D. Noja and D.E. Pelinovsky
Standng waves of the quintic NLS equation on the tadpole graph
Calc. Var. PDE 59 (2020) 173 (31 pages)
Abstract:
The tadpole graph consists of a circle and a half-line attached at a vertex.
We analyze standing waves of the nonlinear Schrodinger equation with quintic power
nonlinearity closed with the Neumann-Kirchhoff boundary conditions. The profile of
the standing wave with the frequency ω is characterized as a global minimizer of the
constrained variational problem, which is equivalent to the minimization of the action
at the Nehari manifold. The set of minimizers includes the set of ground states of the
system, i.e. the global minimizers of the energy at constant mass (L2-norm), but it
is actually wider. While ground states exist only for a certain interval of masses, the
standing waves exist for every ω ∈ (-∞, 0) and correspond to a bigger interval of masses.
It is shown that there exists critical frequencies ω0 and ω1 such that the standing waves
are the ground states for ω ∈ [ω0, 0),
local minimizers of the energy at constant mass for ω ∈ (ω1, ω0),
and saddle points of the energy at constant mass for ω ∈ (-∞,ω1).
Keywords:
Nonlinear Schrodinger equation; tadpole graphs, standing wave solutions,
existence and stability, variational characterization, period function.