On stability properties of the Cubic-Quintic Schrodinger equation with delta-point interaction

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrodinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.