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Integrability of Anti-Self-Dual Vacuum Einstein Equations with Nonzero Cosmological Constant: An Infinite Hierarchy of Nonlocal Conservation Laws

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Abstract

We present an infinite hierarchy of nonlocal conservation laws for the Przanowski equation, an integrable second-order PDE locally equivalent to anti-self-dual vacuum Einstein equations with nonzero cosmological constant. The hierarchy in question is constructed using a nonisospectral Lax pair for the equation under study. As a by product, we obtain an infinite-dimensional differential covering over the Przanowski equation.

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Notes

  1. Recall that an equation is called differentially connected if the only functions that are invariant with respect to all total derivatives on \(\mathscr {E}\) are constants.

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Acknowledgements

The work of IK was partially supported by the RFBR Grant 18-29-10013 and IUM-Simons Foundation. The research of AS was supported in part by the Ministry of Education, Youth and Sport of the Czech Republic (MŠMT ČR) under RVO funding for IČ47813059 and the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028. AS is pleased to thank A. Borowiec, M. Dunajski and K. Krasnov for stimulating discussions. AS also warmly thanks the Institute of Theoretical Physics of the University of Wrocław, and especially A. Borowiec, for the warm hospitality extended to him in the course of his visits to Wrocław, where some parts of the present article were worked on. It is our great pleasure to thank the anonymous referee for useful and relevant comments.

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Authors and Affiliations

  1. V.A. Trapeznikov Institute of Control Sciences RAS, Profsoyuznaya 65, Moscow, 117342, Russia

    I. Krasil’shchik

  2. Independent University of Moscow, B. Vlasevsky 11, Moscow, 119002, Russia

    I. Krasil’shchik

  3. Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01, Opava, Czech Republic

    A. Sergyeyev

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  1. I. Krasil’shchik
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  2. A. Sergyeyev
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Correspondence to A. Sergyeyev.

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Communicated by Nikolai Kitanine.

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Krasil’shchik, I., Sergyeyev, A. Integrability of Anti-Self-Dual Vacuum Einstein Equations with Nonzero Cosmological Constant: An Infinite Hierarchy of Nonlocal Conservation Laws. Ann. Henri Poincaré 20, 2699–2715 (2019). https://doi.org/10.1007/s00023-019-00816-0

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