dorsal/arxiv
View SchemaHigher Order Asymptotics of the Modified Non-Linear Schr\"{o}dinger Equation
| Authors | A. H. Vartanian |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9804013 |
| URL | https://arxiv.org/abs/solv-int/9804013 |
Abstract
Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as $t \to \pm \infty$ $(x/t \sim {\cal O}(1))$ of the solution to the Cauchy problem for the modified non-linear 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 non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent non-linear evolution equations; in particular, the derivative non-linear Schr\"{o}dinger equation, $i \partial_{t} q + \partial_{x}^{2} q - i \partial_{x}(| q |^{2} q) = 0$.
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"abstract": "Using the matrix Riemann-Hilbert factorisation approach for non-linear\nevolution systems which take the form of Lax-pair isospectral deformations, the\nhigher order asymptotics as $t \\to \\pm \\infty$ $(x/t \\sim {\\cal O}(1))$ of the\nsolution to the Cauchy problem for the modified non-linear Schr\\\"{o}dinger\nequation, $i \\partial_{t} u + {1/2} \\partial_{x}^{2} u + | u |^{2} u + i s\n\\partial_{x} (| u |^{2} u) = 0$, $s \\in \\Bbb R_{\u003e 0}$, which is a model for\nnon-linear pulse propagation in optical fibres in the subpicosecond time scale,\nare obtained: also derived are analogous results for two gauge-equivalent\nnon-linear evolution equations; in particular, the derivative non-linear\nSchr\\\"{o}dinger equation, $i \\partial_{t} q + \\partial_{x}^{2} q - i\n\\partial_{x}(| q |^{2} q) = 0$.",
"arxiv_id": "solv-int/9804013",
"authors": [
"A. H. Vartanian"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Higher Order Asymptotics of the Modified Non-Linear Schr\\\"{o}dinger Equation",
"url": "https://arxiv.org/abs/solv-int/9804013"
},
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