Solitary waves of nonlinear nonintegrable equations

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 non-perturbative, are divided in two classes. In the first class, which includes the well-known so-called truncation methods, one a priori assumes a given class of expressions (polynomials, etc.) for the unknown solution; the work involved can easily be done by hand but all solutions outside the given class are certainly missed. In the second class, instead of seeking an expression for the solution, one builds an intermediate equation with equivalent information, namely the first-order autonomous ODE satisfied by the solitary wave; in principle, no solution can be missed, but the work involved requires computer algebra. We present the application to the cubic and quintic complex one-dimensional Ginzburg-Landau equations, and to the Kuramoto-Sivashinsky equation.