Bifurcation Analysis And Exact Solutions Of Some Nonlinear Evolution Equations

Qualitative theory and soliton theory is applied in this paper to study the three-dimensional Kudryashov-Sinelshchikov (3DKS) equation and the Landou-Ginburg-Higgss (LGH) equation. The phase portraits of different situations are given, exact solutions are also given according to some of the orbits of the phase portraits. The work includes the following two aspects:(1) In chapter2, the3DKS equation is studied. Bifurcations and phase portraits are discussed. The parameter χ,which depends on the viscosity, affects the types and stability of the equilibrium points in phase portraits. With the different values of χ, five cases of the phase portraits can be obtained. When the parameter of the equation χ=0, i.e., without the effect of the liquid viscosity, periodic and homoclinic orbits can be found. Also periodic and solitary wave solutions are derived.For the case χ≠0,i,e., in consideration of element of the liquid viscosity, hete-roclinic orbits can be found. Also kink wave solutions are derived. Furthermore, the effects of the parameter χ are discussed. In addition, Bell polynomial approach is applied to obtain the bilinear form of the3DKS equation. Hirota bilinear method is applied to obtain the N-soliton solutions and Backlund transformation.(2) In chapter3, the LGH equation is studied. Bifurcations and phase portraits for this equation are discussed. We found out that the wave speed c can affect the types and stability of the equilibrium points in phase portraits. For the different values of c, two cases of the phase portraits can be obtained. When-1<c<1, heteroclinic and periodic orbits can be found. Also kink and periodic wave solutions are derived. When c<-1or c>1, two kinds of periodic and homoclinic orbits can be found. Also two kinds of periodic and solitary wave solutions are derived.