This article has an erratum: [erratum]
Volume 100, Number 1, October 2012
|Number of page(s)||6|
|Published online||17 October 2012|
Stationary states of NLS on star graphs
1 Dipartimento di Scienze Matematiche, Politecnico di Torino - C.so Duca degli Abruzzi 24, 10129 Torino, Italy, EU
2 Hausdorff Center for Mathematics, Institut für Angewandte Mathematik - Endenicher Allee, 60, 53115 Bonn, Germany, EU
3 Facoltà di Ingegneria, UTIU UniNettuno - Corso V. Emanuele II 39, 00186 Roma, Italy, EU
Received: 26 July 2012
Accepted: 7 September 2012
We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity |ψ|2μψ of focusing type describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the δ potential of strength α on the line, including as a special case (α = 0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (α < 0, representing a potential well) and repulsive (α > 0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity μ < 2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.
PACS: 05.45.Yv – Solitons / 03.75.Kk – Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow / 42.65.Tg – Optical solitons; nonlinear guided waves
© EPLA, 2012
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