The Bilinear Transformation And Exact Solutions Of Soliton Equations

With the development of science and technology,many mathematicians and physicists pay attention to the nonlinear problems which exist widely in the field of natural science and social science.Nonlinear mainly has three big branches: fractal, chaos,Soliton.in the field of fluid mechanics, plasma physics, nonlinear optics, classical field theory and quantum theory has been widely in research and application, so the study soliton has broad prospects and ver important significance.Solving soliton equations especially for exact solutions both in theory and practice is a very important research topic. At the same time, various methods for solving soliton equations have been found, such as the inverse scattering method, Hirota bilinear method, homogeneous balance method, Painlevé analysis and Darboux transformation method. The Hirota bilinear method is very effective and practical which has been widely used in solving soliton equations.The main content of this article is solving the bilinearization equation of the Kdv equation and(1+1)-dimensional dispersive long wave equation.First,depending on bilinear method to research Column Kdv equation and(1+1)-dimensional dispersive long wave equation,transforming Kdv equation and(1+1)-dimensional dispersive long wave equation into bilinearization equation, and then calculating the exact solition solutions of the two equations.Second,using Wronskian technique solve the Wronskian determinant solutions; using Pfaffian technique solve Grammian determinant solutions.Third,depending on the bilinearization equation deduces bilinear B?cklund transformation,and then finds the exact soliton solutions and the Wronskian determinant solutions by using the bilinear Backlund transformation.