The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3

Abstract

In this article we consider the defocusing nonlinear Schrödinger equation, with time-dependent potential, in space dimensions $n=1,2$ and $3$, with nonlinearity $|u{|}^{p-1}u$, $p$ an odd integer, satisfying $p\ge 5$ in dimension $1$, $p\ge 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k\ge 2$ under the action of certain vector fields generating symmetries of the free Schrödinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.