Apply Hirota Bilinear To Solve Explicit Solutions Of Some Soliton Equations

The exact solutions of nonlinear evolution equations(NLEEs)play an important role in the development of soliton theory.By many scholars' research,using the mathematical method can cleverly construct exact solutions,such as lump solution,kink-lump solution and rational solutions.However,it is still important and difficult to solve the exact solu-tions of various nonlinear evolution equations and use solutions to explain some physical phenomena.This article separately takes Backlund transformation and Hirota bilinear method as the research methods to solve some exact solutions to(2+1)-dimensional BKP equations,(2+1)-dimensional BKP-like equations,and(3+1)-dimensional Jimbo-Miwa-like equations.These exact solutions include soliton solution,lump solution,kink-lump solution and rational solutions.The lump solution is the focus of this study,and it is also a hot topic of research in the field of soliton theory in recent years.The main contents of this article include two parts:the first one is to obtain lump solution and soliton solutions of(2+1)-dimensional BKP equation,lump solution and rational solutions of(2+1)-dimensional BKP-like equation;the second one is to get kink-lump solution and rational solutions of(3+1)-dimensional Jimbo-Miwa-like equation.The starting chapter briefly summarizes the research background and relevant re-search methods about solving the nonlinear partial differential equations.The Hirota bilinear method and Backlund transformation are introduced in detail and explored.Af-ter that,the author elaborates on the generation and research status of the lump solution,and then prepares the theory for further study.The second chapter mainly studies the(2+1)-dimensional BKP equation and(2+1)-dimensional BKP-like equation.On the one hand,the soliton solutions of the(2+1)-dimensional BKP equation are solved by Hirota bilinear method.Then under the com-bined effect of the Backlund transformation and the nonlinear superposition formula,we can search the lump solutions of the(2+1)-dimensional BKP equation.On the other hand,we can use the generalized bilinear forms to obtain BKP-like equation and seek lump solution and rational solutions to this equation.The third chapter focuses on kink-lump solution and rational solutions of(3+1)-dimensional Jimbo-Miwa-like equation.Starting from the(3+1)-dimensional Jimbo-Miwa equation,the(3+1)-dimensional Jimbo-Miwa-like equation is obtained by the gen-eralized bilinear form.Considering the dimensionality of this equation,we first reduce the dimension of the Jimbo-Miwa-like equation,and then use Maple software to solve the kink-lump solution and rational solutions.Finally,we give some figures to describe the shape and surface for these solutions.