On the Cauchy problem for the Zakharov system

We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data (u(0), n(0), partial derivative,n(0)) is an element of H-k + H-l + Hl-1, in arbitrary space dimension nu. We define a natural notion of criticality according to which the critical values of (k, l) are (nu/2 -3/2, nu/2-2), Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of (k, l). The results cover the whole subcritical range for nu greater than or equal to 4. For nu less than or equal to 3, they cover only part of it and the lowest admissible values are (k, l) = (1/2, 0) for nu = 2, 3 and (k, l) = (0, -1/2) for nu = 1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of (k, l). (C) 1997 Academic Press.