The Lump Solution And Understanding Of The (2+1) Dimensional PKP Equation

The phenomenon of nonlinear waves has a wide range of applications in many fields such as physics and applied mathematics.With the development of symbolic computing,the solution of nonlinear partial equations has attracted the attention of many scholars and has become a research topic in recent years.They have proposed many solving methods,such as inverse scattering transformation method,Lie group method,Hirota bilinear method,etc.In this paper,we first do a logarithmic transformation to the solution of(2+1)-dimensional potential Kadomtsev-Petviashvili(pKP)equation.Then using the Hirota bilinear method,the equation is transformed into a bilinear equation with the same solution and the positive quadratic solutions and rational solutions are obtained.Finally,from the inverse transformation,we further obtain the lump solutions and rational solutions of(2+1)dimension pKP equation.In addition,through drawing its three-dimensional map,contour map and high-density map,we analyzed the waveform characteristics of these solutions and the trajectory of the wave at different times.This paper consists of three chapters:The first chapter is the introduction.Firstly,some contents of nonlinear partial differential equations are briefly summarized.Then the background and development of nonlinear partial differential equations are introduced.At the end of this chapter,the research background of(2+1)-dimensional pKP equation is introduced.Chapter 2 is the propaedeutics,giving some basic concepts and algorithm principles of the Hirota bilinear method.Firstly,the related concepts and main functions of D-operator are introduced.After that three transformations of bilinear methed are introduced.The third chapter is the core of this paper.Using the Hirota bilinear method,we bilinearized the(2+1)dimension pKP equation to obtain its bilinear equation.Then through the solution of the bilinear equation system and inverse transformation,obtained the lump solutions and rational solutions of the(2+1)dimension pKP equation,and analyze the waveform and moving trajectory of the solutions by drawing the image of the solution.