Nonlinear Coupled Scalar Field Equations Solitary Wave Solutions Of Symbolic Computation

The nonlinear differential equation (NDE) based on physics is an important aspect in the contempary study of nonlinear science and exploring and developing new method to solve the NDE is one of the forefront topic in the studies of the nonlinear physics. The present paper intends to construct a systematic approach to search a type of particular exact solutions to nonlinear wave equations by utilizing the theory of mathematics mechanization proposed by famous mathematician Wu Wentsun. By the approach, two classes of important nonlinear coupled scalar field equations arising in the field of nonlinear physics are studied systematically and a batch of exact solutions containing steady and diverging solitary wave solutions and periodic ones are obtained, which are helpful in clarifying the movement of matter under the nonlinear interactiveties and play an important role in sicentifically explaining of the corresponding physical phenomenon.The basic principle of our algorithm is based on such observation that a majority of meaningful solitary wave solutions can be expressed as the polynomial forms in terms of "bell-shaped" function sech and "kink-shaped" function tanh which possessing localized property. Proceeding from a kind of Riccati equations that the above two functions satisfied, a direct algebra method-REQs for finding solitary wave solutions to nonlinear wave equations is proposed in this paper, which can not only obtain steady solitary wave solutions, diverging ones but also periodic solutions.The core of the mathematics mechanization theory is algebraization and further mechaiza-tion. The essence of the REQs method is to convert the problem of finding the exact solutions for noninear wave equations to the problem of solving large scale nonlinear algebra equations. The Ritt-Wu's method is a powerful tool for solving nonlinear algebra equations. For concrete situations, special tactics are taken to simplify the desired equation and then solve it by using the Ritt-Wu's method. With the aid of computer algebra system Maple, the nonlinear algebra equations corresponding to the nonlinear coupled scalar field equations can be successfully solved this way, and eventually many exact solutions to it can be obtained.