Elliptic general analytic solutions
arXiv:0903.2009 · doi:10.1111/j.1467-9590.2009.00447.x
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.
17 pages, to appear, Studies in Applied Mathematics
References in corpus (4)
- Exact Solutions of the Saturable Discrete Nonlinear Schrodinger Equation
- Meromorphic solutions of a third order nonlinear differential equation
- On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation
- Doubly periodic waves of a discrete nonlinear Schrödinger system with saturable nonlinearity
Cited by in corpus (6)
- Meromorphic solutions of a third order nonlinear differential equation
- Exact meromorphic stationary solutions of the real cubic Swift-Hohenberg equation
- Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation
- New solutions to the complex Ginzburg-Landau equations
- Detection and construction of an elliptic solution to the complex cubic-quintic Ginzburg-Landau equation
- All meromorphic traveling waves of cubic and quintic complex Ginzburg-Landau equations