The Periodic Wave Solutions And Bell Polynomial Application For Soliton Equations

The solvability of equations is not only the focal point of studing the nonlinear partial differential equations but also a hot spot in soliton theory. This article focuses on three kinds of integrable equations:the forced variable-coefficient KdV equation; the variable-coefficient mKdV equation; and (2+1)-dimensional AKNS equation. For the three kinds of equations, we apply Hirota bilinear method and Riemann theta function method to study the soliton solutions and Riemann theta function periodic solutions. Then we consider the asymptotic analysis of the periodic solutions, the relations between the periodic solutions and the well-know soliton solutions are established. These solutions would help us recognize the interaction behaviors of the nonlinear wave. For (2+1)-dimensional AKNS equation, because bilinear forms is difficult to get, we need introduce a auxiliary variable. Based on Bell polynomial’s method, we derive the bilinear forms, and structure out its Backlund transform. Finally, via the Backlund transform, the soliton solutions and periodic solutions are present.In Chapter2We mainly study the soliton solutions and periodic solutions of the forced variable-coefficient KdV equations. Base on the Hirota bilinear method, the bi-linear form and the soliton solutions are present. Like Nakamure studying the periodic wave solutions of soliton equation, we also obtain the Riemann theta function periodic solutions. At last, the asymptotic properties of the periodic wave solutions are analyzed in detail.In Chapter3Because the bilinear form is a coupling, we use the extend Riemann theta function method to explicitly construct periodic solutions of the variable-coefficient mKdV equations. Also, we show that the solitions can be reduced form the periodic wave solutions.In Chapter4We consider the (2+1)-dimensional AKNS equation. First of all, based on the Bell polynomial, the bilinear form are obtained. But as bilinear form have a auxiliary variable form, obtaining the periodic solutions is difficult, so we must obtain the Backlund transform first, then we got the periodic solutions. As the same time, the asymptotic analysis of the periodic solutions are present.