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Berry phases in the reconstructed KdV equation.
Blagoje Oblak1, Gregory Kozyreff2
1Laboratoire de Physique Théorique et Hautes Energies, Sorbonne Université and CNRS UMR 7589, F-75005 Paris, France.
Chaos (Woodbury, N.Y.)
|December 2, 2020
Summary
The KdV equation
Area of Science:
- Nonlinear dynamics
- Mathematical physics
- Fluid mechanics
Background:
- The Korteweg-de Vries (KdV) equation describes nonlinear wave phenomena.
- Lie-Poisson reconstruction offers a fluid particle dynamics perspective.
- Periodic waves on a circle exhibit complex iterated map dynamics.
Purpose of the Study:
- To investigate the geometric origin of drift velocity in the KdV equation's reconstructed motion.
- To analyze the contributions of dynamical, Berry, and anomalous phases.
- To explore phenomena like orbital bifurcations in cnoidal wave solutions.
Main Methods:
- Lie-Poisson reconstruction of the KdV equation.
- Analysis of iterated maps and Poincaré rotation numbers.
- Derivation of a uniformizing map for cnoidal waves.
- Investigation of Virasoro group structure implications.
Main Results:
- Drift velocity is a sum of dynamical, Berry, and anomalous phases.
- Berry and anomalous phases stem from Virasoro group structure.
- Cnoidal waves allow closed-form evaluation of phases via a derived uniformizing map.
- Orbital bifurcations occur in a resonance wedge of the cnoidal parameter space.
Conclusions:
- The drift velocity in the KdV equation has a universal geometric origin.
- Virasoro group structure dictates Berry and anomalous phases.
- Cnoidal waves provide a tractable model for studying these phase contributions and associated bifurcations.
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