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Infinitely many nonlocal conservation laws for the ABC equation with \(A+B+C\ne 0\)

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Abstract

We construct an infinite hierarchy of nonlocal conservation laws for the ABC equation \(A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0\), where A, B, C are nonzero constants and \(A+B+C\ne 0\), using a nonisospectral Lax pair. As a byproduct, we present new coverings for the equation in question. The method of proof of nontriviality of the conservation laws under study is quite general and can be applied to many other integrable multidimensional systems.

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Notes

  1. In the present paper we mean by integrability existence of a nontrivial Lax pair for the system under study, cf. e.g. [1, 11, 39] and references therein for details.

  2. See e.g. [11, Subsection 10.3.3] and [13] and references therein for the general construction leading from a covering of the type (3) to a Lax pair of the type (5), and [11, 26] and references therein for nonisospectral Lax pairs in general.

  3. By a slight abuse of terminology, we speak of a differential equation even though it could actually be a system of differential equations.

  4. Or, more generally, on solutions of some differential equations whose coefficients depend on formal solutions of \(\mathcal {E}\).

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Acknowledgments

The research of AS was supported in part by the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028 and by the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR) under RVO funding for IČ47813059. The work of ISK was partially supported by the Simons-IUM fellowship. OIM gratefully acknowledges financial support from the Polish Ministry of Science and Higher Education. AS is pleased to thank E.V. Ferapontov and R.O. Popovych for stimulating discussions.This research was initiated in the course of visits of OIM to Silesian University in Opava and of AS to the AGH University of Science and Technology. The authors thank the universities in question for warm hospitality extended to them. The authors thank the editor and the anonymous referee for useful suggestions.

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Authors and Affiliations

  1. Independent University of Moscow, B. Vlasyevskiy per. 11, Moscow, 119002, Russia

    I. S. Krasil’shchik

  2. Russian State University for the Humanities, Miusskaya sq. 6, GSP-3, Moscow, 125993, Russia

    I. S. Krasil’shchik

  3. Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 74601, Opava, Czech Republic

    A. Sergyeyev

  4. Faculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, 30059, Kraków, Poland

    O. I. Morozov

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  1. I. S. Krasil’shchik
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  2. A. Sergyeyev
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  3. O. I. Morozov
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Correspondence to A. Sergyeyev.

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Communicated by F. Hélein.

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Krasil’shchik, I.S., Sergyeyev, A. & Morozov, O.I. Infinitely many nonlocal conservation laws for the ABC equation with \(A+B+C\ne 0\) . Calc. Var. 55, 123 (2016). https://doi.org/10.1007/s00526-016-1061-0

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