Symbolic Computation On The Soliton Solution And Soliton Dynamics Of The Nonlinear Equations

In the past decades, nonlinear evolution equations (NLEEs) have played roles in various fields, such as fluid dynamics, plasma physics, field theory and condensed matter physics. Soltion has attracted more and more attention because of its physical characteristics. How to obtain the soliton solutions of the NLEEs becomes a problem. Up to now, there are kinds of methods to obtain the soliton solutions of the NLEEs, such as the homogeneous balance method, Darboux transformation method, Backlund transformation method and Hirota method. Among them, Hirota’s is a straightforward and analytic approach to deal with the NLEEs.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 Hirota method. In chapter two, we investigate the (2+1)-dimensional dispersive long wave system. With the generalized Bell polynomials, its bilinear form is derived. New analytic solutions are obtained via symbolic computation, based on which, the propagation of the water waves are analyzed graphically and the different phenomena of fission and fusion are revealed. Additionally, the bilinear auto-Backlund transformation is obtained by the generalized Bell polynomials.In chapter three, we investigate the quasi-one-dimensional Gross-Pitaevskii equation. Via the Horita method and symbolic computation, analytic bright N -soliton solution is obtained. Soliton dynamics and interaction is analyzed graphically.In chapter four, we investigate soliton dynamics of the Bose-Einstein condensates (BECs) with time-varying control parameters. Via the Hirota method and symbolic computation, the bright N -soliton solution is obtained. Through the analysis on the solution, influence of the time-varying control parameters on the physical quantities of the solitons in the BECs is studied. Properties of the soliton interactions are revealed based on the two- and three-soliton solutions.In chapter five, we investigate the (2+1)-dimensional Kaup system are analyzed via symbolic computation and Hirota’s bilinear method. By the truncating the Painleve expansion, the transformation for the bilinear form is obtained, with which the bilinear form is derived. Via the bilinear form, soliton solutions are obtained. Dynamic behaviors of those solitons are analyzed graphically and the inelastic interactions including soliton fission and fusion are obtained. Finally, the bilinear auto-Backlund transformation is constructed.In chapter six, we investigate the (2+1)-dimensional nonlinear long wave equation of the Boussinesq class. Its bilinear form is derived by virtue of the generalized binary Bell polynomials. Via symbolic computation, the analytic N -soliton solution is obtained. Based on those solutions, the properties of the (2+1)-dimensional long waves are obtained.Finally are our conclusions.