摘要
本文研究了广义Camassa-Holm方程、Degasperis-Procesi方程的整体守恒解,以及新型双Sine-Gordon方程的不连续解。Camassa和Holm利用哈密顿方法获得了一类新型色散波方程,叫Camassa-Holm方程(简称CH方程),它具有双哈密顿结构和无穷多守恒量,是完全可积的。Degasperis-Procesi方程(简称DP方程)是Degasperis和Procesi得到的,它不仅有尖峰解,还有激波解。双Sine-Gordon方程是一个很重要的方程,因为它广泛应用于诸如非线性光学等领域。最近,双Sine-Gordon方程的一些精确解已经得到。
第三章主要研究广义CH方程初值问题的整体守恒解,先将这个方程转化成一个常微分系统。在这个常微分系统中,应用索伯列夫空间的一些不等式、常微分方程相关知识,讨论解的适定性问题;
第四章主要研究了DP方程初值问题的适定性问题,采用了一个不同于CH方程的守恒律来辅助证明,证明了短时期解的存在性,进而证明了它的整体守恒解存在且唯一;
第五章主要研究了新型双Sine-Gordon方程的精确解,我们发现有两个精确解是不连续的,进而通过守恒律方程理论证明了这两个解是非连续的。
In this paper,we study the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation and Degasperis-Procesesi equation,and the exact solutions of the new double Sine-Gordon equation.Camassa and Holm derived a new completely integrable dispersive wave equation for water waves by Hamiltonian method,namely Camassa-Holm equation(i.e.,CH equation).It has bi-hamiltonian structure and infinite sequence of conserved quantities, and it is completely integrable.Degasperis-Procesi equation(i.e.,DP equation) was found by Degasperis and Procesi.It has not only peakon wave solutions,but also shock wave solutions.The double Sine-Gordon equation is a significant equation because of its applications in many fields such as nonlinear optics.Recently,many exact solutions of the double Sine-Gordon equation were found.
In Chapter Three,we prove the existence of global conservative solutions of the Cauchy problem for the generalized Camassa-Holm equation.We change it into an ODE system in a Banach space.In this system,by applying some inequalities in Sobolev space and some knowledge about the ODE theories,we consider the well-posedness of the solutions.
In Chapter Four,we study the well-posedness of the DP equation.We choose a conservative law which is different from CH equation to finish our proof.Then we prove the existence and uniqueness of the short-time solutions.Hence the existence of global conservative solutions with respect to the initial date is obtained.
In Chapter Five,we study the exact solutions for new double Sine-Gordon equation,we find that two exact solutions of the equation are noncontinous.Then we prove that the two solutions are noncontinuous solutions by conservation equation theory.
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