Exact Solutions To Some Nonlinear Differential Equations In Mathematical Physics

With the development of computer algebraic theory, many physicists, mathematicians and computer scientists are attracted to study the construction of exact solutions for nonlinear mathematical physics equations. Finding exact solutions for nonlinear equations is one of main aims in nonlinear science. Since there isn't united method for solving nonlinear differential equations, some special methods are proposed in recent fifty years. With the helps of symbolic softwear like Mathematica and Maple, the construction of exact solutions to some nonlinear differential equations is mechanized. Some software packages have been finished. But most of these special methods miss strict foundations in theory, and have limitations in practice. At the same time, some new methods need to be applied to more equations. In this paper, these problems are studied systematically. For solving some difficult algebraic equations systems, symbolic software Mathematica is applied to them. This is also an important application of computer technology to mathematical physics. First, projective Riccati equation method is discussed in detail. According to the scheme of every projective Riccati equation method, a united scheme is given. Furthermore, mathematical foundation of this scheme is studied, and its limitations in practice are pointed out. Through several theorems, the fact that projective Riccati equation method isn't suitable to some nonlinear differential equations with nonhomogeouns rank is showed. As application, BBM-Burgers equation is solved. Second, using the method of complete discrimination system for polynomial, a number of exact traveling wave solutions to five nonlinear mathematical physics equations are obtained, which includ compound KdV equation with any order nonlinear term, nonlinear coupled scall field equation, 2+1 dimemsional generalized Hirota equation, Maccari's equations system and 2+1 dimensional Bussionesq equation. Some solutions were new. At last, using Liu's trial equation method, many single traveling wave solutions to two nonlinear differential equations, i.e., 1+1 dimensional Cammasa-Holm equation and Jaulent-Miodek equation, are given. However, reducing directly these two equations to elementary integrals is difficult or impossible. By trial equation method, an integerable subequation was separated from the equation considered, and thereby the corresponding solutions are obtained. Amomg those solutions, some are new.