Berry Phases in the Reconstructed KdV Equation

Understand

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 Poincar\'e 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.