The Non-linear Schrödinger Equation with a Periodic Δ–interaction

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distribution at the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbation method and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in H per ([−π, π]) with respect to perturbations which have the same space-periodic 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 ([−π, π]) and unstable in H per ([−π, π]) with respect to perturbations which have the same space-periodic as the wave itself.