Research On The Exact Solutions And Related Properties Of Two Types Of Nonlinear Partial Differential Equation

Nonlinear partial differential equations have been applied in many fields.How to find the solution of nonlinear partial differential equation has become an important research content in soliton theory.The solution methods mainly include Hirota bilinear method,Backlund transformation method,Riemann Hilbert method,etc.Among them,the Hirota bilinear method can solve nonlinear partial differential equation more effectively.This method is to bilinear the nonlinear partial differential equation by using the defined D-operator and variable replacement,then solve the bilinear equation by the trial function method or series perturbation method,and finally obtain the solution of the objective equation by using the initial variable replacement again.Recently,Professor Ma Wenxiu generalized the generalized bilinear differential operator on the basis of the operator Dp-.The innovation of this paper lies in the introduction of generalized bilinear differential operator Dp-.Based on this operator,a new nonlinear partial differential equation is constructed,and various types of exact solutions are solved by using different forms of function construction.In this paper,we first use the Hirota bilinear method to study the 1-soliton solution,2-soliton solution,and 3-soliton solution of the(2+1)dimensional nonlinear partial differential equation,draw the image of the solution,and analyze the waveform and motion trajectory of the solution.In addition,the(2+1)dimensional nonlinear partial differential equation and(4+1)dimensional nonlinear partial differential equation under D3 operators are derived by using the generalized bilinear differential operator Dp and Bell polynomial theory.Using the constructor method and using the symbolic computing software Maple,obtain the lump solution,rational solution,rogue wave solution,and breathing solution of equation(4-2)and(5-5).Finally,through the theory of generalized bilinear differential operator Dp and Bell polynomials,the(2+1)dimensional nonlinear partial differential equations and(4+1)dimensional nonlinear partial differential equations under the operator D5 are derived,namely equation(6-2)and(6-4).Apply the function construction method to solve these two equations and obtain their rational solutions.