Pierce, R.D., Wayne, C.E.
On the validity of mean-field amplitude equations for counterpropagating wavetrains
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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.