Bilinear formalism for Schwarzian KP and Harry Dym hierarchies

TL;DR

This paper reformulates the Schwarzian KP and Harry Dym hierarchies using bilinear formalism, revealing their connections to tau-functions, Bäcklund-Darboux transformations, and multi-component KP hierarchy.

Contribution

It introduces a bilinear integral equation for Schwarzian KP and links Harry Dym hierarchy to the Lax-Sato formulation, expanding understanding of integrable hierarchies.

Findings

Schwarzian KP hierarchy can be expressed as an integral bilinear equation for tau-functions.

Harry Dym hierarchy is derived as a Lax-Sato formulation of Schwarzian KP.

Schwarzian KP hierarchy naturally embeds into multi-component KP hierarchy.

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.