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

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 $p \geq 5$ in dimension $1$, $p \geq 3$ 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 $k \geq 2$ 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.

💡 Research Summary

The paper addresses the large‑data final‑state problem for the defocusing nonlinear Schrödinger equation (NLS) in spatial dimensions (n=1,2,3). The equation studied is \