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 [GRAPHICS] where Delta = partial derivative(x1)(2) + partial derivative(x2)(2), c(1), c(2) is an element of R, u is a complex valued function and phi is a real valued function. When (c(1), c(2)) = (-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,H-0 boolean AND H-0,H-3, where H-m,H-l = {f is an element of L(2); parallel to(1 - partial derivative(x1)(2) - partial derivative(x2)(2))(m/2) (1 + x(1)(2) + x(2)(2))(1/2) f parallel to(L2) < infinity). Furthermore, we prove L(infinity) time decay estimates of solutions to the system such that parallel to u(t)parallel to L infinity less than or equal to C(1 +\t\)(-1).