The Construction Of Solutions For The Multi-component Shallow Water System

The study of shallow water system not only plays an important role in ap-plied mathematics and mathematical physics,but also presents an important subject in the study of nonlinear partial differential equations.In this thesis,we consider the generalized two-component Dullin-Gottwald-Holm shallow wa-ter system and analyse one kind of self-similar solutions of this system and its extended form.By the resulting of Emden equation,we investigate the global existence and finite-time blowup phenomenon.In the field of mathematical physics,the exact solutions are not only helpful for understanding the properties of equations itself,but also effective in char-acterizing some nonlinear phenomenon.Although the constructing method is not unique,the goal is to simplify the nonlinear problems.There are many well-known classical methods,such as Darboux transformation,Inverse scat-tering method,Backlund transformation,Hirota's bi-linear method,Painleve singularity analysis method and separation method.In this thesis,separation method,perturbation method and characteristic method are employed to study the construction and properties of generalized two-component DGH system and its extended form.The thesis is organized as follows:Firstly,we construct the self-similar solutions admitting elliptic symmetry with a drift structure and analyse the global existence and blowup phenomenon of generalized DGH system through separation method.Secondly,we extend the GDGH2 system and construct the self-similar blowup solutions.Furthermore,the perturbation method and characteristic method are used to construct two type of exact solutions,then analysing the properties and significance.