Quasi-classical limit of BKP hierarchy and W-infinity symmeties

TL;DR

This paper extends the quasi-classical limit analysis from KP and Toda hierarchies to the BKP hierarchy, revealing how W-infinity symmetries contract in this limit.

Contribution

It reformulates key tools for the BKP hierarchy and introduces subalgebras of W-infinity algebras as fundamental symmetry structures.

Findings

Quantum W-infinity algebra appears as symmetry of BKP hierarchy

Symmetries contract from quantum to dispersionless form in the quasi-classical limit

Framework for analyzing quasi-classical limits of integrable hierarchies

Abstract

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$ and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and $w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, these $\WB_{1+\infty}$ symmetries are shown to be contracted into $\wB_{1+\infty}$ symmetries of the dispersionless BKP hierarchy.