Traveling Wave Solutions For A Class Of Higher Order Nonlinear Wave Equations

Traveling wave solutions are some special kinds of solutions to nonlinear evolution equations.They can be usually characterized as solutions invariant with respect to translation in space.A variety of physical,chemical and biological phenomena that can be observed in the experiment can be described by the traveling wave solutions of the differential equations,especially those nonlinear equations.For various of waves that can be observed in our daily life,such as water wave,acoustic wave and electromagnetic wave and so on,the traveling wave solutions of the corresponding nonlinear wave equation can be used to describe the process of transmission.Understanding and discovering the intrinsic mechanism of the various waves in these nonlinear equations has become an important research topic in the field of nonlinear wave theory and numerical analysis.At present,the application of dynamical system theorems and method to the study of nonlinear wave equations is one of the most active research field.In this thesis,we firstly study a class of Ito fifth-order mKdV equation,whose corresponding traveling wave equations can be reduced to a fourth-order ordinary differential equations with parameters.For the corresponding fourth-order traveling wave equation,with the aid of computer symbolic computation,we obtain a two-dimensional invariant manifold which is determined by the planar dynamical system under certain parameter conditions.To determine the two-dimensional manifold,we study the bifurcation of the two-dimensional system by using of qualitative analysis and bifurcation theory of planar dynamical systems.Thus,all kinds of smooth bounded travelling wave solutions,including solitary wave solutions,kink wave solutions and periodic wave solutions of different amplitude are derived.Secondly,we study a complex mKdV equation,which is transformed into a second-order ordinary differential equation of real function under certain condition of parameters by means of translation,rotation and scaling transformation.We study the bifurcation and the phase portrait of this planar dynamical system by using the qualitative and bifurcation theory of dynamical systems.Then we derive the bounded solution of the equation under various parameters,and thus some bounded traveling wave solutions including the envelope solution of amplitude of periodic function of the complex mKdV equation are obtained.