The Research Of Hirota Method And Bell Polynomial On Nonlinear Partial Differential Equations

As one of the important topic of soliton theory, the investigation of integrability for anonlinear evolution equation can be regarded as a pre-test and the first step of its exactsolvability. With the continuous development of integrable system, there are many significantproperties, such as Painleve test, Infinite conservation laws, Hamiltonian structure, Lax pairs,Inverse scattering transformation, infinite symmetries that can characterize integrability ofnonlinear equations. This paper mainly describes the properties of Bell polynomials and therelationship between Hirota bilinear operator and it. Then this method is applied to theclassical KP equation and the variable coefficient (2+1) dimensional generalized breakingsoliton equation.The layout of this paper is as follows.In chapter1, we briefly introduce the development history of soliton and some methodsto achieve the exact solutions of nonlinear evolution equations, then we give the main topic ofthis article.In chapter2, we establish the relationship between Hirota bilinear operator and Bellpolynomial based on the study of their related properties.In chapter3, based on Dimensional analysis and symbolic computation, we extendBell polynomial and the Hirota bilinear method to the classical KP equation and the variablecoefficient (2+1) dimensional generalized breaking soliton equation. Then, the bilinearformulism, the bilinear Backlund transformation, the Lax pairs and the infinite conservationlaws of the studied equation are systematically constructed in a quick manner.In chapter4, we give some conclusions.