Asymptotic expansion of the solution to the nonlinear Schrodinger equation with nonlocal interaction

This paper deals with the equation iu(t) + (1/2) Deltau = lambda(\x\(-1) * \u\(2)) u, u(0, x) = u(o)(x). Here, u is it complex-valued function of (t,x) is an element ofR x R-n, n greater than or equal to2; and lambda is a real number. If u(o) is small in L-2,L-s with s> (n/2) + 2, then the solution u(t) behaves asymptotically as u(t, x) = (it)(-n/2) exp((t\x\(2)/2t) - i (S) over tilde (t, x\t)) [GRAPHICS] uniformly in R-n as t-->infinity. Here phi is a suitable function called the modified scattering state, and the functions (S) over tilde, psi (1,j), j = 0, 1, 2, are represented explicitly by using phi. (C) 2001 Academic Press.