Research On The Exact Wave Solutions Of Two Nonlinear Wave Equations

Nonlinear wave equations are mathematical models describing a number of nonlinearphenomena, and solutions of nonlinear wave equations are the important theoretical basesfor study of nonlinear phenomena. Therefore the solutions of the nonlinear wave equationsbecome to be an important part in the nonlinear science. With the development ofscience and technology, more and more methods of solving these nonlinear wave equationsare proposed. In the present thesis, two typical nonlinear wave equations, i.e. ZK-MEW equation and Kaup-Boussinesq equation system, are solved based on the qualitativetheory of diferential equations and the bifurcation method of dynamical systems, andexact traveling wave solutions are then given.ZK-MEW equation is one of the important nonlinear diferential equations in physicalapplications. In the second part of this thesis, frstly, we show the bifurcation phasesportraits for the corresponding plane system of this equation; and give27exact travelingwave solutions, which contain solitary wave solutions, blow-up solutions, periodic blow-up solutions, periodic wave solutions and kink wave solutions; we also discuss the limitsof some solutions under the condition of special parameters. Then we obtain some newexact solutions, such as the rational fraction solutions. Finally, we confrm the correctnessof these solutions by Software Mathematica.In the third part of the paper, we study the Kaup-Boussinesq system, which is afamous shallow water equation system. Using the bifurcation method, We draw the phaseportraits for the corresponding plane system of this equation system, and get25exacttraveling wave solutions, such as, solitary wave solutions, blow-up solutions, periodicblow-up solutions and kink wave solutions. Furthermore, some solutions are new. In theend, solutions has been proved to be correct by Software Mathematica.