dorsal/arxiv
View SchemaExistence threshold for the ac-driven damped nonlinear Schr\"odinger solitons
| Authors | I. V. Barashenkov, E. V. Zemlyanaya |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9906001 |
| URL | https://arxiv.org/abs/patt-sol/9906001 |
| DOI | 10.1016/S0167-2789(99)00055-X |
Abstract
It has been known for some time that solitons of the externally driven, damped nonlinear Schr\"odinger 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.
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"abstract": "It has been known for some time that solitons of the externally driven,\ndamped nonlinear Schr\\\"odinger equation can only exist if the driver\u0027s\nstrength, $h$, exceeds approximately $(2/ \\pi) \\gamma$, where $\\gamma$ is the\ndissipation coefficient. Although this perturbative result was expected to be\ncorrect only to the leading order in $\\gamma$, recent studies have demonstrated\nthat the formula $h_{thr}= (2 /\\pi) \\gamma$ gives a remarkably accurate\ndescription of the soliton\u0027s existence threshold prompting suggestions that it\nis, in fact, exact. In this note we evaluate the next order in the expansion of\n$h_{thr}(\\gamma)$ showing that the actual reason for this phenomenon is simply\nthat the next-order coefficient is anomalously small: $h_{thr}=(2/ \\pi) \\gamma\n+ 0.002 \\gamma^3$. Our approach is based on a singular perturbation expansion\nof the soliton near the turning point; it allows to evaluate $h_{thr}(\\gamma)$\nto all orders in $\\gamma$ and can be easily reformulated for other perturbed\nsoliton equations.",
"arxiv_id": "patt-sol/9906001",
"authors": [
"I. V. Barashenkov",
"E. V. Zemlyanaya"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1016/S0167-2789(99)00055-X",
"title": "Existence threshold for the ac-driven damped nonlinear Schr\\\"odinger solitons",
"url": "https://arxiv.org/abs/patt-sol/9906001"
},
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