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Link between solitary waves and projective Riccati equations. (English) Zbl 0782.35065
Summary: Many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, we deduce a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. We handle several examples. For the Hénon-Heiles Hamiltonian system, we obtain several exact solutions; one of them defines a new solitary wave solution for a coupled system of Boussinesq and nonlinear Schrödinger equations. For a third order dispersive equation with two monomial nonlinearities, we isolate all cases where the general solution is single valued.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
37J35Completely integrable systems, topological structure of phase space, integration methods
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies