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

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.