Elliptic general analytic solutions

TL;DR

This paper introduces an algorithm that analytically finds all elliptic and degenerate elliptic travelling wave solutions of certain nonlinear PDEs, surpassing traditional methods that only produce some solutions.

Contribution

The authors develop a novel algorithm based on classical mathematical results to obtain all elliptic and degenerate elliptic solutions in closed form.

Findings

Successfully applied to Kuramoto-Sivashinsky equation

Derived solutions for cubic and quintic Ginzburg-Landau equations

Provides a comprehensive solution set, unlike previous methods

Abstract

In order to find analytically the travelling waves of partially integrable autonomous nonlinear partial differential equations, many methods have been proposed over the ages: "projective Riccati method", "tanh-method", "exponential method", "Jacobi expansion method", "new ...", etc. The common default to all these "truncation methods" is to only provide some solutions, not all of them. By implementing three classical results of Briot, Bouquet and Poincare', we present an algorithm able to provide in closed form \textit{all} those travellingz waves which are elliptic or degenerate elliptic, i.e. rational in one exponential or rational. Our examples here include the Kuramoto-Sivashinsky equation and the cubic and quintic complex Ginzburg-Landau equations.