Research On Several Methods For Exact Solutions Based On Symbolic Calculation

The research on the exact solutions of nonlinear partial differential equations has always been a hot topic,which provides the foundation and theoretical basis of modern natural science for studying and explaining some nonlinear phenomena in atmosphere,river and Marine environment.Therefore,it is of great practical value and significance to study the exact explicit solutions of nonlinear partial differential equations.In this paper,with the help of computer symbolic calculation software,some important methods for solving nonlinear partial differential equations in soliton theory,such as Lie symmetry analysis method,transformed rational function method,linear superposition principle,Hirota bilinear method and positive quadratic function method are studied,and a large number of new exact explicit solutions are obtained.The main contents of this paper are as follows.In chapter 1,we introduce the research background,significance of this paper,the research contents and methods to be adopted.In chapter 2,the classical Lie point symmetry and similarity reduction of the Mikhal?v-Pavlov equation are obtained by using Lie symmetry analysis method.By solving the reduced equations?including:variable coefficient partial differential equations and constant coefficient partial differential equations?,the exact explicit solutions of the equation are obtained.Finally,the conservation laws of the equation are obtained by using symmetry and Ibragimov's theorem.In chapter 3,based on a transformed rational function method and its extension method and the positive quadratic function method,the exact explicit solutions of some nonlinear equations with Hirota bilinear form and generalized Hirota bilinear form are obtained by the aid of the symbolic calculation software Maple.Firstly,based on a transformed rational function method and different ordinary differential relations,a rich traveling wave solutions of a?3+1?-dimensional nonlinear partial differential equation are obtained.Further,the complexiton solutions of the generalized?3+1?-dimensional Shallow Water equation on the D₅ operator and a?3+1?-dimensional nonlinear partial differential equation are obtained by using the extended transformed rational function method.Secondly,by applying the positive quadratic function method,we get the lump solutions and rogue wave solutions of?3+1?-dimensional nonlinear evolution equation on the D3 operator,and obtain lump solutions and periodic solutions of the equation on D₅ operator.In chapter 4,we mainly study the complexiton solutions of the?2+1?-dimensional Sawada-Kotera equation,the?2+1?-dimensional bidirectional Sawada-Kotera equation and the?3+1?-dimensional potential-Yu-Toda-Sasa-Fukuyama equation by using the Hirota linear superposition principle.Firstly,the resonant multiple wave solutions of the three equations in the real number domain are obtained by using the linear superposition principle.Then,based on the linear superposition property of the resonant multiple wave solutions,the resonant multiple wave solutions are extended to complex field,and the complexiton solutions of the equations are constructed based on a group of two complex valued resonant multiple wave solutions.In addition,the positive complexiton solutions of the?2+1?-dimensional Sawada-Kotera equation and?2+1?-dimensional bidirectional Sawada-Kotera equation are constructed.In the last section of this chapter,the dynamic characteristics of the positive complexiton solutions and the resonant multiple wave solutions are analyzed.Finally,in the three years of postgraduate study,we study and summarize the interesting hotspot methods in soliton theory,and put forward a prospect of the future research work.