具有尖峰孤子解可积系统的非局部对称与守恒律

发布时间：2018-06-02 16:50

本文选题：非局部对称 + 守恒律　；参考：《西北大学》2015年博士论文

【摘要】：如今非线性现象越来越多的出现在自然科学与社会科学中,用来描述该现象的微分方程受到相关数学家和物理学家的关注.本文主要研究了几个偏微分方程的非局部对称和守恒律以及精确解.包括两分量μ-Camassa-Holm方程,带有色散项γux的μ-Camassa-Holm方程,Foursov方程,复Camassa-Holm方程.本文还考虑浅水波方程3×3的谱问题,构造了Degasperis-Procesi方程和Novikov方程的无穷多守恒律,证明了Degasperis-Procesi方程和Novikov方程没有依赖其拟势函数的非局部对称.本文主要内容如下：首先,主要介绍了所研究方程的背景,以及研究非局部对称和求精确解所用的基本理论和方法.其次,我们研究了两分量μ-Camassa-Holm方程的非局部对称和守恒律.该方程是两分量Camassa-Holm方程的一个short-wave极限.先从该方程的几何可积性出发,得到方程的拟势函数进而构造方程的守恒律,并且得到方程无穷多守恒律.其次,根据这个方程仅有平凡的李对称和方程特殊结构,我们研究了它的非局部对称.最后,得到两分量μ-Camassa-Holm方程依赖其势函数的非局部对称.然后考虑带有低阶色散项γux的μ-Camassa-Holm方程,从该方程Lax对出发,借助其波函数,构造方程拟势函数,利用所得的拟势函数得到守恒律以及无穷多守恒量,进而运用李对称方法得到该方程依赖拟势函数的非局部对称.修正的μ-Camassa-Holm方程是作为修正Camassa-Holm方程非局部形式提出来的,从该方程几何可积性出发,求出拟势函数,进一步构造无穷多守恒律,并且证明该方程不具有依赖拟势函数的非局部对称.短脉冲方程是包含一阶线性项ux的修正μ-Camassa-Holm方程的伸缩极限方程,本章最后得到该方程的拟势函数,无穷多守恒律.第三,运用不变子空间的方法结合广义条件对称方法研究b-family系统的精确解.这种解是由三维的不变子空间生成,对应于方程的一类广义条件对称,其中系数依赖于时间的函数满足Emden方程,通过对Emden方程解的结构进行分析得到了方程解的结构,给出了解整体存在性和爆破条件,进而对解的结构给出完整的刻画.然后研究对偶Foursov方程的尖峰孤子解.先得到方程的弱形式,然后利用方程的弱形式证明该方程具有尖峰孤子解.最后讨论复Camassa-Holm系统,这类系统可以从Mobus几何曲线流中得到,我们证明了该方程具有尖峰孤子解.第四,考虑Degasperis-Procesi方程和Novikov方程的无穷多守恒律以及非局部对称,Degasperis-Procesi方程和Novikov方程是典型的具有3×3谱问题的波方程,以构造2×2谱问题方程的非局部对称和守恒律的方法为基础扩展到3×3谱问题的方程.先构造出Degasperis-Procesi方程的两个势函数和两个拟势函数,进一步通过势函数做幂级数展开,比较相同普参数λ的系数达到无穷多守恒律,最终运用李对称方法得到该方程的超定方程组,对方称组分析得到Degasperis-Procesi方程不具有依赖其拟势函数的非局部对称.同样对于Novikov方程,求出其势函数和拟势函数,对势函数进行同样处理,得到无穷多守恒量,最后,知道该方程组不具有依赖其势函数的非局部对称.最后,对尚待解决的问题进行了讨论.
[Abstract]:Nowadays, the nonlinear phenomena appear more and more in the natural and social sciences. The differential equations used to describe the phenomenon are concerned by the relevant mathematicians and physicists. This paper mainly deals with the non local symmetry and conservation laws and exact solutions of several partial differential equations, including the dichotomous -Camassa-Holm equation with the dispersion term. The -Camassa-Holm equation of gamma UX, the Foursov equation, the complex Camassa-Holm equation. This paper also considers the spectral problem of the shallow water wave equation 3 x 3, and constructs the infinitely many conservation laws of the Degasperis-Procesi equation and the Novikov equation. It is proved that the Degasperis-Procesi equation and the Novikov equation do not depend on the non local symmetry of the potential function. First, we introduce the background of the equation and the basic theory and method to study the nonlocal symmetry and precision. Secondly, we study the nonlocal symmetry and conservation law of the two component -Camassa-Holm equation. This equation is a short-wave limit of the two component Camassa-Holm equation. First, the geometry of the equation is derived. Based on the integrability, the quasi potential function of the equation is obtained and the conservation law of the equation is constructed, and the infinitely many conservation laws of the equation are obtained. Secondly, we have studied the nonlocal symmetry of the equation based on the ordinary Lie symmetry and the special structure of the equation. Finally, we get the non local symmetry of the -Camassa-Holm equation of the two points dependent on its potential function. Then we consider the muon -Camassa-Holm equation with the low order dispersion term gamma UX. Starting from the Lax pair of the equation, the pseudo potential function of the equation is constructed by its wave function, and the conservation law and infinitely many conservation quantities are obtained by using the obtained potential function. Then the Lie symmetry method is used to obtain the non local symmetry of the equation depending on the quasi potential function. The modified mu -Camassa-Ho is obtained. The LM equation is derived from the non local form of the modified Camassa-Holm equation. From the geometric integrability of the equation, the pseudo potential function is obtained, and the infinitely many conservation laws are constructed, and it is proved that the equation does not have the non local symmetry dependent on the pseudo potential function. The short pulse equation is a modified mu -Camassa-Holm equation containing the first order linear term UX. In this chapter, the quasi potential function and infinite conservation law of the equation are obtained. Third, the exact solution of the B-family system is studied with the method of invariant subspace combined with the generalized conditional symmetry method. This solution is generated by a three dimensional invariant subspace, corresponding to a class of generalized conditional symmetry of the equation, and the coefficients depend on the time. The function satisfies the Emden equation. By analyzing the structure of the solution of the Emden equation, the structure of the solution of the equation is obtained. The whole existence and blasting conditions are given. Then the structure of the solution is fully depicted. Then the peak soliton solution of the dual Foursov equation is studied. First, the weak form of the square path is obtained, and then the weak form of the equation is proved by the weak form of the equation. The equation has a peak soliton solution. Finally, we discuss the complex Camassa-Holm system, which can be obtained from the Mobus geometric flow. We prove that the equation has a peak soliton solution. Fourth, the infinitely many conservation laws of the Degasperis-Procesi equation and the Novikov equation, and the non local symmetry, the Degasperis-Procesi equation and the Novikov equation are considered. It is a typical wave equation with 3 x 3 spectrum problems, which is extended to the 3 x 3 spectral problem based on the non local symmetry and conservation law of the 2 x 2 spectral equation. It first constructs the two potential functions and two pseudo potential functions of the Degasperis-Procesi equation, and further expands the power series by the potential function number, and compares the same general parameter [lambda]. The coefficients reach infinitely many conservation laws. Finally, we use the Lie symmetry method to get the hyperset equations of the equation. The other is called the group analysis to obtain the non local symmetry of the Degasperis-Procesi equation which does not depend on its potential function. Also, the potential function and potential function are obtained for the Novikov equation, and the potential function is treated equally, and the infinitely many conservation are obtained. Finally, we know that the equations do not have non local symmetry depending on their potential functions. Finally, we discuss the unsolved problems.
【学位授予单位】：西北大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175.29

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