Matrix extension of multidimensional dispersionless integrable hierarchies

TL;DR

This paper develops a matrix extension framework for multidimensional dispersionless integrable hierarchies, providing new Lax pairs, equations, and solutions, with applications to geometry and explicit solvability in Abelian cases.

Contribution

It introduces a comprehensive matrix extension scheme for multidimensional dispersionless integrable systems, including Lax pairs, dressing methods, and geometric connections, especially for dimensions d≥4.

Findings

Extended Lax pairs and equations for matrix hierarchies

Explicit solutions in Abelian extension cases

Connection to geometric structures and Penrose-like formulas

Abstract

We consistently develop a recently proposed scheme of matrix extension of dispersionless integrable systems for the general case of multidimensional hierarchies, concentrating on the case of dimension $d\geqslant 4$. We present extended Lax pairs, Lax-Sato equations, matrix equations on the background of vector fields and the dressing scheme. Reductions, construction of solutions and connections to geometry are discussed. We consider separately a case of Abelian extension, for which the Riemann-Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of Penrose formula in the curved space.