| It is an important subject of modern mathematics and physics to explore the traveling wave solutions of mathematical physics equations and their dynamic properties.On the one hand,this paper studies the traveling wave solutions of rotation-two-component Camassa-Holm(R2CH)equation and coupled Higgs field equation from the perspective of dynamical systems.Specifically,we transform them into planar dynamical systems by using certain skills.Then,by the bifurcation method of dynamic systems,we obtain the bifurcation of phase portraits under corresponding parameter conditions,from which we derive the existence of solitary wave solutions,peakons and kink waves of R2 CH equation and coupled Higgs field equation.Further,through Hamiltonian function of planar systems,we derive their corresponding expressions and then we analyze their dynamical properties.On the other hand,we exploit geometric singular perturbation theory and Melnikov function to study the existence of homoclinic orbits of traveling wave systems corresponding to Gardner equation with Kuramoto-Sivashinsky perturbation.To be specified,we reduce the dimension of the traveling wave system by using the geometric singular perturbation theory,and then prove the existence of its homoclinic orbit through Melnikov function.These results will help us understand the dynamical properties of nonlinear systems and have important theoretical significance.The main structure of this paper is as follows:In the first part,we mainly introduce the basic knowledge and research methods of planar dynamical systems,as well as the research background and research works of some mathematical physics equations.From the second part to the fourth part,we study traveling wave solutions and their dynamical solutions corresponding to R2 CH equation and coupled Higgs field equation,and the existence of homoclinic orbits of Gardner equation with Kuramoto-Sivashinsky perturbation,respectively.In the fifth part,we mainly summarize the main research results of this thesis,and makes prospect for the future research. |
