Variational properties and orbital stability of standing waves for NLS equation on a star graph

摘要

We study standing waves for a nonlinear Schrodinger equation on a star graph G, i.e. N halflines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength alpha <= 0. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form i d/dt Psi(t) = H Psi(t) - |Psi(t)|(2 mu)Psi(t), where H is the Hamiltonian operator which generates the linear Schrodinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every omega > alpha(2)/N-2. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed omega if the nonlinearity is subcritical or critical, and for omega < omega* otherwise. (C) 2014 Elsevier Inc. All rights reserved.