We define a nonlinear Schr\"odinger equation (NLS) with a power nonlinearity $|\psi|^{2\mu}\psi$ of focusing type on the ramified structure given by $N$ edges connected at a vertex with boundary conditions generalizing 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 exist both for attractive ($\alpha<0)$ and repulsive ($\alpha>0)$ interaction and we give explicitly their expression. In the case of attractive interaction at the vertex and nonlinearity $\mu<2$ we characterize the ground state as minimizer of a constrained action and we rigorously 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.
Stationary States of NLS on Star Graphs / Adami, Riccardo; Cacciapuoti, C.; Finco, D.; Noja, D.. - In: EUROPHYSICS LETTERS. - ISSN 0295-5075. - 100:(2012). [10.1209/0295~5075/100/10003]
Stationary States of NLS on Star Graphs
ADAMI, RICCARDO;
2012
Abstract
We define a nonlinear Schr\"odinger equation (NLS) with a power nonlinearity $|\psi|^{2\mu}\psi$ of focusing type on the ramified structure given by $N$ edges connected at a vertex with boundary conditions generalizing 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 exist both for attractive ($\alpha<0)$ and repulsive ($\alpha>0)$ interaction and we give explicitly their expression. In the case of attractive interaction at the vertex and nonlinearity $\mu<2$ we characterize the ground state as minimizer of a constrained action and we rigorously 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.Pubblicazioni consigliate
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https://hdl.handle.net/11583/2503429
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