Research On Several Problems Of Nonlinear Wave Equations

Nonlinear wave equations are important mathematical models for describing naturalphenomena and are one of the forefront topics in the study of mathematical physics,especially in the study of soliton theory. In this doctorial dissertation, we introduce twosimple methods for solving exact solutions by analysis the characteristics of some equa-tions, and study several problems of nonlinear wave equations by using the bifurcationmethod of dynamical systems, conservation laws, orbitally stable theory, and the softwareof Mathematica. The main work of this dissertation are as follows.In Chapter2, a simple method is proposed for constructing general exact solutions ofnonlinear partial diferential equations. We will give some previous results from anotherpoint of view. We choose the Camassa and Holm Degasperis and Procesi（CH DP）equation and the generalized b family equations to illustrate the validity and advantagesof the method.In Chapter3, we introduce a simple approach to reduce the order of equations withhigher order nonlinearity. We choose the generalized forms of the KdV equation, theCamassa Holm（CH） equation and the Zakharov Kuznetsov（ZK） equation to illustratethe validity and advantages of the approach. Then the exact solutions of the reducedequations can be discussed further. The approach used here can be applied to some othernonlinear equations with higher order nonlinearity and is also suitable for some higherdimensional models.In Chapter4, the objective of this paper is to extend some results of pioneers for thenonlinear equationThe equivalent relationship of the traveling wave solutions between the integrable equa-tion and the generalized KdV equation is revealed. Moreover, when k=pq（p≠q and p, q∈Z+）, we obtain some new traveling wave solutions by the bifurcationmethod of dynamical systems.In Chapter5, we study orbital stability of the smooth solitary wave with nonzeroasymptotic value for the mCH equationu_t-u_xxt+2ku_x+au²u_x=2u_xu_xx+uu_xxxUnder the parametric conditions a>0and k <1/8a, an interesting phenomenon is discov- ered, that is, for the stability there exist three bifurcation wave speedssuch that the following conclusions hold.（i） When wave speed belongs to the interval （c₁, c₂） for-63/8a<k <1/8a, the smoothsolitary wave is orbitally stable.（ii） When wave speed belongs to the interval （c₂, c₃） for-63/8a<k <1/8a, the smoothsolitary wave is orbitally unstable.（iii） When wave speed belongs to the interval （c₁, c₃） for k≤-63/8a, the smooth solitarywave is orbitally unstable.