On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

Abstract

We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schrödinger 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. 2000 Mathematics Subject Classification. Primary: 35Q55, 37K40, 37K45; Secondary: 47E05, 35J61. JAIME ANGULO PAVA AND CÉSAR A. HERNÁNDEZ MELO ON STABILITY OF THE CUBIC-QUINTIC SCHRÖDINGER WITH δ 2095 φ ω = φ ω,Z,λ1,λ2 for α = λ1 4 and β = λ2 2096 JAIME ANGULO PAVA AND CÉSAR A. HERNÁNDEZ MELO