dorsal/arxiv
View SchemaAsymptotics of Solutions to the Modified Nonlinear Schr\"{o}dinger Equation: Solitons on a Non-Vanishing Continuous Background
| Authors | A. V. Kitaev, A. H. Vartanian |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9801001 |
| URL | https://arxiv.org/abs/solv-int/9801001 |
Abstract
Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as $t \to \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schr\"{o}dinger equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u + | u |^{2} u + i s \partial_{x} (| u |^{2} u) = 0$, $s \in \Bbb R_{>0}$, which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative nonlinear Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i \partial_{x} (| q |^{2} q) = 0$. As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the presence of the continuous spectrum are calculated.
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"abstract": "Using the matrix Riemann-Hilbert factorization approach for nonlinear\nevolution systems which take the form of Lax-pair isospectral deformations and\nwhose corresponding Lax operators contain both discrete and continuous spectra,\nthe leading-order asymptotics as $t \\to \\pm \\infty$ of the solution to the\nCauchy problem for the modified nonlinear Schr\\\"{o}dinger equation, $i\n\\partial_{t} u + {1/2} \\partial_{x}^{2} u + | u |^{2} u + i s \\partial_{x} (| u\n|^{2} u) = 0$, $s \\in \\Bbb R_{\u003e0}$, which is a model for nonlinear pulse\npropagation in optical fibers in the subpicosecond time scale, are obtained:\nalso derived are analogous results for two gauge-equivalent nonlinear evolution\nequations; in particular, the derivative nonlinear Schr\\\"{o}dinger equation, $i\n\\partial_{t} q + \\partial_{x}^{2} q - i \\partial_{x} (| q |^{2} q) = 0$. As an\napplication of these asymptotic results, explicit expressions for position and\nphase shifts of solitons in the presence of the continuous spectrum are\ncalculated.",
"arxiv_id": "solv-int/9801001",
"authors": [
"A. V. Kitaev",
"A. H. Vartanian"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Asymptotics of Solutions to the Modified Nonlinear Schr\\\"{o}dinger Equation: Solitons on a Non-Vanishing Continuous Background",
"url": "https://arxiv.org/abs/solv-int/9801001"
},
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