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TMF, 1998, Volume 115, Number 2, Pages 199–214 (Mi tmf867)  

This article is cited in 33 scientific papers (total in 33 papers)

The dressing chain of discrete symmetries and proliferation of nonlinear equations

A. B. Borisov, S. A. Zykov

Institute of Metal Physics, Ural Division of the Russian Academy of Sciences

Abstract: In the examples of sine-Gordon and Korteweg–de Vries (KdV) equations, we propose a direct method for using dressing chains (discrete symmetries) to proliferate integrable equations. We give a recurrent procedure (with a finite number of steps in general) that allows the step-by-step production of an integrable system and its $L$$A$ pair from the known $L$$A$ pair of an integrable equation. Using this algorithm, we reproduce a number of known results for integrable systems of the KdV type. We also find a new integrable equation of the sine-Gordon series and investigate its simplest soliton solution of the double $\pi$-kink type.


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English version:
Theoretical and Mathematical Physics, 1998, 115:2, 530–541

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Received: 22.12.1997

Citation: A. B. Borisov, S. A. Zykov, “The dressing chain of discrete symmetries and proliferation of nonlinear equations”, TMF, 115:2 (1998), 199–214; Theoret. and Math. Phys., 115:2 (1998), 530–541

Citation in format AMSBIB
\by A.~B.~Borisov, S.~A.~Zykov
\paper The dressing chain of discrete symmetries and proliferation of nonlinear equations
\jour TMF
\yr 1998
\vol 115
\issue 2
\pages 199--214
\jour Theoret. and Math. Phys.
\yr 1998
\vol 115
\issue 2
\pages 530--541

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    This publication is cited in the following articles:
    1. A. B. Borisov, S. A. Zykov, “Reflectionless sine-Gordon potentials with an infinite spectrum”, Theoret. and Math. Phys., 118:3 (1999), 264–271  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. V. Yurov, “Conjugate chains of discrete symmetries in $(1+2)$ nonlinear equations”, Theoret. and Math. Phys., 119:3 (1999), 731–738  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    4. V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. S. B. Leble, “Covariance of Lax Pairs and Integrability of the Compatibility Condition”, Theoret. and Math. Phys., 128:1 (2001), 890–905  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. A. B. Borisov, “Bдcklund Transformation and Dressing Chains for the Landau–Lifshitz Equation”, Theoret. and Math. Phys., 128:2 (2001), 1025–1033  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. V. Zhiber, V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Russian Math. Surveys, 56:1 (2001), 61–101  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Borisov, AB, “Proliferation scheme for Kaup-Boussinesq system”, Physica D-Nonlinear Phenomena, 152 (2001), 104  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. A. B. Borisov, S. A. Zykov, M. V. Pavlov, “Tzitzéica Equation and Proliferation of Nonlinear Integrable Equations”, Theoret. and Math. Phys., 131:1 (2002), 550–557  mathnet  crossref  crossref  mathscinet  zmath  isi
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    11. M. V. Pavlov, “The Boussinesq equation and Miura type transformations”, J. Math. Sci., 136:6 (2006), 4478–4483  mathnet  crossref  mathscinet  zmath
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    13. M. V. Pavlov, “The description of pairs of compatible first-order differential geometric poisson brackets”, Theoret. and Math. Phys., 142:2 (2005), 244–258  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. Taras V. Skrypnyk, “Quasigraded Lie Algebras and Modified Toda Field Equations”, SIGMA, 2 (2006), 043, 14 pp.  mathnet  crossref  mathscinet  zmath
    15. Sazonov, SV, “Soliton regimes of ultrashort pulse propagation through an array of asymmetric quantum objects”, Journal of Experimental and Theoretical Physics, 103:4 (2006), 561  crossref  adsnasa  isi  scopus  scopus  scopus
    16. JETP Letters, 83:11 (2006), 483–487  mathnet  crossref  isi
    17. Skrypnyk, T, “Integrable deformations of the mKdV and SG hierarchies and quasigraded Lie algebras”, Physica D-Nonlinear Phenomena, 216:2 (2006), 247  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    18. S. V. Sazonov, N. V. Ustinov, “Integrable models of the dynamics of longitudinal-transverse acoustic pulses in a paramagnetic crystal”, Theoret. and Math. Phys., 151:2 (2007), 632–647  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. A. V. Kiselev, “Algebraic properties of Gardner's deformations for integrable systems”, Theoret. and Math. Phys., 152:1 (2007), 963–976  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. R. D. Murtazina, “Nelineinye giperbolicheskie uravneniya i kharakteristicheskie algebry Li”, Tr. IMM UrO RAN, 13, no. 4, 2007, 103–118  mathnet  elib
    21. Sazonov, SV, “Dynamics of ultrashort optical solitons in a system of nonsymmetric quantum objects”, Journal of Optical Technology, 74:11 (2007), 730  crossref  mathscinet  isi  scopus  scopus  scopus
    22. Ustinov, NV, “Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem”, European Physical Journal B, 58:3 (2007), 311  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    23. Sazonov, SV, “New kinds of acoustic solitons”, Journal of Physics A-Mathematical and Theoretical, 40:26 (2007), F551  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    24. A. G. Meshkov, “Nonlocal symmetries in two-field divergent evolutionary systems”, Theoret. and Math. Phys., 156:3 (2008), 1268–1279  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    25. R. D. Murtazina, “Kharakteristicheskie algebry Li medlennogo rosta i uravnenie MSG”, Ufimsk. matem. zhurn., 1:3 (2009), 111–118  mathnet  zmath
    26. A. G. Meshkov, V. V. Sokolov, “Hyperbolic equations with third-order symmetries”, Theoret. and Math. Phys., 166:1 (2011), 43–57  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    27. A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, “Kharakteristicheskie koltsa Li i integriruemye modeli matematicheskoi fiziki”, Ufimsk. matem. zhurn., 4:3 (2012), 17–85  mathnet  mathscinet
    28. 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  adsnasa  isi  scopus  scopus  scopus
    29. Kiseley A.V., Krutov A., “Gardner's Deformations as Generators of New Integrable Systems”, Physics and Mathematics of Nonlinear Phenomena 2013, Journal of Physics Conference Series, 482, IOP Publishing Ltd, 2014, 012021  crossref  isi  scopus
    30. Sazonov S.V., Ustinov N.V., “New Integrable Model of Propagation of the Few-Cycle Pulses in An Anisotropic Microdispersed Medium”, Physica D, 366 (2018), 1–9  crossref  mathscinet  zmath  isi  scopus
    31. Yurov A.V., Yurov V.A., “The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons””, Symmetry-Basel, 10:4 (2018), 82  crossref  isi  scopus  scopus  scopus
    32. Sazonov V S., Ustinov V N., “Propagation of Few-Cycle Pulses in a Nonlinear Medium and An Integrable Generalization of the Sine-Gordon Equation”, Phys. Rev. A, 98:6 (2018), 063803  crossref  isi  scopus
    33. Ustinov N.V., “Integrable Generalizations of the Sine-Gordon, Short Pulse, and Reduced Maxwell-Bloch Equations”, J. Math. Phys., 60:1 (2019), 013503  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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