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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Volume 266, Pages 227–236
(Mi tm1876)
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This article is cited in 10 scientific papers (total in 10 papers)
Differential Transformations of Parabolic Second-Order Operators in the Plane
S. P. Tsareva, E. S. Shemyakovab a Institute of Mathematics, Siberian Federal University, Krasnoyarsk, Russia
b Research Institute for Symbolic Computation, J. Kepler University, Linz, Austria
Abstract:
Darboux's classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form $Lu=(D^2_x+a(x,y)D_x+b(x,y)D_y+c(x,y))u=0$. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.
Received in December 2008
Citation:
S. P. Tsarev, E. S. Shemyakova, “Differential Transformations of Parabolic Second-Order Operators in the Plane”, Geometry, topology, and mathematical physics. II, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 266, MAIK Nauka/Interperiodica, Moscow, 2009, 227–236; Proc. Steklov Inst. Math., 266 (2009), 219–227
Linking options:
https://www.mathnet.ru/eng/tm1876 https://www.mathnet.ru/eng/tm/v266/p227
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Abstract page: | 411 | Full-text PDF : | 61 | References: | 78 |
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