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Dynamics in Siberia - 2019
March 1, 2019 11:30–12:00, Novosibirsk, Sobolev Institute of Mathematics of Russian Academy of Sciences, Conference Hall

Plenary talks

2D-reductions of the Mikhalev–Pavlov equation and their nonlocal symmetries

I. S. Krasil'shchik

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Abstract: The Mikhalev–Pavlov equation (MPE) reads $u_{yy} = u_{tx} + u_y u_{xx} - u_x u_{xy}$ and belongs to the class of integrable linearly degenerate equations. It admits an infinite-dimensional Lie algebra of symmetries and all its 2D-symmetry reductions were described in [3]. Among these reductions, the following ones are of a special interest: (1) $u_{yy} = (u_y + 2x)u_{xx} +(y - u_x )u_{xy} - u$, (2) $u_{yy} = (u_y+y)u_{xx}-u_xu_{xy}-2$ (the last one is equivalent to the Gibbons-Tsarev equation). Under the reductions, the isospectral Lax pair of MPE transforms to rational differential coverings of the form
$$\label{eq:1} w_x=\frac{a_2w^2+a_1w+a_0}{w^2+c_1w+c_0},\qquad w_y=\frac{b_2w^2+b_1w+b_0}{w^2+c_1w+c_0}$$
for the reduced equations, where $a_i$, $b_i$, and $c_i$ are functions in $u$ and its derivatives, [1]. The standard “reversion procedure”, makes it possible to introduce a fake spectral parameter to \eqref{eq:1} and construct infinite series of conservation laws together with the corresponding infinite-dimensional coverings. Using the known description of the Lax pair for MPE, [2], it is proved that the algebras of nonlocal symmetries for reductions are isomorphic to the Witt algebra, cf. [4].
References
[1] H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Vojčá, Integrability properties of some equations obtained by symmetry reductions, J. of Nonlinear Math. Phys., 22, 2015, Issue 2, 210–232, https://doi.org/10.1080/14029251.2015.102358
[2] H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Vojčá, Nonlocal symmetries of integrable linearly degenerate equations: a comparative study, Theor. and Math. Phys., 196, 2018, Issue 2, 1089–1110, https://doi.org/10.1134/S0040577918080019
[3] H. Baran, I.S. Krasil'shchik, O.I. Morozov, P. Vojčá, Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems, J. of Nonlinear Math. Phys., 21, 2014, Issue 4, 643–671, https://doi.org/10.1080/14029251.2014.975532
[4] P. Holba, I.S. Krasil'shchik, O.I. Morozov, P. Vojčá, Reductions of the universal hierarchy and rddym equations and their symmetry properties, Lobachevskii J. of Math., 39,2018, Issue 5, 673–681, https://doi.org/10.1134/S1995080218050086

Language: English

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