An improved local well-posedness result for the one-dimensional Zakharov system

AbstractThe 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schrödinger data u0∈Hk,pˆ and wave data (n0,n1)∈Hl,pˆ×Hl−1,pˆ under certain assumptions on the parameters k,l and 1<p⩽2, where ‖u0‖Hk,pˆ:=‖〈ξ〉ku0ˆ‖Lp′, generalizing the results for p=2 by Ginibre, Tsutsumi and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<−1/2, i.e. data u0∉L2 and (n0,n1)∉H−1/2×H−3/2, which was excluded in the case p=2