The Study Of The Analytical Solutions,Symmetry And Conservation Laws For Nonlinear Partial Differential Equations

Many complex natural phenomena can be explained by nonlinear equations,so it is quite meaningful to study the analytical solutions for nonlinear equations.Inverse scattering method,Hirota bilinear method,Darboux transformation and Lie symmetry analysis etc.are effective methods to solve nonlinear equations,we can construct breather solutions,rogue wave solutions and resonant solutions and other analytical solutions by using these methods.In this paper,we study resonant solutions and rogue wave solutions of(2+1)-dimensional Sawada-Kotera equation,lump solutions and their interaction solutions of(2+1)-dimensional dispersive long wave equation,breather solutions and rogue wave solutions of(3+1)-dimensional Jimbo-Miwa equation,two types of semi-rational solutions of the coupled nonlinear Schr ¨odinger equations and analytical solutions,symmetry and conservation law of the coupled Kd V system.In the first chapter,we mainly introduce research background and meaning of nonlinear evolution equations,then introduce the research methods and contents of the paper.In the second chapter,we derive the Hirota bilinear form of(2+1)-dimensional Sawada-Kotera equation by Hirota bilinear method,then with the help of linear superposition principle,the resonant solutions on the real and complex fields are constructed.Then,by using a kind of ansatz method,we construct a type of rogue wave solutions:first-order rogue wave and second-order rogue waves.In the last,the spatial and density images of resonant solutions and rogue waves are drawn to illustrate the behavior characteristics of these solutions.In the third chapter,we mainly use painlevé analysis method to construct lump solutions and their interaction solutions of the(2+1)-dimensional dispersive long wave equation.Firstly,we expend the equation by truncated painlevé,the painlevé B¨acklund equation is obtained.Then,by taking appropriate seed solutions,equation about(,,)are derived.When (,,)is a quadratic function,we obtain the lump solutions of the equation.When (,,)is a combination of a quadratic function and an exponential function,the interaction solutions between lump solution and one linesoliton are constructed.When (,,)is the combination of a quadratic function and two exponential functions,the interaction solutions between lump solution and a pair of line solitons are obtained.The spatial characteristics of these solutions are analyzed by images.In the fourth chapter,we firstly obtain the bilinear form of(3+1)-dimensional Jimbo-Miwa equation,then with the help of a ansatz form of -soliton solutions,the breather solutions of the equation are obtained by complex conjugate method.In addition,we derive first-order rogue waves,second-order rogue waves and third-order rogue waves of(3+1)-dimensional Jimbo-Miwa equation by a ansatz form.Finally,we analyze the breather solutions and rogue wave solutions by selecting different parameter values.In the fifth chapter,generalized Darboux transformation is obtained by extending the classical Darboux transformation.Then two kinds of semi-rational solutions for a coupled nonlinear Schr ¨odinger equation are constructed by using the generalized Darboux transformation.For the first kind of semi-rational solutions,we analyze the regular solitons,two degenerate solitons and three degenerate solitons.For the second kind of semi-rational solutions,we construct several types of situations by taking several different types of parameters.In the last,two kinds of semi-rational solutions are analyzed by their spatial images and density images.In the sixth chapter,we study a coupled Kd V system by using the Lie symmetry analysis method.Firstly,the Lie symmetry,the one-parameter transformation group and optimal system of the system are obtained.In addition,through the analysis of optimal system,the similarity reductions and the invariant solutions are constructed.Finally,the conservation laws of the coupled Kd V system are also constructed by applying a new conservation theorem.In the last chapter,summary the research work of the paper and prospect the future research plan.