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Orbital instability of standing waves for NLS equation on star graphs


Author: Adilbek Kairzhan
Journal: Proc. Amer. Math. Soc. 147 (2019), 2911-2924
MSC (2010): Primary 35Q55, 37K45; Secondary 35B35
DOI: https://doi.org/10.1090/proc/14463
Published electronically: March 15, 2019
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Abstract: We consider a nonlinear Schrödinger (NLS) equation with any positive power nonlinearity on a star graph $ \Gamma $ ($ N$ half-lines glued at the common vertex) with a $ \delta $ interaction at the vertex. The strength of the interaction is defined by a fixed value $ \alpha \in \mathbb{R}$. In the recent works of Adami et al., it was shown that for $ \alpha \neq 0$ the NLS equation on $ \Gamma $ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $ \alpha <0$ that in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable.

In this paper, we analyze stability of standing waves for both $ \alpha <0$ and $ \alpha >0$. By extending the Sturm theory to Schrödinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For $ \alpha <0$, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For $ \alpha >0$, we prove the orbital instability of all standing waves.


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Additional Information

Adilbek Kairzhan
Affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
Email: kairzhaa@math.mcmaster.ca

DOI: https://doi.org/10.1090/proc/14463
Keywords: Nonlinear Schr\"odinger equation, orbital instability, NLS on graphs, Sturm theory
Received by editor(s): January 25, 2018
Received by editor(s) in revised form: September 21, 2018
Published electronically: March 15, 2019
Communicated by: Catherine Sulem
Article copyright: © Copyright 2019 American Mathematical Society