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Miura-reciprocal transformations and localizable Poisson pencils

P. Lorenzoni, S. Shadrin, R. Vitolo

TL;DR

This work extends the Dubrovin–Zhang program by establishing a systematic Miura–reciprocal framework for classifying deformations of localizable semisimple Poisson pencils, and by showing that every orbit admits a local representative. It develops explicit change-of-variable formulae, derives the Ferapontov–Pavlov transformation for Poisson bivectors, and proves the closure of localizable weakly non-local pencils under Miura–reciprocal actions. Using theta-formalism and bi-Hamiltonian cohomology, it proves vanishing and central-invariant results for dispersive deformations, proving that the orbit space is parametrized by $N$ smooth functions and that central invariants can be read from the pencil’s symbol. It also establishes Doyle–Potëmin form invariance under the projective subgroup, linking to Mokhov’s conjecture and applications to Dubrovin–Zhang hierarchies, thereby broadening the classification of integrable bi-Hamiltonian PDEs to nonlocal settings with concrete invariants.

Abstract

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group.

Miura-reciprocal transformations and localizable Poisson pencils

TL;DR

This work extends the Dubrovin–Zhang program by establishing a systematic Miura–reciprocal framework for classifying deformations of localizable semisimple Poisson pencils, and by showing that every orbit admits a local representative. It develops explicit change-of-variable formulae, derives the Ferapontov–Pavlov transformation for Poisson bivectors, and proves the closure of localizable weakly non-local pencils under Miura–reciprocal actions. Using theta-formalism and bi-Hamiltonian cohomology, it proves vanishing and central-invariant results for dispersive deformations, proving that the orbit space is parametrized by smooth functions and that central invariants can be read from the pencil’s symbol. It also establishes Doyle–Potëmin form invariance under the projective subgroup, linking to Mokhov’s conjecture and applications to Dubrovin–Zhang hierarchies, thereby broadening the classification of integrable bi-Hamiltonian PDEs to nonlocal settings with concrete invariants.

Abstract

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group.
Paper Structure (20 sections, 19 theorems, 104 equations)

This paper contains 20 sections, 19 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. A variety of jet space transformation groups
  3. Action of the transformation groups
  4. Weakly non-local Poisson bi-vectors of localizable shape
  5. Localizability
  6. Projective group and Doyle--Potëmin form
  7. Acknowledgments
  8. Formulae for the action
  9. The Ferapontov--Pavlov formula
  10. Weakly non-local bi-vectors of localizable shape
  11. Schouten bracket for weakly non-local operators of localizable shape
  12. The two approaches
  13. Identification of the two approaches
  14. Pencils of weakly non-local bi-vectors of localizable shape
  15. Bi-Hamiltonian cohomology
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $X^i\delta_{u^i}=Y^i\delta_{w^i}$ be a variational vector field in the coordinate systems $(x,u^{i,\sigma})$ and $(y,w^{i,\sigma})$, respectively, where the latter coordinates systems are related by a holonomic reciprocal differential substitution $y=P$, $w^i = Q^i$. Then the following change of where

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Conjecture 1.6: See MokhovSurvey1998
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • ...and 34 more