

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


2Dreductions 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 infinitedimensional
Lie algebra of symmetries and all its 2Dsymmetry 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
GibbonsTsarev equation). Under the reductions, the isospectral Lax pair of
MPE transforms to rational differential coverings of the form
\begin{equation}\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}
\end{equation}
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
infinitedimensional 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 3dimensional
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

