Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation

TL;DR

This paper introduces the ll-Bogoyavlensky hierarchy, extending the ll-reduced KP hierarchy, and provides a group-theoretic and Lax formalism-based framework for understanding its equations and solutions, including solitons.

Contribution

It presents a new hierarchy extending the ll-reduced KP hierarchy, characterized by a toroidal Lie algebra, and derives Hirota bilinear equations and soliton solutions.

Findings

Reproduces Hirota bilinear equations similar to recent work.

Derives equations from a Lax formalism.

Constructs generalized breaking soliton solutions.

Abstract

We introduce an extension of the \ell-reduced KP hierarchy, which we call the \ell-Bogoyavlensky hierarchy. Bogoyavlensky's 2+1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of \ell=2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra sl_\ell^{tor}. This reproduces essentially the same Hirota bilinear equations as those recently introduced by Billig and Iohara et al. We can further derive these Hirota bilinear equation from a Lax formalism of the hierarchy.This Lax formalism also enables us to construct a family of special solutions that generalize the breaking soliton solutions of Bogoyavlensky. These solutions contain the N-soliton solutions, which are usually constructed by use of vertex operators.