Berry Phases in the Reconstructed KdV Equation

Abstract

We consider the KdV equation on a circle and its Lie-Poisson reconstruction, which is reminiscent of an equation of motion for fluid particles. For periodic waves, the stroboscopic reconstructed motion is governed by an iterated map whose Poincaré rotation number yields the drift velocity. We show that this number has a geometric origin: it is the sum of a dynamical phase, a Berry phase, and an "anomalous phase". The last two quantities are universal: they are solely due to the underlying Virasoro group structure. The Berry phase, in particular, was previously described in [arXiv:1703.06142] for two-dimensional conformal field theories, and follows from adiabatic deformations produced by the propagating wave. We illustrate these general results with cnoidal waves, for which all phases can be evaluated in closed form thanks to a uniformizing map that we derive. Along the way, we encounter "orbital bifurcations" occurring when a wave becomes non-uniformizable: there exists a resonance wedge, in the cnoidal parameter space, where particle motion is locked to the wave, while no such locking occurs outside of the wedge.

arXiv comment43 pages, 8 figures. v2: typos fixed, refs updated, minor clarifications added along with an extra appendix. Published in "Chaos"

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Cited by in corpus (7)

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• Anisotropic Quantum Hall Droplets
• Shallow Water Memory: Stokes and Darwin Drifts
• Adiabatic Deformations of Quantum Hall Droplets
• Berry Phases in the Bosonization of Nonlinear Edge Modes
• Topological Bifurcations and Reconstruction of Travelling Waves