Article

Stationary States of NLS on Star Graphs

04/2011;
Source: arXiv

ABSTRACT 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 strength \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, 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.

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8 Dec 2012

Keywords

\delta potential

attractive

free propagation

generalized nonlinear Schr\"odinger equation

ground state

minimizer

N edges

NLS

orbital stability

potential barrier

power nonlinearity \mu<2

propagation

ramified structure

repulsive

solitons

special case

star graph

stationary states

strength \alpha