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 Ufimsk. Mat. Zh., 2012, Volume 4, Issue 3, Pages 104–154 (Mi ufa158)

Integrable evolution equations with a constant separant

A. G. Meshkova, V. V. Sokolovb

a Orel State Technical University, Orel, Russia
b Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow. reg., Russia

Abstract: The survey contains results of classification for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on neccesary integrability conditions that follow from the existence of the formal recursion operator for integrable equations. Recursion formulas for the whole infinite sequence of these conditions are presented for the first time. The most of the classification statements can be found in papers by S. I. Svinilupov and V. V. Sokolov but the proofs never been published before. The result concerning the fifth order equations is stronger then obtained before.

Keywords: evolution differential equation, integrability, higher symmetry, conservation law, classification.

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UDC: 517.957

Citation: A. G. Meshkov, V. V. Sokolov, “Integrable evolution equations with a constant separant”, Ufimsk. Mat. Zh., 4:3 (2012), 104–154

Citation in format AMSBIB
\Bibitem{MesSok12} \by A.~G.~Meshkov, V.~V.~Sokolov \paper Integrable evolution equations with a~constant separant \jour Ufimsk. Mat. Zh. \yr 2012 \vol 4 \issue 3 \pages 104--154 \mathnet{http://mi.mathnet.ru/ufa158} 

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This publication is cited in the following articles:
1. De Sole A., Kac V.G., “Non-Local Poisson Structures and Applications to the Theory of Integrable Systems”, Jap. J. Math., 8:2 (2013), 233–347
2. Meshkov A., Sokolov V., “Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries”, Lett. Math. Phys., 104:3 (2014), 341–360
3. V. E. Adler, “Necessary integrability conditions for evolutionary lattice equations”, Theoret. and Math. Phys., 181:2 (2014), 1367–1382
4. Meshkov A.G., Sokolov V.V., “Integrable Evolution Hamiltonian Equations of the Third Order With the Hamiltonian Operator D-X”, J. Geom. Phys., 85 (2014), 245–251
5. M. Yu. Balakhnev, “Differential Substitutions for Vectorial Generalizations of the mKdV Equation”, Math. Notes, 98:2 (2015), 204–209
6. Meshkov A.G., Sokolov V.V., “Integrable Hamiltonian Equations of Fifth Order With Hamiltonian Operator D-X”, Russ. J. Math. Phys., 22:2 (2015), 201–214
7. V. E. Adler, “Integrability Test For Evolutionary Lattice Equations of Higher Order”, J. Symb. Comput., 74 (2016), 125–139
8. Ufa Math. J., 9:3 (2017), 158–164
9. A. V. Bochkarev, A. I. Zemlyanukhin, “The geometric series method for constructing exact solutions to nonlinear evolution equations”, Comput. Math. Math. Phys., 57:7 (2017), 1111–1123
10. I. T. Habibullin, A. R. Khakimova, “On a Method For Constructing the Lax Pairs For Integrable Models Via a Quadratic Ansatz”, J. Phys. A-Math. Theor., 50:30 (2017), 305206
11. A. G. Meshkov, V. V. Sokolov, “On Third Order Integrable Vector Hamiltonian Equations”, J. Geom. Phys., 113 (2017), 206–214
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