SHARP WELL-POSEDNESS RESULTS FOR THE SCHRODINGER-BENJAMIN-ONO SYSTEM

This work is concerned with the Cauchy problem for a coupled Schrodinger-Benjamin-Ono system { i partial derivative(t)u + partial derivative(2)(x)u = alpha uv, t is an element of[-T,T], x is an element of R, partial derivative(t)u + nu H partial derivative(2)(x)v = beta partial derivative(x) (vertical bar u vertical bar(2)), u(0, x) = phi, v(0, x) = psi, (phi, psi) is an element of H-s'(R). In the non-resonant case (vertical bar nu vertical bar not equal 1), we prove local well-posedness for a large class of initial data. This improves the results obtained by Bekiranov, Ogawa and Ponce (1998). Moreover, we prove C-2-ill-posedness at low-regularity, and also when the difference of regularity between the initial data is large enough. As far as we know, this last ill-posedness result is the first of this kind for a nonlinear dispersive system. Finally, we also prove that the local well-posedness result obtained by Pecher (2006) in the resonant case (vertical bar nu vertical bar = 1) is sharp except for the end-point.