LINK BETWEEN SOLITARY WAVES AND PROJECTIVE RICCATI-EQUATIONS

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 Henon-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 Schrodinger equations. For a third order dispersive equation with two monomial nonlinearities, we isolate all cases where the general solution is single valued.