- 94-380 Pierce, R.D., Wayne, C.E.
- On the validity of mean-field amplitude equations for counterpropagating wavetrains
(232K, uuencoded, tar-compressed Postscript file)
Nov 29, 94
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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.