Integrals Of Motion Of The Reduced Three-wave Interaction System, And Several Types Of Nonlinear Wave Equations Exact Solitary Wave Solutions

The paper includes two parts. In the first part, the integrability of differential equations is introduced. By using the Darboux method of integrability to some ordinary differential equations, under some given parameters conditions, a lot of integrals of motion have been found. Using the Painleve method of singularity analysis, two integras of motion of the reduced three-wave interaction system were found.The main result of this part is the following, by using the method of characteristic curves for solving linear partial differential equations, the whole classification of the integrals of motion of the reduced three-wave interaction system is obtained with the condition of some parameters. Rigorous proof is given. This part consists of three sections. In section 1, the introduction and statement of the main results are stated; in section 2, the main tool of this part using is narrated, that is, the method of characteristic curves for solving linear partial differential equations; in section 3, Rigorous proof is given. Rigid proof is presented.In the second part, the exact solitary wave solutions for some nonlinear wave equations is investigated. The Mathematics- Mechanization method is applied the field of differential equations. Many algorithm for constructing solitary wave solutions for a class of nonlinear wave equations are given, and implemented in a computer algebraic system, such as the hyperbolic tangent function method and the hyperbolic function method etc. Exact solitary wave solutions of a great deal of nonlinear equations are gained.The mostly conclusion of this part is as follows, on the conditon of travelling wave, the exact solitary wave solutions to some nonlinear wave equations such as Sawada-Kotera equation, Kaup-Kupershmidt equation, the fifth order KdV equation, Fisher-Kolmogorov equation, on the help of the computer algebraic system (Maple), are explicitly established by making use of the hyperbolic function method. This part is maken up of three sections. In section 1, some nonlinear wave equations of this part discussing are recommended; in section 2, the elementary tool of this part utilizing is mentioned, namely, the hyperbolic function method; in section 3, seme exact solitary wave solutions to these nonlinear wave equations are attained.