Global Well-Posedness and Non-linear Stability of Periodic Traveling Waves for a Schrodinger-Benjamin-Ono System

Abstract

The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for \emph{low-regularity} initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrödinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called {\it dnoidal}, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.

38 pages; typos corrected and global well-posedness theorem (in the continuous case) reworked to follow closely the arguments of Colliander, Holmes and Tzirakis