Research On Several Solutions Of Nonlinear Partial Differential Equations

As one of the main contents of nonlinear science,nonlinear partial differential equa-tions are mathematical models used to describe the complex physical phenomena that the objective world generates with space and time.For decades,through the efforts made by relevant researchers,many methods have been created for solving nonlinear partial differential equations,such as Dabour transformation,symmetry reduction,homotopy perturbation.This paper will focus on extending and developing some of the solutions,which will be helpful to obtain more types of new solutions.Specifically,it includes the following aspects:Chapter 1:There is an introduction about background and corresponding knowl-edge of nonlinear partial differential equations.At the same time,a brief description of the research results obtained in this paper is given.Chapter 2:The function expansion method is extended.Firstly,the solution is extended from the original positive power to the negative power,and then all variables in the expansion are completely separated,which enriches the exact solution of the nonlin-ear partial differential equation.Finally,the(~?/~2)-expansion method and the(F/G)-expansion method are used to solve the(2+1)-dimensional Broer-Kaup-Kupershmidt e-quation and the(2+1)-dimensional fractional order Nizhnik-Novikov-Veselov equation,and the structural excitation solutions of their special solitons are also given.Chapter 3:The Hirota bilinear derivative method is used to combine the Lump so-lution of the generalized(3+1)-dimensional shallow water wave equation with the respira-tory wave solution,which shows that the Lump-type soliton is kinked by the solitary wave engulfing process.Then,the single soliton and Lump solution of the(2+1)-dimensional Sawada-Kotera equation are combined and superimposed to explore the particle charac-teristics of the two types of solutions during the interaction,such as collision,rebound,absorption and splitting.In addition,the Lump-type soliton only appears in an instant due to the influence of the double-striped solitons,and then disappears immediately,so the Lump-type soliton becomes an resonance strange wave.In order to get a profound understanding of this type of strange wave,the characteristic quantity of the new strange wave will be obtained through the combination of theoretical calculation and digital com-bination,such as movement trace,existence time,area and volume.Chapter 4:The first-order analytical approximate solution of the fractional Klein-Gordon equation under strong and weak nonlinear condition is solved by the renormal-ization method.When no special consideration is given to the parameter size,the lin-earization and correction method is directly adopted to get the first-order approximate solution of the equation,the results of which are also compared.Chapter 5:Summary and outlook.