ON THE GENERALIZED BENJAMIN-ONO-EQUATION

We study well-posedness of the initial value problem for the generaliz ed Benjamin-Ono equation partial derivative(t)u + u(k)partial derivati ve(x)u - partial derivative(x)D(x)u = 0, k is-an-element-of Z+, in Sob olev spaces H(s)(R). For small data and higher nonlinearities (k great er-than-or-equal-to 2) new local and global (including scattering) res ults are established. Our method of proof is quite general. It combine s several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Be njamin-Ono to nonlinear Schrodinger equations (or systems) of the form partial derivative(t)u - i partial derivative(x)2u + p(u, partial der ivative(x)u, uBAR, partial derivative(x)uBAR) = 0.