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Zbl 0782.35065
Conte, R.; Musette, M.
Link between solitary waves and projective Riccati equations. (English)
[J] J. Phys. A, Math. Gen. 25, No.21, 5609-5623 (1992). ISSN 0305-4470

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 2000:: *35Q53 KdV-like equations
35Q51 Solitons
37J35 Completely integrable systems, etc.
37K10 Completely integrable systems etc.

Keywords: Hénon-Heiles Hamiltonian system; Boussinesq; nonlinear Schrödinger equations

Cited in: Zbl 1081.34003

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