Existence threshold for the ac-driven damped nonlinear Schrödinger solitons
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.