On additional symmetries of the KP hierarchy and Sato's B\"acklund transformation

TL;DR

This paper proves that the additional symmetries of the KP hierarchy, whether defined via pseudodifferential operators or tau-functions, are equivalent, and introduces a new simple formula for their generator.

Contribution

It provides a concise proof of the equivalence of two definitions of KP hierarchy symmetries and presents a new formula for the symmetry generator.

Findings

Confirmed the equivalence of symmetry definitions

Derived a new simple formula for the generator

Simplified understanding of KP hierarchy symmetries

Abstract

A short proof is given to the fact that the additional symmetries of the KP hierarchy defined by their action on pseudodifferential operators, according to Fuchssteiner-Chen-Lee-Lin-Orlov-Shulman, coincide with those defined by their action on $\tau$-functions as Sato's B\"acklund transformations. Also a new simple formula for the generator of additional symmetries is presented.