Lagrangian extensions of multi-dimensional integrable equations. I. The five-dimensional Mart{\'{\i}}nez Alonso--Shabat equation

TL;DR

This paper explores a Lagrangian extension of a five-dimensional integrable equation, analyzing its symmetry algebra, recursion operators, and Lax pairs, revealing complex algebraic structures and integrability properties.

Contribution

It introduces a Lagrangian extension of the 5d Martínez Alonso--Shabat equation, describes its symmetry algebra, constructs recursion operators, and develops new Lax pairs, advancing understanding of high-dimensional integrable systems.

Findings

Symmetry algebra is complex and described via deformations.

Constructed two families of recursion operators, with one family being hereditary.

Developed two new parametric Lax pairs involving higher derivatives.

Abstract

We study a Lagrangian extension of the 5d Mart\'inez Alonso--Shabat equation $\mathcal{E}$ \begin{equation*} u_{yz}=u_{tx}+u_y\,u_{xs}-u_x\,u_{ys} \end{equation*} that coincides with the cotangent equation $\mathcal{T^*E}$ to the latter. We describe the Lie algebra structure of its symmetries (which happens to be quite nontrivial and is described in terms of deformations) and construct two families of recursion operators for symmetries. Each family depends on two parameters. We prove that all the operators from the first family are hereditary, but not compatible in the sense of the Nijenhuis bracket. We also construct two new parametric Lax pairs that depend on higher-order derivatives of the unknown functions.