Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} .