Study On The Automated Derivation Of Several Kinds Of Wave Solutions For Nonlinear Evolution Equations

The study of many problems in natural science and engineering technologies can be derived as the problem of solving nonlinear evolution equations ultimately. Nonlinear evolution equations can describe related things or phenomena reasonably. Therefore, constructing their solutions is very important for the study of nonlinear problems. Nowadays, constructing the solutions of nonlinear evolution equations needs massive computation, but simple handwritten calculation cannot satisfy the increasing com-puting requirements. With the rapid development of computer technologies, people began to research computer method based on symbol. Solving nonlinear evolution e-quations using the Computer Algebra Systems promoted the development of nonlinear science greatly. In this paper, we took the nonlinear evolution equation as the research object, improved and generalized the simplified Hirota method. Based on the symbolic computation system Maple, we developed the software package ZASP, which can solve nonlinear evolution equations automatically. The work we did can be listed as follows:1. The Hirota’s bilinear method is a direct algebraic way to obtain wave solutions of nonlinear evolution equations. Based on the work of former study, this thesis re-viewed ideas and steps of the method for solving nonlinear evolution equations. The assumptions of the auxiliary function in the simplified Hirota method is obtained. These work provides the basis for proposing a new algorithm for constructing wave solutions of nonlinear evolution equations.2. The simplified Hirota method was given on the basis of Hirota’s bilinear method. The method bypassed the bilinearization of nonlinear evolution equation. The solitary wave or soliton solution of the specific type of the original equation is con-structed by assuming the auxiliary function. However, the simplified Hirota method cannot solve every equation which can be solved by the traditional Hirota’s bilinear method. For example, the sine-Gordon equation that contains trigonometric function and the Schrodinger equation transformation whose transformation contains complex function. In this paper, the solution process of the simplified Hirota method is gen- eralized, and a new algorithm for solving nonlinear evolution equations is proposed based on the Hirota’s method and the generalized simplified Hirota method.3. We developed a software package, namely ZASP, in symbolic computation system Maple based on the new algorithm. ZASP can deduce wave solutions of single nonlinear evolution equations and nonlinear coupled equations automatically.