Nonlocal Symmetries And Exact Solutions For Camassa-Holm Type Systems

Nonlinear phenomena generally exist in mathematics,physics and other disciplines.With its in-depth study,non-linear science has gradually developed into an important comprehensive discipline.Shallow water system plays an important role in the research of many partial differential equations,which can be simplified by constructing solutions to nonlinear problems.The structure of the solutions can not only describe the nonlinear phenomena effectively,but also help us to understand the properties of the equations more comprehensively.The main methods used in this thesis are symmetric group method,separation method,perturbation method and characteristic method.In this thesis,we ues these methods to construct the solutions of the ?-camassa-holm equation,the GDGH2 system and the extend form.Furthermore,we analyse the properties of solutions.The thesis is organized as follows:Firstly,the non-local symmetry of ?-Camassa-Holm equation is constructed by using the geometric integrability.On the basis of the new non-local symmetry,the initial value problem is solved.Secondly,the self-similar solutions of the GDGH2 system and the extended form are constructed by using the method of separation method,perturbation method and characteristic method,then discussing the global existence and blow-up of the solutions.