We study the nonlinear Schrödinger equation (NLS) on a star graph G. At the vertex an interaction occurs described by a boundary condition of delta type with strength α ∈ R. We investigate the orbital instability of the standing waves e iωt Φ(x) of the NLS-δ equation with attractive power nonlinearity on G when the profile Φ(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-δ equation with repulsive nonlinearity.2010 Mathematics Subject Classification. Primary: 35Q55, 81Q35, 37K40, 37K45; Secondary: 47E05.
We study strong instability (by blow-up) of the standing waves for the nonlinear Schrödinger equation with δ-interaction on a star graph Γ. 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 δ-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-δ ′ equation on the line.
Abstract. We study analytically the orbital stability of the standing waves with a peak-Gausson profile for a nonlinear logarithmic Schrö-dinger equation with δ-interaction (attractive and repulsive). A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing wave. This is overcome by the perturbation method, the continuation arguments, and the theory of extensions of symmetric operators.Mathematics Subject Classification (2010). Primary 35Q51, 35J61; Secondary 47E05.
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