We prove the invariance of the Gibbs measure for the periodic Schrodinger-Benjamin-Ono system (when the coupling parameter |\gamma| \ne 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L^2 setting for the Benjamin-Ono part.) We also show the ill-posedness result in H^s(\mathbb{ T}) \times H^{ s-{ 1/2}}(\mathbb{ T}) for s < { 1/2} when |\gamma| \ne 0, 1 and for any s \in \mathbb{ R} when |\gamma| =1.
Categories: math.AP