Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity

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