On Discrete Symmetries of the Multi-Boson KP Hierarchies

TL;DR

This paper reveals that multi-boson KP hierarchies have discrete symmetries connecting them to Toda systems, with these symmetries generated by similarity transformations of the Lax operator, highlighting their canonical nature.

Contribution

It introduces a new understanding of the discrete symmetries in multi-boson KP hierarchies and their relation to Toda systems, including the concept of square-root lattices and covariance under Bäcklund transformations.

Findings

Discrete symmetries link KP hierarchies to Toda systems.

Spectral equations determine symmetry structures.

Introduction of square-root lattice and new pseudo-differential operators.

Abstract

We show that the multi-boson KP hierarchies possess a class of discrete symmetries linking them to the discrete Toda systems. These discrete symmetries are generated by the similarity transformation of the corresponding Lax operator. This establishes a canonical nature of the discrete transformations. The spectral equation, which defines both the lattice system and the corresponding Lax operator, plays a key role in determining pertinent symmetry structure. We also introduce a concept of the square-root lattice leading to a family of new pseudo-differential operators with covariance under additional B\"{a}cklund transformations.