Variable-Coefficients Soliton Equation With Bell Polynomial Method

Soliton equations play an important role in many science and engineering fields. They can be used to describe the nonlinear phenomena in the field of plasma physics, optical fiber communication, solid mechanics and fluid mechanics, etc. Due to their extensive applications, Soliton receive public attention. By studying the analytical solution and graphics of the characteristics of dynamic mechanism reflected by soliton equations. Considering the inhomogeneity of the propagation medium and the uniformity of the boundary, the variable-coefficients soliton equation is considered more accurately to reflect the real phenomenon than the constant-coefficient soliton equation. In this article, first, the constant-coefficient soliton equation is studied with Bell polynomial method and bilinear method; then we promote and apply it to the variable-coefficient of soliton equation such as shallow water areas; finally, through the Bell polynomial method and bilinear method, we obtain soliton solution of high-dimensional and high-order soliton equation.In the first part of the paper, the soliton equation and the history of soliton equations are introduced, then several research methods of soliton equation are expounded, Bell polynomial method and bilinear method are on the point.In the second part, the generalized shallow water wave equation which changed with time and space is investigated. First of all, equation bilinear form is derived with the Bell polynomial method; Secondly, by bilinear method, the equation of single soliton, double solitons and N-soliton solution are obtained; finally, the simulation of transmission process is realized by computer symbols.In the third part, the generalized variable-coefficients shallow water wave equation is studied on the basis of the second part. The Bell polynomial method and bilinear method are applied to the variable-coefficients soliton equation, and the form of the soliton solutions is obtained and analyzed with the drawing function of computer.The fourth part is to explore, first of all, the soliton solution of (3+1)-dimensional Boussinesq equation are given by Bell polynomial method and bilinear method; Second, the pictures are drawn by computer symbol; finally, the interactions between single soliton, double soliton and three soliton are analyzed.In the last part of the article, the innovation and difficult points of this article are summarized: first, the bilinear form of the generalized shallow water equation is obtained by Bell polynomial method. Second, the solitons solution of the generalized variable-coefficients shallow water wave equation is given by the Bell polynomial method and bilinear method on the basis of the generalized shallow water wave equation, and the simulation environment is more close to reality environment. Finally, the Bell polynomial method is applied in the high-dimension, high-order soliton equation. The development prospect of the soliton equation is discussed.