On Tautological Flows of Partial Difference Equations

TL;DR

This paper introduces the tautological flow method to analyze the integrability of partial difference equations by relating them to their continuous PDE counterparts, revealing symmetries and bihamiltonian structures.

Contribution

The paper presents a novel tautological flow method for studying integrability of PΔEs, connecting discrete equations to PDE structures and enabling direct symmetry and bihamiltonian structure searches.

Findings

Proves the discrete q-KdV equation as a symmetry of the q-deformed KdV hierarchy.

Demonstrates direct search for symmetries and structures using approximated tautological flows.

Establishes the relationship between discrete and continuous integrability structures.

Abstract

We propose a new analyzing method, which is called the tautological flow method, to analyze the integrability of partial difference equations (P$\Delta$Es) based on that of partial differential equations (PDEs). By using this method, we prove that the discrete $q$-KdV equation is a discrete symmetry of the $q$-deformed KdV hierarchy and its bihamiltonian structure, and we also demonstrate how to directly search for continuous symmetries and bihamiltonian structures of P$\Delta$Es by using the approximated tautological flows and their quasi-triviality transformation.