Well-posedness of the periodic nonlinear Schr\"odinger equation with concentrated nonlinearity

TL;DR

This paper develops the first rigorous solution theory for the periodic nonlinear Schr"odinger equation with concentrated nonlinearity, establishing existence, uniqueness, and energy conservation for solutions in and below the energy space.

Contribution

It introduces novel approximation schemes to prove well-posedness of the equation with concentrated nonlinearity on the torus, including solutions below the energy space.

Findings

Existence of global energy-conserving solutions in H^1

Uniqueness of solutions in the energy space and below

First rigorous solution theory for this class of equations

Abstract

We study the solution theory of the nonlinear Schr\"odinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our approach is based on two approximation schemes, namely the concentrated limit of a smoothed nonlinear Schr\"odinger equation and the inviscid limit of a concentrated complex Ginzburg--Landau equation. We also prove the existence and uniquness of solutions below the energy space. To our knowledge, this is the first rigorous solution theory for a periodic nonlinear Schr\"odinger equation with a concentrated nonlinearity.