Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold

Haonan Qu, Qiulan Zhao

TL;DR

This work investigates the asymmetric rational reductions of the 2D-Toda hierarchy, deriving local bihamiltonian structures for types $(2,1)$ and $(1,2)$ at full dispersion and computing their central invariants. It constructs a three-dimensional generalized Frobenius manifold $M$ with non-flat unity from the $(2,1)$ reduction and shows that its dispersionless flows form the Principal Hierarchy of $M$, with explicit Legendre transformations connecting $M$ to the bi-graded Toda and constrained KP Frobenius manifolds. The authors establish explicit linear reciprocal transformations linking the RR2T to both extended bi-graded Toda and constrained KP hierarchies, providing a unified topological framework corroborated by spectral and Miura-type analyses. The results illuminate deep connections between RR2T, Frobenius geometry, and well-known integrable hierarchies, and suggest conjectures about central invariants for broader RR2T families and avenues for exploring symmetric reductions and topological deformations.

Abstract

We study the local bihamiltonian structures of the asymmetric rational reductions of the 2D-Toda hierarchy (RR2T) of types $(2,1)$ and $(1,2)$ at the full-dispersive level, and construct a three-dimensional generalized Frobenius manifold with non-flat unity associated with the $(2,1)$-type. Furthermore, we explicitly relate the $(2,1)$-type RR2T to the bi-graded Toda and constrained KP hierarchies via linear reciprocal and Miura-type transformations.

Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold

TL;DR

This work investigates the asymmetric rational reductions of the 2D-Toda hierarchy, deriving local bihamiltonian structures for types and at full dispersion and computing their central invariants. It constructs a three-dimensional generalized Frobenius manifold with non-flat unity from the reduction and shows that its dispersionless flows form the Principal Hierarchy of , with explicit Legendre transformations connecting to the bi-graded Toda and constrained KP Frobenius manifolds. The authors establish explicit linear reciprocal transformations linking the RR2T to both extended bi-graded Toda and constrained KP hierarchies, providing a unified topological framework corroborated by spectral and Miura-type analyses. The results illuminate deep connections between RR2T, Frobenius geometry, and well-known integrable hierarchies, and suggest conjectures about central invariants for broader RR2T families and avenues for exploring symmetric reductions and topological deformations.

Abstract

We study the local bihamiltonian structures of the asymmetric rational reductions of the 2D-Toda hierarchy (RR2T) of types and at the full-dispersive level, and construct a three-dimensional generalized Frobenius manifold with non-flat unity associated with the -type. Furthermore, we explicitly relate the -type RR2T to the bi-graded Toda and constrained KP hierarchies via linear reciprocal and Miura-type transformations.

Paper Structure

This paper contains 15 sections, 9 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Asymmetric RR2T and their bihamiltonian structures
  3. The definition of the RR2T of type $(2,1)$
  4. Bihamiltonian formalism of the RR2T of type $(2,1)$
  5. Relation to the RR2T of type $(1,2)$
  6. Bihamiltonian structures for the two asymmetric RR2T
  7. Central invariants for the two asymmetric RR2T
  8. From the $(2,1)$-type RR2T to a generalized Frobenius manifold
  9. Generalized Frobenius manifold
  10. Principal Hierarchy
  11. Generalized Legendre transformations
  12. Linear reciprocal transformations
  13. Relation to the extended bi-graded Toda hierarchy
  14. Relation to the constrained KP hierarchy
  15. Conclusions

Key Result

Theorem 2.2

The RR2T of $(2,1)$-type GAL-1--GAL-6 can be represented by the following Hamiltonian systems: for $i\in\{2,3\}$, $k\geq 0$ or $i=0$, $k\leq -1$. Here the Hamiltonian operator $\mathcal{P}_1$ is given in HamP1, and the Hamiltonians have the expressions where $M=A^{-1}B$ is defined as in Lax-M.

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof : Proof of Theorem \ref{['thm:HamFormalism-1']}
  • Theorem 2.5
  • proof
  • ...and 18 more