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

Jaime Angulo Pava; César A. Hernández Melo

doi:10.3934/cpaa.2019094

Article Contents

2019, Volume 18, Issue 4: 2093-2116. Doi: 10.3934/cpaa.2019094

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

Jaime Angulo Pava^1, , and
César A. Hernández Melo^2,

1.
Department of Mathematics, IME-USP, Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil
2.
Department of Mathematics, DMA-UEM, Av. Colombo, 5790 Jd. Universitário, CEP 87020-900, Maringá, PR, Brazil

* Corresponding author
* Corresponding author

Received: May 2018

Revised: August 2018

Published: January 2019

The first author is supported by Grant CNPq/Brazil.

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Abstract

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.

Keywords:

Nonlinear stability,
standing wave solutions,
nonlinear Schródinger equation with point interaction,
analytic perturbation theory,
self-adjoint extensions.

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

Citation:

Full Text(HTML)

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

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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 $

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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) $

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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) $

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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 $

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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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