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

This article is cited in 15 scientific papers (total in 15 papers)

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
Received: 20.01.2012

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
\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

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    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  crossref  mathscinet  zmath  isi  scopus
    2. Meshkov A., Sokolov V., “Vector Hyperbolic Equations on the Sphere Possessing Integrable Third-Order Symmetries”, Lett. Math. Phys., 104:3 (2014), 341–360  crossref  mathscinet  zmath  isi  elib  scopus
    3. V. E. Adler, “Necessary integrability conditions for evolutionary lattice equations”, Theoret. and Math. Phys., 181:2 (2014), 1367–1382  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    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  crossref  mathscinet  zmath  isi  scopus
    5. M. Yu. Balakhnev, “Differential Substitutions for Vectorial Generalizations of the mKdV Equation”, Math. Notes, 98:2 (2015), 204–209  mathnet  crossref  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    7. V. E. Adler, “Integrability Test For Evolutionary Lattice Equations of Higher Order”, J. Symb. Comput., 74 (2016), 125–139  crossref  mathscinet  zmath  isi  scopus
    8. Ufa Math. J., 9:3 (2017), 158–164  mathnet  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  crossref  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    11. A. G. Meshkov, V. V. Sokolov, “On Third Order Integrable Vector Hamiltonian Equations”, J. Geom. Phys., 113 (2017), 206–214  crossref  mathscinet  zmath  isi  scopus
    12. R. N. Garifullin, R. I. Yamilov, “Ob integriruemosti reshetochnykh uravnenii s dvumya kontinualnymi predelami”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 159–164  mathnet  mathscinet
    13. Carillo S., “Kdv-Type Equations Linked Via Backlund Transformations: Remarks and Perspectives”, Appl. Numer. Math., 141:SI (2019), 81–90  crossref  mathscinet  zmath  isi  scopus
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    15. Carillo S., Lo Schiavo M., Schiebold C., “Abelian Versus Non-Abelian Backlund Charts: Some Remarks”, Evol. Equ. Control Theory, 8:1, SI (2019), 43–55  crossref  mathscinet  zmath  isi  scopus
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