dorsal/arxiv
View SchemaOn the validity of mean-field amplitude equations for counterpropagating wavetrains
| Authors | R. D. Pierce, C. E. Wayne |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9411002 |
| URL | https://arxiv.org/abs/patt-sol/9411002 |
| DOI | 10.1088/0951-7715/8/5/007 |
Abstract
We rigorously establish the validity of the equations describing the evolution of one-dimensional long wavelength modulations of counterpropagating wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We consider both periodic amplitude functions and localized wavepackets. For the localized case, the wavetrains are completely decoupled at leading order, while in the periodic case the amplitude equations take the form of mean-field (nonlocal) Schr\"odinger equations rather than locally coupled partial differential equations. The origin of this weakened coupling is traced to a hidden translation symmetry in the linear problem, which is related to the existence of a characteristic frame traveling at the group velocity of each wavetrain. It is proved that solutions to the amplitude equations dominate the dynamics of the governing equations on asymptotically long time scales. While the details of the discussion are restricted to the class of model equations having a leading cubic nonlinearity, the results strongly indicate that mean-field evolution equations are generic for bimodal disturbances in dispersive systems with \O(1) group velocity.
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"abstract": "We rigorously establish the validity of the equations describing the\nevolution of one-dimensional long wavelength modulations of counterpropagating\nwavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We\nconsider both periodic amplitude functions and localized wavepackets. For the\nlocalized case, the wavetrains are completely decoupled at leading order, while\nin the periodic case the amplitude equations take the form of mean-field\n(nonlocal) Schr\\\"odinger equations rather than locally coupled partial\ndifferential equations. The origin of this weakened coupling is traced to a\nhidden translation symmetry in the linear problem, which is related to the\nexistence of a characteristic frame traveling at the group velocity of each\nwavetrain. It is proved that solutions to the amplitude equations dominate the\ndynamics of the governing equations on asymptotically long time scales. While\nthe details of the discussion are restricted to the class of model equations\nhaving a leading cubic nonlinearity, the results strongly indicate that\nmean-field evolution equations are generic for bimodal disturbances in\ndispersive systems with \\O(1) group velocity.",
"arxiv_id": "patt-sol/9411002",
"authors": [
"R. D. Pierce",
"C. E. Wayne"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1088/0951-7715/8/5/007",
"title": "On the validity of mean-field amplitude equations for counterpropagating wavetrains",
"url": "https://arxiv.org/abs/patt-sol/9411002"
},
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