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TMF, 1999, Volume 121, Number 1, Pages 165–176 (Mi tmf803)  

This article is cited in 18 scientific papers (total in 18 papers)

Third version of the dressing method

A. B. Shabat

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: The two previous versions of the dressing method in soliton theory were elaborated in collaboration with V. E. Zakharov. In our works published in the 1970s, we did not consider discrete equations on a lattice, Painlevé-type equations, and finite-band potentials, and I now make up for this gap. Although the suggested elementary discretization scheme for the linear spectral problem is rather general, the presentation of the material is mainly confined to the framework of the Zakharov–Shabat spectral problem.


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English version:
Theoretical and Mathematical Physics, 1999, 121:1, 1397–1408

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Received: 28.04.1999

Citation: A. B. Shabat, “Third version of the dressing method”, TMF, 121:1 (1999), 165–176; Theoret. and Math. Phys., 121:1 (1999), 1397–1408

Citation in format AMSBIB
\by A.~B.~Shabat
\paper Third version of the dressing method
\jour TMF
\yr 1999
\vol 121
\issue 1
\pages 165--176
\jour Theoret. and Math. Phys.
\yr 1999
\vol 121
\issue 1
\pages 1397--1408

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    This publication is cited in the following articles:
    1. M. Yu. Kulikov, V. S. Novikov, “Reduction of the dressing chain of the Schrödinger operator”, Theoret. and Math. Phys., 123:3 (2000), 768–775  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. 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
    3. Shabat A., “Dressing chains and lattices”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After Needs '79, 2000, 331–342  crossref  mathscinet  zmath  isi
    4. V. E. Adler, V. G. Marikhin, A. B. Shabat, “Lagrangian Chains and Canonical Bäcklund Transformations”, Theoret. and Math. Phys., 129:2 (2001), 1448–1465  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Shabat, AB, “Discretization of the Schrodinger spectral problem”, Inverse Problems, 18:4 (2002), 1003  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    6. Alonso, LM, “Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type”, Physics Letters A, 300:1 (2002), 58  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    7. Alonso, LM, “Energy-dependent potentials revisited: a universal hierarchy of hydrodynamic type”, Physics Letters A, 299:4 (2002), 359  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    8. Willox, R, “Painlevé equations from Darboux chains: I. P-III-P-V”, Journal of Physics A-Mathematical and General, 36:42 (2003), 10615  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    9. Yurov, AV, “Discrete symmetry's chains and links between integrable equations”, Journal of Mathematical Physics, 44:3 (2003), 1183  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    10. S. B. Leble, “Necessary Covariance Conditions for a One-Field Lax Pair”, Theoret. and Math. Phys., 144:1 (2005), 985–994  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. F. Musso, A. B. Shabat, “Elementary Darboux Transformations and Factorization”, Theoret. and Math. Phys., 144:1 (2005), 1004–1013  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Ruijsenaars, SNM, “Isometric reflectionless eigenfunction transforms for higher-order A Delta Os”, Journal of Nonlinear Mathematical Physics, 12 (2005), 565  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    13. M. D. Vereschagin, S. D. Vereschagin, A. V. Yurov, “Trekhmernoe preobrazovanie Mutara”, Matem. modelirovanie, 18:5 (2006), 111–125  mathnet  mathscinet  zmath
    14. Butler S., Joshi N., “An inverse scattering transform for the lattice potential KdV equation”, Inverse Problems, 26:11 (2010), 115012  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    15. Lue Da-Zhao, Cui Yan-Ying, Liu Chang-He, Zhang Meng, “Novel Interaction Solutions to Kadomtsev-Petviashvili Equation”, Commun Theor Phys (Beijing), 54:3 (2010), 484–488  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    16. Butler S., “Multidimensional Inverse Scattering of Integrable Lattice Equations”, Nonlinearity, 25:6 (2012), 1613–1634  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    17. Tsuda T., “From KP/Uc Hierarchies to Painlevé Equations”, Int. J. Math., 23:5 (2012), 1250010  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus  scopus
    18. 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
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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