The Method of ${\cal M}_{n}$-Extension: The KdV Equation

TL;DR

This paper introduces a generalized method called ${ m f M}_n$-extension for integrable equations, exemplified through multiple extensions of the KdV equation, providing new recursion operators and reduction techniques.

Contribution

The paper develops a generalized ${ m f M}_n$-extension framework for integrable equations, offering explicit constructions for the KdV equation and its multi-field extensions.

Findings

Five different ${ m f M}_3$-extensions of KdV are presented.

A compact form of the ${ m f M}_n$-extension for KdV is derived.

Nonlocal reductions of the ${ m f M}_3$-extension are discussed.

Abstract

In this work we generalize ${\cal M}_{2}$-extension that has been introduced recently. For illustration we use the KdV equation. We present five different ${\cal M}_{3}$-extensions of the KdV equation and their recursion operators. We give a compact form of ${\cal M}_{n}$-extension of the KdV equation and recursion operator of the coupled KdV system. The method of ${\cal M}_{n}$-extension can be applied to any integrable scalar equation to obtain integrable multi-field system of equations. We also present unshifted and shifted nonlocal reductions of an example of ${\cal M}_{3}$-extension of KdV.