On the orbital instability of excited states for the NLS equation with the <i>δ</i>-interaction on a star graph

Abstract

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.

On the standing waves of the NLS-log equation with a point interaction on a star graph

“…See the proof of Lemma 2.3 in [6] (or Theorem 3.4 in [5]) for the detailed explanation of this fact.…”

On stability properties of the Cubic-Quintic Schródinger equation with <inline-formula><tex-math id="M1">\begin{document}$\delta$\end{document}</tex-math></inline-formula>-point interaction

“…The key point of this method is to use that the mapping datasolution associated to model (1) is at least of class C 2 (see Theorem 2.1 above). We note that the results in [35] have been applied successfully in the case of Schrödinger models on start graphs [10]- [11] and in [14]- [15] for models of KdV-type.…”

“…The case λ 2 = 0 and Z = 0 has been also studied substantially in the literature (see [3,8,12,16,20,22,23], [27,28,32,36,37] and references therein). The case of Schrödinger models on star graphs with δ conditions on the vertex also have been studied recently in Adami et.al ( [1,2] and references therein) and An-gulo&Goloshchapova ( [10,11,12]). The case of either other defect type or nonlinearity have been studied in [3,9,12,13,43].…”

“…Proof. The proof of local existence of solutions can be obtained via the abstract result in Theorem 3.7.1 in [21] and from the properties established in (10). Here by convenience, we will use standard arguments of Banach's fixed point theorem.…”

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

“…In the last years, the study of nonlinear dispersive models in a metric graph has attracted a lot of attention of mathematicians, physicists, chemists and engineers, see for details [9,10,14,34,35] and references therein. In particular, the framework prototype (graphgeometry) for description of these phenomena have been a star graph G, namely, on metric graphs with N half-lines of the form (0, +∞) connecting at a common vertex ν = 0, together with a nonlinear equation suitably defined on the edges such as the nonlinear Schrödinger equation (see Adami et al [1,2] and Angulo and Goloshchapova [5,6]). We note that with the introduction of nonlinearities in the dispersive models, the network provides a nice field, where one can look for interesting soliton propagation and nonlinear dynamics in general.…”