The Traveling Wave Solutions And Properties For Some Kinds Of Nonlinear Mathematical Physics Equations

In this thesis,the traveling wave solutions and properties of some nonlinear mathematical physics equations are studied by means of dynamic system method,geometric singular perturbation theory,Melnikov’s function,invariant manifold theory and symbolic computation.The main work includes the following four parts:In Chapter 3,the traveling wave solutions of a class of K(m,n)equation with generalized evolution terms are studied via using dynamical system method.Firstly,bifurcations of the traveling wave solutions are studied under the framework of singular system,and the parameter conditions of the bifurcations,the bifurcation phase diagrams and some exact traveling wave solutions are obtained,thus the work of the literatures[97-107]have been generalized.A non-singular traveling wave system is obtained by introducing variable transformation.The similarities and differences of traveling wave solutions and dynamic properties between singular and non-singular traveling wave systems are studied,it is revealed that variable transformation may change the type and dynamic properties of traveling wave solutions,thus the work of the literatures[97-107]have been developed.In Chapter 4,the perturbed double Sine-Gordon equation is studied by means of dynamical system method,geometric singular perturbation theory,Melnikov’s function and invariant manifold theory.Firstly,when 0<c<1,the coordinate translation transformation is introduced to transform the traveling wave system(4.11)into the traveling wave system(4.21).With the help of the theorem A of the literature[119],it is proved that the least positive period T*(h)of periodic orbits to the traveling wave system(4.21)is monotonous in the interval(-2/(1-c2),2/(1-c2))Further,we study the least positive period T(h)of periodic orbits to the traveling wave system(4.11)and obtain T*(h)=T(h+1/(2-2c2),then it is proved T(h)is monotonous in interval(-3/(2-2c2),5/(2-2c2).Thus the first open problem proposed in the literature[118]is solved.Secondly,when 0<c<1,with the help of mathematical software and the calculation technique of definite integral of trigonometric function,the Melnikov’s function corresponding to the heteroclinic orbits of the approximate Hamilton system(4.38)is solved,thus the existence of kink and anti-kink solutions of the perturbed double Sine-Gordon equation is proved.The principle of wave velocity selection is given.Thus the second open problem proposed in the literature[118]is solved.In Chapter 5,the traveling wave solution of the perturbed Sine-Cosine-Gordon equation is studied by means of dynamical system method,geometric singular perturbation theory,Melnikov’s function,invariant manifold theory and symbolic calculation.Firstly,by means of dynamical system method and symbolic calculation,the bifurcation phase diagrams of the traveling wave solutions and exact expression of partial traveling wave solutions are obtained.Secondly,the perturbed Sine-Cosine-Gordon equation is compressed into a two-dimensional invariant manifold by means of the singular perturbation theory,thus transformed into an approximate Hamilton system.Finally,the existence of periodic solutions,solitaries,kink and anti-kink solutions of the perturbed Sine-Cosine-Gordon equation are proved by means of Melnikov’s function.There are few studies on the existence of traveling wave solutions for the perturbed Sine-Cosine-Gordon equation,the study develops the traveling wave solution of the perturbed Sine-Cosine-Gordon equation.In Chapter 6,the(2+1)-dimensional KdV equation with variable coefficients is studied by means of Hirota bilinear method.Firstly,1-soliton,2-soliton,3-soliton and N-soliton are obtained by means of exponential functional formal solutions.Secondly,by selecting complex conjugate parameter pairs,1-breather and 2-breather are obtained from 2-soliton and 4-soliton,respectively.By using long-wave limit technique and selecting appropriate parameters,1-lump,2-lump and 3-lump are obtained from 2-soliton,4-soliton and 6-soliton,and the dynamic property of lump solutions moving in a straight line with time are given.Finally,the interaction solutions of soliton,breather and lump are discussed,which show that the wave collision is elastic.In[132,133,136-139],β=3αand β(t)=3α(t)have been thoroughly studied.The variable coefficient relationship we consider is κ=α(t)/β(t),where κ is a non-zero constant,thus the work of the above-mentioned literatures is generalized and developed.