SMALL SOLUTIONS TO NONLINEAR SCHRODINGER-EQUATIONS

It is shown that the initial value problem for the nonlinear Schroding er equations partial derivative(t)u = iDELTAu + P(u, del(x), u, uBAR, del(x) uBAR), t is-an-element-of R, x is-an-element-of R(n), where P(. ) is a polynomial having no constant or linear terms, is locally well posed for a class of ''small'' data u0. The main ingredients in the pr oof are new estimates describing the smoothing effect of Kato type for the group {e(itDELTA)}-infinity(infinity). This method extends to sys tems and other dispersive models.