Melnikov analysis for a singularly perturbed DSII equation

摘要

Rigorous Melnikov analysis is accomplished for Davey-Stewartson II equation under singular perturbation. Unstable fiber theorem and center-stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well posedness is necessary for such a Melnikov analysis. The main open issue regarding a proof of the existence of a homoclinic orbit to the perturbed Davey-Stewartson II equation is discussed in the Appendix.