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

^(a) riccardo.adami@polito.it
^(b) cacciapuoti@him.uni-bonn.de
^(c) d.finco@uninettunouniversity.net
^(d) diego.noja@unimib.it

Received: 26 July 2012
Accepted: 7 September 2012

Abstract

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.