Physics – Mathematical Physics
Scientific paper
2011-04-19
Physics
Mathematical Physics
Major changes, Th. 1 and Th. 2 revised, 4 pages, 2 figures
Scientific paper
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.
Adami Riccardo
Cacciapuoti Claudio
Finco Domenico
Noja Diego
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