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October 2018, 38(10): 5039-5066. doi: 10.3934/dcds.2018221
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
Received November 2017 Revised June 2018 Published July 2018
Fund Project: 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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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.
Keywords: Nonlinear Schrödinger equation, point interaction, self-adjoint extension, deficiency indices, orbital stability, standing wave, star graph, power nonlinearity, analytic perturbation.
Mathematics Subject Classification: Primary: 35Q55, 81Q35, 37K40, 37K45; Secondary: 47E05.
Citation: Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
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Jaime Angulo Pava Nataliia Goloshchapova