The Non-Linear Schr\"odinger Equation with a periodic {\bf{$\delta$}}--interaction

TL;DR

This paper investigates the existence and stability of standing wave solutions for a periodic nonlinear Schr"odinger equation with a point defect modeled by a Dirac delta, revealing stability conditions depending on the defect's nature.

Contribution

It provides a detailed analysis of standing wave solutions with Jacobi elliptic profiles and characterizes their stability for both attractive and repulsive point defects.

Findings

Stable standing waves for attractive defects in $H^1_{per}$.

Stable in even subspace, unstable in full space for repulsive defects.

Existence of positive periodic solutions with elliptic function profiles.

Abstract

We study the existence and stability of the standing waves for the periodic cubic nonlinear Schr\"odinger equation with a point defect determined by a periodic Dirac distribution at the origin. This equation admits a smooth curve of positive periodic solutions in the form of standing waves with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbation method and continuation argument, we obtain that in the case of an attractive defect the standing wave solutions are stable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of $H^1_{per}$ and unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself.