Elliptic general analytic solutions
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 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.