Symbolic Computation On The Darboux Transformation And Soluiton Solutions For The WBK Equation And Ralated Equations

The soliton theory as an important branch of applied mathematics and mathematical physics is an important part of the nonlinear science. One of the main topics of the soliton theory is to search for the soliton solutions of nonlinear evolution equations (NLEEs). Up to now, there are many kinds of methods to obtain the soliton solutions of the NLEEs, such as homogeneous balance method, Backlund transformation method, Hirota method and Darboux transformation method. Among the various approaches, the Darboux transformation is a very powerful tool for constructing soliton solutions of the NLEEs from a trivial seed.In the first chapter of this dissertation, we introduce the history and development of the soliton theory. Then by means of several examples we explain some methods of contructing the solion solutions of the NLEEs, especially focus on the Darboux transformation method.In chapter two, we derive the Lax pair for the Whitham-Broer-Kaup (WBK) equation, and construct the Darboux transformation. By applying the obtained Darboux transformation, some new soliton solutions together with their physical structures are presented for such a model.In chapter three, we investigate the variable-coefficient WBK equation. First, the Painleve analysis method is used to find the restriction of the coefficients for the system to be completely integrable. Under the constraint conditions, the Lax pair of the system is derived, and then the Darboux transformation is constructed. At last, new soliton solutions of different physical structures are obtained by using the Darboux transformation method.In chapter four, a new (2+1)-dimensional variant Boussinesq system with its spectral problems is presented in this dissertation, which has a close connection with the WBK soliton hierarchy. Based on the associated spectral problems, the Darboux transformation is constructed with the help of symbolic computation. Then, by using the Darboux transformation, some new one- and two-soliton solutions of the (2+1)-dimensional variant Boussinesq system are obtained and graphically illustrated.Finally are conclusions and prospects about our work.