Asymptotic Expansion of the Solution to the Nonlinear Schrِdinger Equation with Nonlocal Interaction

This paper deals with the equationiut+(1/2) Δu=λ(|x|−1*|u|2) u,u(0, x)=u0(x).Here, u is a complex-valued function of (t, x)∈R×Rn, n⩾2, and λ is a real number. If u0 is small in L2, s with s>(n/2)+2, then the solution u(t) behaves asymptotically asu(t, x)=(it)−n/2 exp((i |x|2/2t)−iS(t, x/t))×φ(x/t)+t−1 ∑j=02 ψ1, j(x/t)(log t)j+o(t−(n/2)−1)uniformly in Rn as t→∞. Here φ is a suitable function called the modified scattering state, and the functions S, ψ1, j, j=0, 1, 2, are represented explicitly by using φ.