Global Solutions For Two Kinds Of Mathematical Physical Equations

The thesis concerns with the properties of global solutions for two kinds of mathemat-ical physical equations, precisely speaking, studies the nonlinear dynamics of relativistic membrane in the Minkowski space and solitary waves of the Nwogu's Boussinesq equation in water wave theory. It is organized as follows.Chapter 1 briefly recalls the present situation of the study on the two kinds of mathe-matical physical equations. The central problems under consideration and the main results obtained are stated.Chapter 2 concentrates on the nonlinear dynamics of relativistic membrane. By variational method and geometrical method, the system of equations for the relativistic membrane in the Minkowski space R1+n (n≥3) is derived. It can also be reduced to a (1+2)-dimensional quasilinear hyperbolic system and possesses many important properties such as non-strict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. An interesting phenomenon is found and proved, that is all plane wave solutions to this system are light-like extremal sub-manifolds and vice versa except for a type of special solution.Chapter 3 furthermore studies the nonlinear dynamics of relativistic membrane and mainly focus on the relationships between the equations for the relativistic membrane moving in the Minkowski space R1+n (n≥3) derived by us and that in canonical form in literature. The equivalence between them is proved and another geometric explanation about it through Noether's second theorem is also given. Moreover, for the motion of relativistic string in the Minkowski space, the similar argument about the equivalence is also presented.Chapter 4 investigates the solitary waves and periodic waves of Nwogu's Boussinesq equation in water wave theory. This model contains one independent parameter which is related to that the horizontal velocities at what level are chosen as the horizontal velocity variables. We employ the bifurcation method to qualitatively analyze the existence conditions of solitary waves and periodic waves for this model. Meanwhile, we find that the cusp periodic waves appear.Chapter 5 pays attention to the head-on collsions between two solitary waves of the Nwogu's Boussinesq equation. We first apply the perturbation method to this model and derive an approximate solution. Then the mechanics of the head-on collision, especially the impacts of the independent parameter on the phase shifts and the maximum run-up amplitude of two colliding waves is investigated. Comparison between our results and that of the integrable classical Boussinesq equation is also given...