Elliptic General Analytic Solutions

Abstract

To find analytically the traveling 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 that they provide only some solutions, not all of them. By implementing three classical results of Briot, Bouquet, and Poincaré, we present an algorithm able to provide in closed form all those traveling waves that 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.