Solitary waves of nonlinear nonintegrable equations

TL;DR

This paper reviews methods for finding closed-form solitary wave solutions of nonlinear nonintegrable PDEs, classifying approaches into truncation and first-order ODE methods, and applies them to specific equations.

Contribution

It classifies nonperturbative methods for solitary wave solutions and applies these techniques to several important nonlinear equations.

Findings

Successful application to Ginzburg-Landau equations

Application to Kuramoto-Sivashinsky equation

Comparison of method effectiveness

Abstract

Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first class, which includes the well known so-called truncation methods, one \textit{a priori} assumes a given class of expressions (polynomials, etc) for the unknown solution; the involved work can easily be done by hand but all solutions outside the given class are surely missed. In the second class, instead of searching an expression for the solution, one builds an intermediate, equivalent information, namely the \textit{first order} autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the involved work requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.