Shallow Water Wave Model Equation

The bilinear method, also called the direct method, was invented by Japanese mathematician Ryogo Hirota. It can be used to solve many nonlinear partial differential equations. The basic idea of this method is to transform a nonlinear partial differential equation into a bilinearized equation. The advantage of bilinear method is that the nonlinear partial differential equation discussed does not necessarily possess a Lax pair. So it can be used to study some non-integrable partial differential equations. But what kind of nonlinear partial differential equation can be bilinearized? What is the relationship between a nonlinear partial differential equation and its bilinearized equation? And are they equivalent or equivalent locally? Until now these fundamental questions have not been answered.The Backlund transformation was named after Swedish geometer Albert Victor Backlund. When studying surfaces of constant negative Gauss curvature in3-dimension Euclidean space, Backlund first discovered his celebrated transformation which allows the iterative construction of surfaces of constant negative Gauss curvature. The basic idea of Backlund transformation is that by solving two or more compatible ordinary differential equations, one can get solutions of a nonlinear partial differential equation. With the development of integrable system theory, Backlund transformation has become an important tool for solving some nonlinear partial differential equations. Many famous nonlinear partial differential equations, such as KdV equation and KP equation, admit Backlund transformations which can be used to generate soliton solutions and nonlinear superposition formulas.In this thesis, by taking the model equation for shallow water waves which was introduced by Hirota and Satsuma as an example, we study the above mentioned two problems. The solutions of the model equation for shallow water waves can be obtained from the corresponding bilinearized equation, which can be solved by the perturbation method. Conversely, the solutions of the bilinearized equation can be constructed from that of model equation for shallow water waves. Therefore model equation for shallow water waves and its bilinearized equation are locally equivalent. We give two examples to illustrate how to obtain solutions of bilinearized equation from a trivial solution of model equation for shallow water waves. In addition, we give two different Backlund transformations. One is a classical Backlund transformation for the model equation of shallow water waves. The other is a Backlund transformation from the model equation of shallow water waves to its bilinearized equation (we called it a new type of Backlund transformation), which is defined by a second order ordinary differential equation with its initial values satisfying some constrained condition.Our methods in this thesis are applicable to some other nonlinear partial differential equations.