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Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

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Abstract

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation uty = uxuxy − uyuxx, (2) the Pavlov equation uyy = utx + uyuxx − uxuxy, (3) the universal hierarchy equation uyy = utuxy − uyutx, and (4) the modified Veronese web equation uty = utuxy − uyutx. 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.

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Authors and Affiliations

  1. Mathematical Institute, Silesian University in Opava, Opava, Czech Republic

    H. Baran & P. Vojčák

  2. Trapeznikov Institute of Control Sciences, Moscow, Russia

    I. S. Krasilshchik

  3. Independent University of Moscow, Moscow, Russia

    I. S. Krasilshchik

  4. Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

    O. I. Morozov

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  1. H. Baran
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  2. I. S. Krasilshchik
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  3. O. I. Morozov
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  4. P. Vojčák
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Correspondence to H. Baran.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 196, No. 2, pp. 169–192, August, 2018.

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.

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Baran, H., Krasilshchik, I.S., Morozov, O.I. et al. Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study. Theor Math Phys 196, 1089–1110 (2018). https://doi.org/10.1134/S0040577918080019

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