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

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 an orbital instability of the standing waves $e^{i\omega t}\mathbf{\Phi }(x)$ of NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $\Phi (x)$ 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 orbital stability of the unique standing wave solution of NLS-$\delta$ equation with repulsive nonlinearity.

28 pages. arXiv admin note: text overlap with arXiv:1507.02312

References in corpus (2)

• Spectral stability of shifted states on star graphs
• Orbital instability of standing waves for NLS equation on Star Graphs

Cited by in corpus (12)

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• Blow-up and strong instability of standing waves for the NLS-$\delta$ equation on a star graph
• Dynamical and variational properties of the NLS-${\delta }_s^{\prime }$ equation on the star graph
• Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation
• Unstable kink-soliton profiles for the sine-Gordon equation on a $\mathcal{Y}$-junction graph with $\delta$-interaction
• Orbital instability of standing waves for NLS equation on Star Graphs
• Variational and stability properties of coupled NLS equations on the star graph
• On stability properties of the Cubic-Quintic Schrodinger equation with delta-point interaction
• Controllability for Schrödinger type system with mixed dispersion on compact star graphs