The Traveling Wave Solutions For Several Kinds Of Nonlinear Mathematical Physics Equations

This doctoral dissertation is devoted to study the traveling wave solutions of severalkinds of famous nonlinear mathematical physics equations, such as hyperelastic rod equa-tion, Camassa-Holm-γequation, modified Camassa-Holm equation, modified Degasperis-Procesi equation, quantum Zakharrov-Kuznetsov equation and so on.By using the qualitative theory of di?erential equations, the bifurcation methodsof dynamical systems, simulation method and homotopy perturbation method, auxiliaryequation method, bilinear method, we obtain new exact solitary wave solutions, multiplesoliton solutions, periodic wave solutions et al and the approximate explicit solutionsof these equations. At the same time, we have proved the orbital stability of multiplepeakons of Camassa-Holm-γequation based on some pervious stability theory. The mainresearch work are as follows.In Chapter 1, the historical background, research developments, the main methodsfor solving nonlinear equations and achievements of solitons are summarized. In the end,we give a simply introduction of the main content of this paper.In Chapter 2, by using the qualitative theory of di?erential equations and bifurcationmethod of dynamical systems, we study the compressible hyperelastic rod equation, notonly smooth solitary wave solutions are obtained, but also a new phenomenon is found thatis when the initial value tends to a certain value, the periodic wave loses the smoothnessand becomes a periodic shock wave. But when the initial value equals to this value, theperiodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems,this represents that one of orbits can pass through the singular line.In Chapter 3, we study the orbital stability of multiple peakons of Camassa-Holm-γequation. First of all, we give a full introduction on the orbital stability of peakon solutionof Camassa-Holm equation. Basing on the pervious result, such as the conclusion ofConstantin's, we have proved the orbital stability of multiple peakons of Camassa-Holm-γequation.In Chapter 4, by using the homotopy perturbation method and auxiliary equationmethod, we study modified Camassa-Holm equation and modified Degasperis-Procesiequations. Both approximate solutions and exact solutions are obtained. Meanwhile,the comparison between the approximate solutions and exact solutions are made. Thisillustrates that the approximate solutions are e?ciency in engineering problems.In Chapter 5, by using auxiliary equation method and bilinear method, we studythe quantum Zakharrov-Kuznetsov equation. A series of exact solutions are obtained. Specially, we use the bilinear method to obtain 1-soliton, 2-soliton, 3-soliton solutionsof quantum Zakharrov-Kuznetsov equation. And then it can be extended to N-solitonsolutions, where N≥3.In Chapter 6, we give a simply introduction of 2-component Camassa-Holm equation.Finally, after giving the summary of this dissertation, some problems are proposedfor further research and explore.