This article is cited in 7 papers

Matrix extension of multidimensional dispersionless integrable hierarchies

A. A. Belavin^ab, S. E. Parkhomenko<sup>a

a</sup> Landau Institute for Theoretical Physics of Russian Academy of Sciences
^b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia

Abstract: We consistently develop a recently proposed scheme of matrix extensions of dispersionless integrable systems in 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, the construction of solutions, and connections to geometry are discussed. We separately consider the case of an Abelian extension, for which the Riemann–Hilbert equations of the dressing scheme are explicitly solvable and give an analogue of the Penrose formula in curved space.

Keywords: dispersionless integrable system, gauge field, self-dual Yang–Mills equations.

Received: 05.07.2021
Revised: 05.07.2021

DOI: 10.4213/tmf10145

 English version: Theoretical and Mathematical Physics, 2021, 209:1, 1367–1386

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