Tri-Hamiltonian structure of an asymmetric generalized Ablowitz-Ladik hierarchy and a Frobenius manifold

2026/06/24 by Hanxue Zhang, Qiulan Zhao, Xinyue Li · 1 voice
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.DG

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Abstract

We construct a local tri-Hamiltonian structure of the asymmetric (3,1)-type generalized Ablowitz-Ladik (gAL) hierarchy at the full-dispersive level and rigorously prove its validity using the supervariable technique. All central invariants of the corresponding bi-Hamiltonian structures are computed. In addition, we construct a Frobenius manifold M arising from the dispersionless limit of this hierarchy and show that the dispersionless limits of the first flows of the (3,1)-type gAL hierarchy belong to the Principal Hierarchy of M.

Citations

• Bihamiltonian structure of the (n,1)-type rational reductions of the 2D-Toda hierarchy
• Asymmetric rational reductions of 2D-Toda hierarchy and a generalized Frobenius manifold
• Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchies
• The extended Ablowitz-Ladik hierarchy and a generalized Frobenius manifold
• Solutions of the loop equations of a class of generalized Frobenius manifolds
• Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies
• Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed
• Rational reductions of the 2D-Toda hierarchy and mirror symmetry
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case
• Exact Poisson pencils, τ-structures and topological hierarchies
• Integrable hierarchies and the mirror model of local CP1
• Jacobi Structures of Evolutionary Partial Differential Equations
• The Equivariant Gromov--Witten Theory of CP¹ and Integrable hierarchies
• On Hamiltonian perturbations of hyperbolic systems of conservation laws
• Deformations of Semisimple Bihamiltonian Structures of Hydrodynamic Type
• Hamiltonian operators and l*-coverings
• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Compatible Poisson brackets of hydrodynamic type
• 'Universality' of the Ablowitz-Ladik hierarchy
• Geometry of 2d topological field theories

Discussions

• Hanxue Zhang, Qiulan Zhao, Xinyue Li: Tri-Hamiltonian structure of an asymmetric generalized Ablowitz-Ladik hierarchy and a Frobenius manifold https://arxiv.org/abs/2606.25846 https://arxiv.org/pdf/26 [bsky, 0 points, 0 comments]

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