2D reductions of the equation u_yy = u_tx + u_yu_xx − u_xu_xy and their nonlocal symmetries

P. HolbaMathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01Opava, Czech Republic. [email protected]

I.S. Krasil'shchikIndependent University of Moscow, B. Vlasevsky 11, 119002Moscow, Russia, Russian State University for Humanities, Miusskaya sq. 6, Moscow, GSP-3, 125993, Russia;Trapeznikov Institute of Control Sciences, 65 Profsoyuznaya street, Moscow117997, RussiaCorrespondence[email protected]

O.I. MorozovFaculty of Applied Mathematics, AGH University of Science and Technology, Al. Mickiewicza 30, Kraków30-059, Poland. [email protected]

P. VojčákMathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01Opava, Czech Republic. [email protected]

Abstract

We consider the 3D equation u_yy = u_tx + u_yu_xx − u_xu_xy and its 2D symmetry reductions: (1) u_yy = (u_y + y) u_xx − u_xu_xy − 2 (which is equivalent to the Gibbons-Tsarev equation) and (2) u_yy = (u_y + 2x)u_xx + (y − u_x)u_xy − u_x. Using the corresponding reductions of the known Lax pair for the 3D equation, we describe nonlocal symmetries of (1) and (2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.

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2010 Mathematics Subject Classification:

35B06

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2D reductions of the equation uyy = utx + uyuxx − uxuxy and their nonlocal symmetries

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2D reductions of the equation u_yy = u_tx + u_yu_xx − u_xu_xy and their nonlocal symmetries