The Hirota Method On Soliton Solutions For The Variable Coefficient Nonlinear Evolution Equations

The widely application in nonlinear science has become a research upsurge to solve the nonlinear evolution equations. Because of the constant coefficient just describes a kind of ideal and approximation condition, the variable coefficients have attracted much attention. Meanwhile, the external forces or the variable coefficients which are affected by space and time can better reflect the real environment and interaction mechanism. But they makes more difficult study the nonlinear problems. Until now, there is few papers about it. To find the soliton solutions of the equations, which are affected by space and time or the external forces, is the paper’s purpose. The integrability of the high-dimensional or higher order variable coefficient nonlinear evolution equation need to be examined through the Painleve analysis; On this basis, the Hirota method can be used to make in-depth exploration and study in the variable coefficient systems with different geographical environment.In Chapter1. we first introduce the significance of the variable-coefficient nonlinear evolution systems and the development of the soliton. Then we explain many methods of soliton theory, and the Hirota method and Painleve analysis are on the point.In Chapter2, the (2+1)-dimensional variable-coefficient Zakharov-Kuznetsov equation in plasmas with different geographical environment is investigated. The integrability of this model is examined through the Painleve analysis. According to the Hirota method, the bilinear form of this equation is derived. Based on the obtained bilinear form, the soliton solution is constructed. Then the properties of soliton from different geographical environment are detailed graphically.Chapter3investigates the variable-coefficient fKdV equation with different external force According to the Hirota method, the bilinear form of this equation and the constraint conditions are derived. Based on the obtained bilinear form, the soliton solution is constructed. Moreover, the simulations of the soliton solutions of this equation are carried out.Chapter4summarizes the research results and the innovation points of the study. Firstly, with the development of the Painleve analysis and Hirota method, we can obtain the integrable condition of the high-dimensional nonlinear system, which the variable coefficient of the linear term is affected by time and space. Secondly, under the research results of the constant coefficient fKdV equation, the Hirota method is developed, getting the soliton solution of the higher order variable coefficient fKdV system with forcing term. Thirdly, the dynamic properties of soliton collisions of the high-dimensional or higher order variable coefficient nonlinear system are illustrated. Finally, looks forward to the research prospects of the variable coefficient nonlinear evolution equations.