On the validity of mean-field amplitude equations for counterpropagating wavetrains

TL;DR

This paper rigorously proves the validity of mean-field amplitude equations for counterpropagating wavetrains in the sine-Gordon model, revealing their dominance in long-term dynamics and uncovering the role of hidden symmetries.

Contribution

It establishes the rigorous validity of mean-field amplitude equations for counterpropagating wavetrains in a hyperbolic model, highlighting the influence of symmetries and extending understanding of dispersive systems.

Findings

Amplitude equations dominate long-term dynamics.

Periodic and localized cases exhibit different coupling behaviors.

Hidden symmetries influence the form of amplitude equations.

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.