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 distributionat 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 perturbationmethod and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in Hper1([−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 Hper1([−π, π]) and unstable in Hper1([−π, π]) with respect to perturbations which have the same space-periodic as the wave itself.