Constrained lattice-field hierarchies and Toda system with Block symmetry

TL;DR

This paper develops new symmetry structures for lattice-field hierarchies, constrains these systems to derive novel difference equations, and reveals that the Toda hierarchy possesses a Block Lie symmetry, enriching the understanding of integrable systems.

Contribution

It introduces additional W- and ghost symmetries for lattice hierarchies, constructs new integrable difference equations, and demonstrates the Block Lie symmetry in the Toda hierarchy.

Findings

New integrable difference equations derived from symmetry constraints

The Toda hierarchy exhibits a Block Lie symmetry

Constrained systems unify into the original Toda hierarchy

Abstract

In this paper, we construct the additional $W$-symmetry and ghost symmetry of two-lattice field integrable hierarchies. Using the symmetry constraint, we construct constrained two-lattice integrable systems which contain several new integrable difference equations. Under a further reduction, the constrained two-lattice integrable systems can be combined into one single integrable system, namely the well-known one dimensional original Toda hierarchy. We prove that the one dimensional original Toda hierarchy has a nice Block Lie symmetry.