Blow-up and strong instability of standing waves for the NLS-$\delta$ equation on a star graph

Abstract

We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with $\delta$-interaction on a star graph $\Gamma$. The key ingredient is a novel variational technique applied to the standing wave solutions being minimizers of a specific variational problem. We also show well-posedness of the corresponding Cauchy problem in the domain of the self-adjoint operator which defines $\delta$-interaction. This permits to prove virial identity for the $H^1$- solutions to the Cauchy problem. We also prove certain strong instability results for the standing waves of the NLS-${\delta }^{\prime }$ equation on the line.

arXiv comment30 pages

References in corpus (6)

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Cited by in corpus (6)

• Standing waves on quantum graphs
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• On the nonlinear Dirac equation on noncompact metric graphs
• Blow-up and instability of standing waves for the NLS with a point interaction in dimension two
• Dynamical and variational properties of the NLS-${\delta }_s^{\prime }$ equation on the star graph
• Failure of scattering to standing waves for a Schrödinger equation with long-range nonlinearity on star graph