Extension to semidirect product groups

Construct extensions of the symplectic structures discussed for incompressible and compressible singular vorticities to the semidirect product groups ${\rm Diff}_\sigma(S)\ltimes C^\infty(S)$ and ${\rm Diff}(Q)\ltimes C^\infty(Q)$.

Background

The paper studies symplectic structures arising from coadjoint orbits of volume-preserving diffeomorphism groups and of the full diffeomorphism group. It also describes dual-pair constructions and generalized Marsden–Weinstein structures for singular 1-form densities.

The authors explicitly leave unresolved how these structures should be extended when the diffeomorphism groups are enlarged to the indicated semidirect products with function spaces, which would incorporate additional scalar or advected-field components.

References

Here are several open questions related to the introduction of symplectic structures on the augmented triples.

  1. What could be an interesting (and possible integrable in lower dimensions) analog of the binormal equation for membranes in this compressible setting?
  2. Is it possible to obtain such an analog of the binormal equation by means of a reduction or ansatz from the compressible Euler equation on $Q$, similar to the localized induction approximation in the incompressible setting?
  3. What could be an extension of the above symplectic structures to the semidirect product groups ${\rm Diff}_\sigma(S)\ltimes C\infty(S)$ and ${\rm Diff} (Q)\ltimes C\infty(Q)$.
— Generalized Marsden-Weinstein symplectic structures  (2610.01401 - Khesin et al., 1 Oct 2026) in Section "Dynamics on membranes and open questions", subsection "The binormal equation and its analogs" (Section 5.1)