Final state problem for the cubic nonlinear Schr"odinger equation with repulsive delta potential

TL;DR

This paper studies the long-time behavior of solutions to the cubic nonlinear Schrödinger equation with a repulsive delta potential, proving the existence of solutions that asymptotically match a given profile using the distorted Fourier transform.

Contribution

It introduces a method to construct solutions to delta-NLS that converge to prescribed asymptotic profiles, utilizing the distorted Fourier transform for the delta potential case.

Findings

Existence of solutions converging to prescribed asymptotic profiles.

Application of distorted Fourier transform to delta potential problems.

Advancement in understanding long-time dynamics of delta-NLS.

Abstract

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schr"odinger equation with repulsive delta potential (delta-NLS). We shall prove that for a given asymptotic profile, there exists a solution to (delta-NLS) which converges to the given asymptotic profile as time goes infinity. To show this result we exploit the distorted Fourier transform associated to the Schr"odinger equation with delta potential.