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On the Validity of Mean--eld Amplitude Equations for Counterpropagating Wavetrains
| Content Provider | Semantic Scholar |
|---|---|
| Copyright Year | 1995 |
| 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 de-coupled at leading order, while in the periodic case the amplitude equations take the form of mean-eld (nonlocal) Schrr odinger equations rather than locally coupled partial diierential 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-eld evolution equations are generic for bimodal disturbances in dispersive systems with O(1) group velocity. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://linux46.ma.utexas.edu/mp_arc/c/94/94-380.ps.gz |
| Alternate Webpage(s) | http://www.ma.utexas.edu/mp_arc/c/94/94-380.ps.gz |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | ARID1A wt Allele Aharonov–Bohm effect Cubic function Dispersive partial differential equation Linear programming Nonlinear system Short Interspersed Nucleotide Elements Solutions Velocity (software development) Wave packet travel wavelength |
| Content Type | Text |
| Resource Type | Article |
