Matrix extension of multidimensional dispersionless integrable hierarchies

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 $d⩾4$. 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.

14 pages, to be published in Teor. i Mat. Fiz. (a Russian version)

References in corpus (2)

• Matrix extension of the Manakov-Santini system and integrable chiral model on Einstein-Weyl background
• Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background