Existence threshold for the ac-driven damped nonlinear Schrödinger solitons
Abstract
It has been known for some time that solitons of the externally driven, damped nonlinear Schrödinger equation can only exist if the driver's strength,
h
, exceeds approximately
(2/π)γ
, where
γ
is the dissipation coefficient. Although this perturbative result was expected to be correct only to the leading order in
γ
, recent studies have demonstrated that the formula
h
thr
=(2/π)γ
gives a remarkably accurate description of the soliton's existence threshold prompting suggestions that it is, in fact, exact. In this note we evaluate the next order in the expansion of
h
thr
(γ)
showing that the actual reason for this phenomenon is simply that the next-order coefficient is anomalously small:
h
thr
=(2/π)γ+0.002
γ
3
. Our approach is based on a singular perturbation expansion of the soliton near the turning point; it allows to evaluate
h
thr
(γ)
to all orders in
γ
and can be easily reformulated for other perturbed soliton equations.