Gravity solitary waves with polynomial decay to exponentially small ripples at infinity.
by
Lombardi Eric
Institut de Mathematiques de Toulouse
Coauthors: Gerard Iooss (Universite de Nice)

In this paper, we study the travelling gravity waves of in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness . We prove the existence of periodic travelling waves of arbitrary small amplitude and the existence of generalized solitary waves with exponentially small ripples at infinity and with polynomial decay rate.

The proof is based on a spatial dynamical formulation of the problem which admits a bifurcation which cannot be reduced to finite dimensions using the standard center manifold reduction since in this case the whole real line is embedded in the continuous spectrum of the linear operator. To study such a bifurcation, we proved a normal form lemma in infinite dimensions which ensures that full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillator. The existence of a family of homoclinic connections to these periodic orbits is then obtained by the study of the analytic continuation of the solutions in the complex field which enables one to obtain exponentially small upper bounds of the oscillatory integrals giving the size of the oscillations at infinity.

Date received: May 30, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caum-66.