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 TMF, 2018, Volume 196, Number 2, Pages 169–192 (Mi tmf9452)

Nonlocal symmetries of integrable linearly degenerate equations: A comparative study

H. Barana, I. S. Krasil'shchikbc, O. I. Morozovd, P. Vojčáka

a Mathematical Institute, Silesian University in Opava, Opava, Czech Republic
b V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia
c Independent University of Moscow, Moscow, Russia
d Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

Abstract: We continue the study of Lax integrable equations. We consider four three-dimensional equations{: (1)} the rdDym equation $u_{ty}=u_xu_{xy}- u_yu_{xx}$, $(2)$ the Pavlov equation $u_{yy}=u_{tx}+u_yu_{xx}-u_xu_{xy}$, $(3)$ the universal hierarchy equation $u_{yy}=u_tu_{xy}-u_yu_{tx}$, and $(4)$ the modified Veronese web equation $u_{ty}=u_tu_{xy}-u_yu_{tx}$. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.

Keywords: partial differential equation, integrable linearly degenerate equation, nonlocal symmetry, recursion operator.

 Funding Agency Grant Number Dobrushin Foundation Ministry of Science and Higher Education (Poland) Netherlands Enterprise Agency IČ47813059 The research of I. S. Krasilshchik was supported in part by a grant “Dobrushin Professorship–2017.” The research of O. I. Morozov was supported by the Polish Ministry of Science and Higher Education. The research of H. Baran and P. Vojčák was supported by RVO funding for IČ47813059.

DOI: https://doi.org/10.4213/tmf9452

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English version:
Theoretical and Mathematical Physics, 2018, 196:2, 1089–1110

Bibliographic databases:

Citation: H. Baran, I. S. Krasil'shchik, O. I. Morozov, P. Vojčák, “Nonlocal symmetries of integrable linearly degenerate equations: A comparative study”, TMF, 196:2 (2018), 169–192; Theoret. and Math. Phys., 196:2 (2018), 1089–1110

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf9452
• https://doi.org/10.4213/tmf9452
• http://mi.mathnet.ru/eng/tmf/v196/i2/p169

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This publication is cited in the following articles:
1. Holba P., Krasil'shchik I.S., Morozov O.I., Vojcak P., “Reductions of the Universal Hierarchy and Rddym Equations and Their Symmetry Properties”, Lobachevskii J. Math., 39:5 (2018), 673–681
2. 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 Poincare, 20:8 (2019), 2699–2715
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