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Elliptic general analytic solutions. (English) Zbl 1185.35221
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”, 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 they 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. The examples in the paper are based on the Kuramoto-Sivashinsky equation and the cubic and quintic complex Ginzburg-Landau equations.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q56 Ginzburg-Landau equations
35C07 Traveling wave solutions
Software:
algcurves
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References:
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