Integrable KdV Hierarchies on T^2=S^1\times S^1

TL;DR

This paper develops a framework for integrable KdV hierarchies on the torus, explicitly constructs their Lax pairs, and explores the relationship between KdV and Burgers models in this setting.

Contribution

It introduces a setup for integrable KdV hierarchies on the torus and explicitly constructs their Lax pairs, establishing a connection between KdV and Burgers systems.

Findings

Explicit Lax pair operators for KdV and Burgers models on T^2

Mapping between KdV and Burgers systems on the torus

Confirmation of integrability of these models on T^2

Abstract

Following our previous works on extended higher spin symmetries on the torus we focus in the present contribution to make a setup of the integrable KdV hierarchies on $T^{2} = S^{1} \times S^{1}$. Actually two particular systems are considered, namely the KdV and the Burgers non linear integrable model associated to currents of conformal weights (2, 2) and (1, 1) respectively. One key steps towards proving the integrability of these systems is to find their Lax pair operators. This is explicitly done and a mapping between the two systems is discussed.