New Solutions To Some Nonlinear Evolution Equations

During the past thirty years, nonlinear science is deeply studied and widely applied in natural sciences, such as biology, chemistry, mathematics, communication, condensed matters physics, field theory, low temperature physics, hydrodynamics, plasma physics, optics and so on. Therefore, a large number of nonlinear equations are arised in the area and questions are asked, such as how to solve these nonlinear partial differential equations, what kind of properties possess for these nonlinear equations, how to solve and classify these nonlinear equations, etc.In the first chapter, the developments of soliton theory and some methods for finding the exact solutions of nonlinear evolution equations are introduced.In the second chapter, based on a new ansatz and symbolic computation, a new extended Riccati equation rational expansion method is proposed. It is applied to the Ito system and Whitham-Broer-Kaup equations, and we get some new solutions of these (1+1)-dimensional NLEEs.In Chapter 3, the method is applied to the (2+1)-dimensional breaking s- oliton equations, (2+1)-dimensional Nizhnik equations and (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov system. Several new kinds of exact solutions of these (2+1)-dimensional NLEEs are found by this method.