On Exact Solutions,Integrability And Darboux Transformation To Some Nonlinear Differential Equations

In this paper,the exact solutions of some nonlinear differential equations are stud-ied based on several different methods.The content of this paper is as follows:The first chapter introduces the relevant research background and its significance.In chapter 2,the nonlocal symmetries of variant Boussinesq systems are intro-duced.Firstly,based on the truncated Painleve expansion,the nonlocal symmetry,non-auto Backlund transformation,and Schwarzian form of the variant Boussinesq system are obtained.In order to obtain nonlocal symmetries groups of variant Boussinesq sys-tems,the new dependent variables are introduced,and the corresponding finite group transformation is obtained by solving the initial value problem of the equation.Sec-ondly,according to the definition of CRE,it is proved that the equation is CRE solv-able.The interaction solution between the soliton and the cnoidal wave of the equation is given by assuming the appropriate solution.Finally,according to the classical Lie symmetry analysis,the similarity reduction solutions of variant Boussinesq systems are obtained.Chapter 3,based on Bell polynomials and Hirota bilinear,the bilinear form for the(2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation is ob-tained.On this basis,the soliton solutions of the equation are obtained.Based on the knowledge of Riemannian theta functions,the periodic wave solutions of the equation are obtained.The relationship between the periodic wave solution and the soliton so-lution is graphically analyzed and it is proved that under certain limit conditions the periodic wave solution of the equation can degenerate into a soliton solution.Chapter 4,by using the method of obtaining bilinear in chapter 3,the bilinear forms of the(2+1)-dimensional B-type Kadomtsev-Petviashvili equation and the(3+1)-dimensional Kadomtsev-Petviashvili equation are obtained.On this basis,the appropri-ate extended homoclinic test function is selected to obtain the breather wave solution and the rogue wave solution of the two equations,respectively.The relationship be-tween the breather wave solution and rogue wave solution is further studied,and the results imply that the extreme behavior of the breather wave can reduce to the rogue wave under certain restrictions.Chapter 5,firstly,based on the Lax pair of the generalization of the coupled non-linear schrodinger equation,the Darboux transformation of the equation is obtained.By using the Darboux transformation of the generalization of the coupled nonlinear schrodinger equation,the soliton solutions.the breather wave solutions and rogue wave solutions of the equation are obtained.Furthermore,the higher-order rogue wave of the coupled Hirota equations are obtained by applying the generalized Dar-boux transformation.Chapter 6 gives a brief summary and prospect of this paper.