Theoretical and Mathematical Physics

, Volume 121, Issue 1, pp 1397–1408 | Cite as

Third version of the dressing method

  • A. B. Shabat


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.


Riccati Equation Spectral Problem Volterra Model Toda Chain Periodic Closure 


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  1. 1.
    V. E. Zakharov and A. B. Shabat,Funct. Anal. Appl.,8, 226–235 (1974);13, 166–174 (1980).CrossRefMATHGoogle Scholar
  2. 2.
    A. B. Shabat,Inverse Problems,6, 303–308 (1992).CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    A. P. Veselov, and A. B. Shabat,Funct. Anal. Appl.,27, No. 2, 81–96 (1993).CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    V. E. Adler,Physica D,73, 335–351 (1994).CrossRefADSMathSciNetMATHGoogle Scholar
  5. 5.
    V. E. Adler, and A. B. Shabat,Theor. Math. Phys.,111, 647–657 (1997);112, 935–948 (1997).MathSciNetMATHGoogle Scholar
  6. 6.
    V. G. Marikhin, and A. B. Shabat,Theor. Math. Phys.,118, 173–182 (1999).CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    A. B. Shabat, and R. I. Yamilov,Leningrad Math. J.,2, 377–400 (1991).MathSciNetMATHGoogle Scholar
  8. 8.
    A. N. Leznov, A. B. Shabat, and R. I. Yamilov,Phys. Lett. A,174, 397–402 (1993).CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    E. Schrödinger,Proc. Roy. Irish Acad.,A46, 9–16 (1940).MATHGoogle Scholar
  10. 10.
    L. D. Landau, and E. M. Lifshitz,Course of Theoretical Physics [in Russian], Vol. 3,Quantum Mechanics: Nonrelativistic Theory, Nauka, Moscow (1989); English transl. prev. ed., Oxford Univ. Press, Oxford (1975).Google Scholar
  11. 11.
    A. P. Veselov,Mat. Zametki,45, No. 3, 118–120 (1989).MathSciNetMATHGoogle Scholar
  12. 12.
    V. L. Vereshchagin,Mat. Zametki,48, No. 2, 145–148 (1990).MathSciNetMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • A. B. Shabat
    • 1
  1. 1.Landau Institute for Theoretical PhysicsRASMoscow OblastRussia

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