Study On Exact Solution And Integrability Of Nonlinear Evolution Equation With Symbolic Computation

Nonlinear evolution equation is an important mathematical model for describing physical phenomenon and an important field in the contempary study of nonlinear physics, especially in the study of soliton theory. The research on the explicit solution and integrability are helpful in clarifying the movement of matter under the nonlinear interactiveties and play an important role in scientifically explaining of the corresponding physical phenomenon and engineering application. Many research topics, such as searching for exact explicit solutions, multi-soliton solution, the Painleve test et al., often involve a large amount of tedious algebra auxiliary reasoning or calculations which can become unmanageable in practice. In recent years, the development of symbolic computation accelerates the research of nonlinear evolution equation greatly. Many new methods for constructing exact solutions of nonlinear evolution equations are proposed. This dissertation mainly stduies some aspects of nonlinear evolution equations with the aid symbolic computation, which include searching exact explicit solutions of nonlinear evolution equations by means of several direct algebraic methods proposed in recent years, the Painleve analysis and its application, the interraltions between the algebraic methods with the Painleve analysis. This dissertation consists of the following three parts.Part I is devoted to study the explicit solutions for nonlinear evolution equations by some algebraic methods proposed in recent years.Mixing exponential method proposed by Hereman for finding the solitary wave solutions to a nonlinear evolution equation is developed and perfected. Correspondingly, an extended mixing exponential method is obtained by expressing the solutions as an infite series of the real or complex exponential solutions of the underlying linear equations and improving the solving of recursion relations. The effectiveness of the extended approach is demonstrated by application to some nonlinear evolution equations with physical interest. Not only are steady solitary wave solutions recovered, but also the diverging and the periodic solutions are obtained.Based on Riccati method, deformation mapping method and unified algebraic method, a generalized deformation mapping method is presented for finding travelling wave solutions to nonlinear evolution equations. According to "rank", the nonlinear evolution equations are classified into two kinds of equations. The essence of this method is to take full advantage of two different solvable first-order ordinary differential equations, and convert the problem of finding travelling wave solutions for nonlinear PDE to the problem of solving nonlinear algebraic equations. The nonlinear system is solved by using Wu elimination and a series of travelling wave solutions are then obtained. Several illustrative equations such as coupled mKdV equations, coupled Drinfel'd-Sokolov-Wilson equations, variant Boussinesq sytem etc. are investigated by this metho. A series of travelling wave solutions are obtained in a systematic way, which covered exponential function solutions, polynomial function solutions, rational function solutions, trigonometric function solutions, Jacobi elliptic function solutions, Weierstrass elliptic function solutions, solitary wave solution, combined-form solitary wave solutions and so on. The generalized deformation mapping method is completely algorithmic, so it can be implemented in computer algebra.The homogeneous balance method for constructing solitary wave solutions and soliton solu-tions is further developed on obtaining quasi-solution by step-by-step principle. The main advantage of the extended approach is to avoid the problem of "intermediate expression swell". The effectiveness of the method is demonstrated by application to the generalized Boussinesq equation and the bidirectional Kaup-Kupershmidt equation. The one-soliton, two-soliton and three-soliton solutions with multiple collisions are derived for these two equations with the assistance of Maple. Both...