Integrability of invariant metrics on the diffeomorphism group of the circle

TL;DR

This paper investigates the integrability of invariant Sobolev metrics on the diffeomorphism group of the circle, revealing that only specific cases exhibit bi-Hamiltonian structures.

Contribution

It demonstrates that only the H^0 and H^1 Sobolev inner products lead to bi-Hamiltonian vector fields on the dual of the Lie algebra, highlighting their special integrability properties.

Findings

Only X_0 and X_1 are bi-Hamiltonian with respect to a modified Lie-Poisson structure.

Higher Sobolev inner products do not produce bi-Hamiltonian structures.

The results clarify the unique role of H^0 and H^1 metrics in the integrability of the system.

Abstract

Each H^k Sobolev inner product defines a Hamiltonian vector field X_k on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X_0 and X_1 are bi-Hamiltonian relatively to a modified Lie-Poisson structure.