Bilinear formalism for Schwarzian KP and Harry Dym hierarchies

2026/04/30 by Vadim Prokofev, Anton Zabrodin · 1 voice
Physics and Astronomy · #math-ph #nlin.SI

paper · pdf · doi:10.1016/j.geomphys.2026.105913

Abstract

We consider the Schwarzian KP and Harry Dym hierarchies in the framework of the bilinear formalism which is well known for such integrable hierarchies as KP, modified KP, BKP, Toda lattice and other. We show that, similarly to the bilinear formulation of the modified KP hierarchy, the Schwarzian KP can be reformulated as an integral bilinear equation for a pair of KP tau-functions with the property that any linear combination of them is again a tau function of the KP hierarchy. The Harry Dym hierarchy is then obtained as the Lax-Sato formulation of the SchKP one. The close connection with Backlund-Darboux transformations for integrable hierarchies is also discussed. Besides, it is shown that the SchKP hierarchy has a natural embedding into the multi-component KP hierarchy.

Citations

• Classical facets of quantum integrability
• Free fermions and tau-functions
• The Multicomponent KP Hierarchy: Differential Fay Identities and Lax Equations
• Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
• Coupled Modified KP Hierarchy and Its Dispersionless Limit
• Menelausapos theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
• Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies
• Method of Squared Eigenfunction Potentials in Integrable Hierarchies of KP Type
• Analytic-bilinear approach to integrable hierarchies. I.Generalized KP hierarchy
• INTEGRABLE HIERARCHIES AND DISPERSIONLESS LIMIT
• The n-component KP hierarchy and representation theory
• Solitons and Infinite Dimensional Lie Algebras

Discussions

• Vadim Prokofev, Anton Zabrodin: Bilinear formalism for Schwarzian KP and Harry Dym hierarchies https://arxiv.org/abs/2605.00235 https://arxiv.org/pdf/2605.00235 https://arxiv.org/html/2605.00235 [bsky, 0 points, 0 comments]

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• Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions