The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure

Abstract

A Lie-algebraic approach to constructing the Lax–Sato integrable superanalogs of heavenly equations by use of the loop Lie algebra of superconformal vector fields on a 1|N-dimensional supertorus is proposed. In the framework of this approach integrable superanalogs of the Mikhalev–Pavlov heavenly equation are obtained for all \(N\in {\mathbb {N}} {\setminus } \{ 4,5 \}\) as well as Shabat type reductions for all \(N\in {\mathbb {N}}\). The Lax–Sato integrable superanalogs of the generalized Liouville heavenly equations are found by means of the Lie algebra of holomorphic in “spectral” parameter superconformal vector fields on a 1|N-dimensional complex supertorus.

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Acknowledgements

The authors cordially thank Prof. Maciej Błaszak, Prof. Błażej Szablikowski and Prof. Jan Cieśliński for fruitful discussions during the International Conference in Functional Analysis dedicated to the 125th anniversary of Stefan Banach held on September 18–23, 2017, in Lviv, Ukraine.

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Correspondence to Yarema Prykarpatsky.

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Hentosh, O., Prykarpatsky, Y. The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure. European Journal of Mathematics 6, 232–247 (2020). https://doi.org/10.1007/s40879-019-00329-4

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Keywords

  • Heavenly type equations
  • Lax–Sato integrability
  • Superconformal vector fields
  • Adler–Kostant–Symes theory
  • Casimir invariants

Mathematics Subject Classification

  • 37K05
  • 37K30
  • 37K65
  • 35Q75
  • 35Q35
  • 17B80
  • 58C50