Solitary and Periodic Solutions of Nonlinear Nonintegrable Equations

The singular manifold method and partial fraction decomposition allow one to find some special solutions of nonintegrable partial differential equations (PDE) in the form of solitary waves, traveling wave fronts and periodic pulse trains. The truncated Painlev'e expansion is used to reduce a nonlinear PDE to a multilinear form. Some special solutions of the latter equation represent solitary waves and traveling wave fronts of the original PDE. The partial fraction decomposition is used to obtain a periodic wave train solution as an infinite superposition of the "corrected" solitary waves. We would like to thank G. Ierley, J.Weiss and G.Whitham for useful discussions. 1. Introduction The existence of solitons and periodic wave trains is an important question in the study of nonlinear evolution equations. The methods of finding such solutions for integrable equations are well-known: the solitary solutions can be found by Hirota bilinear method [3,4] and the periodic solutions can be r..