On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph

Jaime Angulo Pava; Nataliia Goloshchapova

doi:10.3934/dcds.2018221

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2018, Volume 38, Issue 10: 5039-5066. Doi: 10.3934/dcds.2018221

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On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph

Jaime Angulo Pava and
Nataliia Goloshchapova^,

Rua do Matão, 1010, Cidade Universitária, São Paulo - SP, 05508-090, Brazil

* Corresponding author: nataliia@ime.usp.br
* Corresponding author: nataliia@ime.usp.br

Received: October 31, 2017

Revised: May 31, 2018

Published: September 2018

J. Angulo was supported partially by Grant CNPq/Brazil. N. Goloshchapova was supported by FAPESP under the project 2016/02060-9.

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Abstract

We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate the orbital instability of the standing waves $e^{i\omega t}{\bf \Phi}(x)$ of the NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile ${\bf \Phi}(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove the orbital stability of the unique standing wave solution to the NLS-$\delta$ equation with repulsive nonlinearity.

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Mathematics Subject Classification: Primary: 35Q55, 81Q35, 37K40, 37K45; Secondary: 47E05.

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