Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA

Abstract

The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized. The class of solitary wave solutions is extended to analytic (rather than rational) functions of the real exponential solutions of the linearized equation. Expanding the solutions in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed. Methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed. The algorithm is now suited to solving nonlinear equations by any symbolic manipulation program. This direct method is illustrated by constructing exact solutions of a generalized KdV equation, the Kuramoto-Sivashinski equation and a generalized Fisher equation.