The Method of ${\mathcal{M}}_n$-Extension: The KdV Equation

In this work we generalize ${\mathcal{M}}_2$-extension that has been introduced recently. For illustration we use the KdV equation. We present five different ${\mathcal{M}}_3$-extensions of the KdV equation and their recursion operators. We give a compact form of ${\mathcal{M}}_n$-extension of the KdV equation and recursion operator of the coupled KdV system. The method of ${\mathcal{M}}_n$-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 ${\mathcal{M}}_3$-extension of KdV.