Constrained energy minimization and orbital stability for the NLS equation on a star graph

Abstract

Abstract. We consider a nonlinear Schrödinger equation with focusing nonlinearity of power type on a star graph G, written as i∂ t Ψ(t) = HΨ(t) − |Ψ(t)| 2µ Ψ(t) , where H is the selfadjoint operator which defines the linear dynamics on the graph with an attractive δ interaction, with strength α < 0, at the vertex. The mass and energy functionals are conserved by the flow. We show that for 0 < µ < 2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m * it attains its minimum value at a certainΨ m ∈ H 1 (G), while for m > m * there is no minimum. Moreover, the set of minimizers has the structure M = {e iθΨ m , θ ∈ R}. Correspondingly, for every m < m * there exists a unique ω = ω(m) such that the standing waveΨ ω e iωt is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is non trivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to α = 0.

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

“…2(b). The ground states of nonlinear star graphs and their stability (with respect to the time-dependent NLSE dynamics) have been the subject of recent research [4][5][6][7][8].…”

Non-Kirchhoff Vertices and Nonlinear Schrödinger Ground States on Graphs

“…Existence and non-existence results are obtained taking advantage of this equivalence and the interplay between pointwise and standard nonlinearities becomes evident. In particular, if q < p 2 + 1, then existence of ground states holds for small masses and does not hold for large masses and this behaviour is similar to the one described in [4] in the case of a linear delta interaction at the origin: this suggests that when the nonlinear delta interaction is not too strong, then it has qualitatively the same effect of the linear delta on existence results. If instead q > p 2 , then ground states exist for large masses and do not exist for small masses.…”

“…In the following, we will refer to global minimizers of (7) or (8) as ground states, regardless of the functional they minimize. The difference between the two approaches reflects on the parameter ω ∈ R in the equation (4). In fact, it can be an unknown of the problem and be interpreted as a Lagrange multiplier like in the former approach or it can be given, like in the latter.…”

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

“…The first results in this direction (e.g., [1,2,3,4]) considered the so-called infinite N -star graph (see, e.g., Figure 2), in the case where −∆ G is the Laplacian with δ-type vertex conditions, that is (−∆ δ,α G u) |Ie := −u e ∀e ∈ E, ∀u ∈ dom(−∆ δ,α G ), dom(−∆ δ,α G ) := u ∈ H 2 (G) : u satisfies (3) and (4) , where u e 1 (v) = u e 2 (v),…”