Well-posedness and stability in the periodic case for the Benney system

Abstract

We establish local well-posedness results in weak periodic function spaces for the Cauchy problem of the Benney system. The Sobolev space H^(1/2) x L^2 is the lowest regularity attained and also we cover the energy space H^1 x L^2, where global well posedness follows from the conservation laws of the system. Moreover, we show the existence of a smooth explicit family of periodic travelling waves of dnoidal type and we prove, under certain conditions, that this family is orbitally stable in the energy space.