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Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs

Alexandr Buryak, Jianghao Xu, Di Yang

TL;DR

This work analyzes bihamiltonian structures in DR hierarchies associated to rank-1 F-CohFTs, linking them to the DH framework and Miura invariants. It develops a systematic bihamiltonian test, revealing a 2-parameter family of bihamiltonian DR hierarchies, with notable one-parameter subfamilies corresponding to Miura-equivalences with the Camassa–Holm and Degasperis–Procesi hierarchies. The authors prove a bihamiltonian result for the Hodge hierarchy and provide a detailed study of Miura invariants, including how transformations can yield CH- or DP-type perturbations from the DR setup. These findings illuminate profound connections between moduli-space topology, DR/DZ hierarchies, and classical integrable systems, and suggest a universal role for rank-1 F-CohFTs in organizing bihamiltonian perturbations. The work also highlights practical procedures to compute Miura invariants and to realize CH/DP-type reductions via carefully chosen R-matrices and Miura transformations.

Abstract

Double ramification (DR) hierarchies associated to rank-$1$ F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a $2$-parameter family. Remarkably, our computations suggest that there is a $1$-parameter subfamily of the rank-$1$ F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another $1$-parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy.

Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs

TL;DR

This work analyzes bihamiltonian structures in DR hierarchies associated to rank-1 F-CohFTs, linking them to the DH framework and Miura invariants. It develops a systematic bihamiltonian test, revealing a 2-parameter family of bihamiltonian DR hierarchies, with notable one-parameter subfamilies corresponding to Miura-equivalences with the Camassa–Holm and Degasperis–Procesi hierarchies. The authors prove a bihamiltonian result for the Hodge hierarchy and provide a detailed study of Miura invariants, including how transformations can yield CH- or DP-type perturbations from the DR setup. These findings illuminate profound connections between moduli-space topology, DR/DZ hierarchies, and classical integrable systems, and suggest a universal role for rank-1 F-CohFTs in organizing bihamiltonian perturbations. The work also highlights practical procedures to compute Miura invariants and to realize CH/DP-type reductions via carefully chosen R-matrices and Miura transformations.

Abstract

Double ramification (DR) hierarchies associated to rank- F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a -parameter family. Remarkably, our computations suggest that there is a -parameter subfamily of the rank- F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another -parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy.
Paper Structure (11 sections, 7 theorems, 91 equations, 1 figure)

This paper contains 11 sections, 7 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on evolutionary PDEs of DZ type
  3. Evolutionary PDEs of DZ type and Miura-type transformations
  4. Central invariant of a pair of compatible Poisson operators
  5. Perturbations of the RH hierarchy
  6. Integrable systems associated to rank-$1$ F-CohFTs
  7. CohFTs, F-CohFTs, and the DR hierarchies
  8. Local DR hierarchy
  9. Bihamiltonian $\tau$-symmetric hierarchies
  10. Bihamiltonian tests for the DR hierarchies for rank-$1$ F-CohFTs
  11. More on Miura invariants of the DR hierarchy

Key Result

Corollary 2.1

Consider a PDE $\frac{\partial u}{\partial t}=Q$ that is bihamiltonian with respect to a pair of compatible Poisson operators $(\mathcal{P}_1,\mathcal{P}_2)$ satisfying $\mathcal{P}_1^{[0]}\ne 0$ and $\mathcal{P}_2^{[0]}\ne\lambda \mathcal{P}_1^{[0]}$ for any $\lambda\in\mathbb{C}$. Then $Q$ is uniq

Figures (1)

  • Figure 1: Relations in the $2$-parameter family $(a_2,b_2)$ for four examples.

Theorems & Definitions (24)

  • Corollary 2.1
  • Example 2.2
  • Definition 2.3: DLYZ16
  • Lemma 2.4: BR25DLYZ16
  • Lemma 2.5: see Remark 3.3 and Proposition 3.4 in BR25
  • Conjecture 2.6: DLYZ16
  • Example 2.7
  • Conjecture 2.8: ALM
  • Lemma 3.1
  • proof
  • ...and 14 more