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.
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Acknowledgments
The author is grateful to E. A. Kusnetsov for the useful comment concerning the topic of this work.
Funding
This research was performed in the framework of State Assignment of the Ministry of Science and Higher Education of the Russian Federation, topic 0029-2021-0004 (Quantum field theory).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 209, pp. 3–15 https://doi.org/10.4213/tmf10125.
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Bogdanov, L.V. Matrix extension of multidimensional dispersionless integrable hierarchies. Theor Math Phys 209, 1319–1330 (2021). https://doi.org/10.1134/S0040577921100019
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DOI: https://doi.org/10.1134/S0040577921100019