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On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction

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    * Corresponding author 
The first author is supported by Grant CNPq/Brazil.
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  • We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schródinger 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.

    Mathematics Subject Classification: Primary: 35Q55, 37K40, 37K45; Secondary: 47E05, 35J61.


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  • Figure 1.  The peak solutions $ \phi_{\omega, Z} $ for $ Z<0 $ and $ Z>0 $

    Figure 2.  Function $ \omega\to-||\phi_{\omega, Z}||^2 $ for specific values of $ Z $. Left picture, $ \omega\in(-6, -1) $, $ Z = -0.86 $. Right picture, $ \omega\in(-6, -1) $, $ Z = -0.9 $

    Figure 3.  Function $ (\omega, Z)\to-\partial_\omega||\phi_{\omega, Z}||^2 $, for $ \omega\in(-50, -2) $. Left picture for $ Z\in(-0.9, -0.8) $. Right picture for $ Z\in(-0.8, -0.7) $

    Figure 4.  Function $ \omega\to-\partial_\omega||\phi_{\omega, Z}||^2 $ for specific values of $ Z $. Left picture, $ Z = -0.86602 $ and $ \omega\in(-10, -0.8) $. Right picture, $ Z = -0.86603 $ and $ \omega\in(-5, -1) $

    Figure 5.  The function $ \omega\to-||\phi_{\omega, Z}||^2 $, for $ \lambda_1 = 2 $, $ \lambda_2 = -1 $ and $ \omega\in (-\frac34, -\frac14) $. Left picture, $ Z = -1 $. Right picture, $ Z = 1 $

    Figure 6.  The function $ \omega\to-||\phi_{\omega, Z}||^2 $, for $ \lambda_1 = 4 $, $ \lambda_2 = -2 $ and $ \omega\in (-\frac32, -\frac14) $. Left picture, $ Z = -1 $. Right picture, $ Z = 1 $

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