Matrix extension of multidimensional dispersionless integrable hierarchies
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 $d≥slant 4$. We present extended Lax pairs, LaxSato 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 RiemannHilbert equations of the dressing scheme are explicitly solvable and give an analogue of the Penrose formula in curved space.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 October 2021
 DOI:
 10.1134/S0040577921100019
 arXiv:
 arXiv:2105.14264
 Bibcode:
 2021TMP...209.1319B
 Keywords:

 dispersionless integrable system;
 gauge field;
 selfdual YangMills equations;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 37K10;
 37K15;
 37K25;
 35Q75
 EPrint:
 14 pages, to be published in Teor. i Mat. Fiz. (a Russian version)