Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems (equation presented) where Δ = ∂²_x1 + ∂²_x2, c₁, C₂ ∈ R, u is a complex valued function and φ is a real valued function. When (c₁,c₂) = (-1,2) the system (*) is called DSI equation in the inverse scattering literature. Our purpose in this paper is to prove the local existence of a unique solution to (*) in the Sobolev space H²(R²) without the smallness condition on the data which were assumed in previous works [7], [17], [19], [26]. Our result is a positive answer to Question 7 in [24].