Global Well-Posedness and Non-linear Stability of Periodic Travelling Waves Solutions for a Schrödinger-Benjamin-Ono System

The objective of this paper is two-fold: firstly, we develop a local and global (in time) wellposedness theory of 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 lowregularity initial data in both periodic and continuous cases; secondly, a family of new periodic travelling 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 of the Jacobian elliptic function called dnoidal, and, moreover, we prove that all these periodic travelling waves are nonlinearly stable by perturbations with the same wavelength.