Global existence and asymptotic behaviour in time of small solutions to the elliptic-hyperbolic Davey-Stewartson system

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson system {i∂_tu + △u = c₁\u\²u + c₂u∂_xiφ (t, x) ∈ ℝ³ (∂_x1² - ∂_x2²)φ = ∂_x1|u|² u = u(t, x) φ = φ(t, x) u(0, x) = φ(x) where △ = ∂_x1² + ∂_x2², c₁, c₂ ∈ ℝ, u is a complex valued function and φ is a real valued function. When (c₁,c₂) = (-1, 2) the above system is called a DSI equation in the inverse scattering literature. Our purpose in this paper is to prove global existence of small solutions to this system in the usual weighted Sobolev space H^3,0 ∩ H^0,3, where H^m,l = {f ∈ L²; ∥(1 - ∂_x1² - ∂_x2²)^m/2(1 + x₁² + x₂²)^l/2f∥_L2 < ∞}. Furthermore, we prove L∞ time decay estimates of solutions to the system such that ∥u(t)∥_L∞ ≤ C(1 + |t|)^-1.