：《江苏师范大学》2017年硕士论文

【摘要】：孤立子方程的无穷多守恒律和精确解是可积系统研究的重要课题,本文研究B ous sinesq方程的复解特别是PT不变解、构造全离散的Liouville方程和ABS分类中的特殊Q1方程的无穷守恒律.论文分为两部分.第二章构造Boussinesq方程的复解和PT不变解.首先,Boussinesq方程的实变量Jacobi椭圆函数实解被构造,这个解的特殊情形给出Boussinesq方程的单孤子解.在此基础上,这个解进一步复化成一个新的实变量的Jacobi椭圆函数复解,其特殊情形给出Boussinesq方程的Jacobi椭圆函数的周期PT对称解和复单孤子解.其次,应用Hirota方法,Boussinesq方程的单孤子解被直接得到.通过假设其中的相位为复数,这个单孤子解就给出了 Boussinesq方程的复的单孤子解.我们证明相位为纯虚数时的复的单孤子解是PT不变的,同时证明这个复的单孤子解(不管它是否PT对称)的全质量和能量都是实的.Boussinesq方程的一类PT不变的双孤子复解被给出.第三章研究如何使用Lax对形式对角化方法构造全离散的Liouville方程和ABS分类中的特殊Q1方程的无穷守恒律.Lax对形式对角化表明存在形式级数使得在奇异点附近规范后的Lax矩阵可规范等价于系数为对角矩阵的形式无穷级数,此时全离散的Lax表示自然给出守恒律方程,由此可构造全离散可积系统的守恒量.作为应用,全离散的Liouville方程和ABS分类中的特殊Q1方程的无穷守恒律被得到.
[Abstract]:The infinite conservation laws and exact solutions of soliton equations are important subjects in the study of integrable systems. In this paper, we study the complex solutions of B ous sinesq equations, especially the PT invariant solutions, and construct the infinite conservation laws of the completely discrete Liouville equations and the special Q1 equations in the ABS classification. The paper is divided into two parts. In chapter 2, the complex solution and PT invariant solution of Boussinesq equation are constructed. First, the real solution of the Jacobi elliptic function of the real variable of the Boussinesq equation is constructed. In the special case of this solution, the single soliton solution of the Boussinesq equation is given. On this basis, the solution is further complex into a new real variable complex solution of Jacobi elliptic function. In the special case, the periodic PT symmetric solution and complex soliton solution of Jacobi elliptic function of Boussinesq equation are given. Secondly, the single soliton solution of the Boussinesq equation is obtained directly by using the Hirota method. By assuming that the phase is complex, this soliton solution gives the complex soliton solution of the Boussinesq equation. We prove that the complex soliton solution with pure imaginary phase is PT invariant, and that the total mass and energy of the complex soliton solution (whether it is PT symmetric or not) are real. Boussinesq equation has a class of PT invariant double soliton complex solutions. In chapter 3, we study how to use Lax pair form diagonalization method to construct completely discrete Liouville equation and the infinite conservation law of special Q1 equation in ABS classification. Lax pair form diagonalization indicates the existence of formal series such that the gauge is near the singular point. Lax matrix can be gauge equivalent to the form infinite series whose coefficient is diagonal matrix. In this case, the fully discrete Lax represents the conservation law equation, and the conserved quantity of the fully discrete integrable system can be constructed. As applications, the infinite conservation laws of the fully discrete Liouville equation and the special Q1 equation in the ABS classification are obtained.
【学位授予单位】：江苏师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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