The Davey Stewartson system in weak \(L^p\) spaces. (Q1945766)

The main result of this paper is the existence and uniqueness of solutions for the elliptic-elliptic and hyperbolic-elliptic cases of the Davey-Stewartson system in weak \(L^p\) spaces with small initial data. The analysis is parallel to the one given in [\textit{T. Cazenave} et al., Commun. Contemp. Math. 3, No. 1, 153--162 (2001; Zbl 0991.35084)] for the non-linear Schrödinger equation with proper modifications to deal with the non-local term in the non-linearity. An application is given in the elliptic-elliptic case to the existence of self-similar solution.