1104.3839 (Riccardo Adami et al.)

Stationary States of NLS on Star Graphs    [PDF]

Riccardo Adami, Claudio Cacciapuoti, Domenico Finco, Diego Noja

We consider a generalized nonlinear Schr\"odinger equation (NLS) with a power
nonlinearity $|\psi|^{2\mu}\psi$ 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 $\delta$ potential of strenght $\alpha$ on the line, including
as a special case ($\alpha=0$) the free propagation. We show that nonlinear
stationary states describing solitons sitting at the vertex exist both for
attractive ($\alpha<0$, representing a potential well) and repulsive
($\alpha>0$, a potential barrier) interaction. In the case of sufficiently
strong attractive interaction at the vertex and power nonlinearity $\mu<2$,
including the standard cubic case (Gross-Pitaevskii equation), 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.

View original: http://arxiv.org/abs/1104.3839