The Non-Linear Schrödinger Equation with a periodic {\bf{$\delta$}}--interaction

Abstract

We study the existence and stability of the standing waves for the periodic cubic nonlinear Schrödinger 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 $H_{per}^1$ 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 $H_{per}^1$ and unstable in $H_{per}^1$ with respect to perturbations which have the same period as the wave itself.

arXiv comment56 pages