Research On Bilinearity,Symmetries And Analytical Solutions Of Some Nonlinear Differential Equations

The study of analytical solutions and symmetries for nonlinear differential equations can help to explain important physical phenomena.In this work,we investigate several types of nonlinear differential equations.Based on Bell polynomials and Hirota bilinear method,several different analytic solutions are constructed,and the propagation characteristics of the obtained solutions are analyzed.In addition,nonlocal symmetries and conservation laws of the equation are established with the help of the theory of symmetry and the Painlev′e truncation expansion method.The main contents of this work are as follows.In chapter 1,the research background and significance of soliton theory and related methods are briefly introduced,and then the main contents of this thesis are also showed.In chapter 2,we first introduce some preliminary knowledge of Hirota bilinear derivatives and Bell polynomials.Then we successfully obtain the bilinear expression of the(3+1)-dimensional KP equation,and found its soliton solutions.By developing the homoclinic breather test approach,homoclinic breather wave and rogue wave solutions of the(3+1)-dimensional KP equation and(3+1)-dimensional generalized nonlinear equation are constructed based on the obtained bilinear expression,respectively.According to the analysis,we can find that the homoclinic breather wave solution can be degraded to rogue wave solution under certain condition.In chapter 3,the bilinear formalism of the generalized(3+1)-dimensional KPBoussinesq equation,the(2+1)-dimensional KdV equation and the(2+1)-dimensional generalized KdV equation is derived based on the Hirota bilinear method,and then their lump solution and interaction solutions are also constructed successfully.For the(3+1)-dimensional KP-Boussinesq equation and the(2+1)-dimensional KdV equation,we obtain the lump solution and the mixed solutions between lump solution and soliton solution by introducing appropriate density function.Meanwhile,the lump solution,the hybrid solutions of lump solution and soliton solution and the hybrid solutions of lump solution and breather wave solution for the(2+1)-dimensional generalized KdV equation are studied by extending the long wave limit method.What’s more,the dynamic evolution characteristics of the above solutions are analysed.In chapter 4,the Painlev′e truncation expansion method is used to obtain nonlocal symmetry of the generalized Ito equation.By introducing new variables and solving the corresponding initial value problems,we can obtain the corresponding symmetric group transformation.The CRE method is used to prove that the equation is solvable.Then,its interaction solutions between soliton wave and cnoidal wave is constructed.Furthermore,the conservation laws are systematically deduced of the equation by extending the Lie symmetry analysis method and the adjoint equation.In chapter 5,we consider the perturbed nonlinear Schr¨odinger equation.By extending the ansatz method,we get the bright soliton solution and the dark soliton solution of the equation.The existence condition of the above solutions is also presented.Additionally,their dynamic propagation properties are analysed by plotting.In the similar way,the complexitons and power series solutions of the perturbed nonlinear Schr¨odinger equation are also constructed.In chapter 6,a brief summary of this thesis and related prospects are given for later work.