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

TL;DR

This paper investigates the blow-up and instability of standing waves in nonlinear Schrödinger equations with delta interactions on star graphs, introducing a new variational approach and analyzing well-posedness and instability results.

Contribution

It introduces a novel variational technique for analyzing standing wave stability in NLS-$\delta$ equations on star graphs and establishes well-posedness and instability results.

Findings

Proves strong instability (blow-up) of standing waves on star graphs.

Develops a new variational method for analyzing stability.

Establishes well-posedness and virial identities for the problem.

Abstract

We study strong instability (by blow-up) of the standing waves for the nonlinear Schr\"odinger 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-$\delta'$ equation on the line.