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 $(2/\pi )\gamma$, where $\gamma$ is the dissipation coefficient. Although this perturbative result was expected to be correct only to the leading order in $\gamma$, recent studies have demonstrated that the formula $h_{thr}=(2/\pi )\gamma$ 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 $h_{thr}(\gamma )$ showing that the actual reason for this phenomenon is simply that the next-order coefficient is anomalously small: $h_{thr}=(2/\pi )\gamma +0.002{\gamma }^3$. Our approach is based on a singular perturbation expansion of the soliton near the turning point; it allows to evaluate $h_{thr}(\gamma )$ to all orders in $\gamma$ and can be easily reformulated for other perturbed soliton equations.

8 pages in RevTeX; 5 figures in ps format included in the text. To be published in Physica D

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