Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies

TL;DR

This paper introduces a new semi-discrete Kadomtsev--Petviashvili equation with integrable properties and derives semi-discrete Drinfel'd--Sokolov hierarchies related to various Kac--Moody Lie algebras, including explicit Lax pairs and solutions.

Contribution

It constructs a novel semi-discrete KP equation with integrability features and derives new semi-discrete Drinfel'd--Sokolov hierarchies from it, including Lax pairs and exact solutions.

Findings

New semi-discrete KP equation with integrability

Derived semi-discrete Drinfel'd--Sokolov hierarchies for multiple Lie algebras

Established existence of exact solutions via direct linearisation

Abstract

We propose a novel semi-discrete Kadomtsev--Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation framework. By performing reductions on the semi-discrete Kadomtsev--Petviashvili equation, new semi-discrete versions of the Drinfel'd--Sokolov hierarchies associated with Kac--Moody Lie algebras $A_r^{(1)}$, $A_{2r}^{(2)}$, $C_r^{(1)}$ and $D_{r+1}^{(2)}$ are successfully constructed. A Lax pair involving the fraction of $\mathbb{Z}_\mathcal{N}$ graded matrices is also found for each of the semi-discrete Drinfel'd--Sokolov equations. Furthermore, the direct linearisation construction guarantees the existence of exact solutions of all the semi-discrete equations discussed in the paper, providing another insight into their integrability in addition to the analysis of Lax pairs.