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Towards an inverse scattering theory for two-dimensional nondecaying potentials

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Abstract

The inverse scattering method is considered for the nonstationary Schrödinger equation with the potentialu (x 1,x 2) nondecaying in a finite number of directions in thex plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the Bäcklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.

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References

  1. V. S. Dryuma,JETP Letters,19, 381 (1974).

    Google Scholar 

  2. V. E. Zakharov and A. B. Shabat,Funct. Anal. Appl.,8, 226 (1974).

    Article  Google Scholar 

  3. B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Doklady,192, 539 (1970).

    ADS  Google Scholar 

  4. M. J. Ablowitz and P. A. Clarkson, “Solitons, nonlinear evolution equations and inverse scattering,” in:Lecture Notes Series, Vol. 49, Cambridge Univ., Cambridge (1991).

    Google Scholar 

  5. B. G. Konopelchenko,Solitons in Multidimensions, World Scientific, Singapore (1993).

    MATH  Google Scholar 

  6. V. E. Zakharov and S. V. Manakov,Sov. Sci. Rev. A,1, 133 (1979); S. V. Manakov,Physica D,3, 420 (1981).

    Google Scholar 

  7. A. S. Fokas and M. J. Ablowitz,Stud. Appl. Math.,69, 211 (1983).

    MathSciNet  Google Scholar 

  8. M. J. Ablowitz and J. Villarroel,Stud. Appl. Math.,85, 195 (1991).

    MathSciNet  Google Scholar 

  9. M. Boiti, F. Pempinelli, and A. Pogrebkov,Inverse Problems,10, 505 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  10. M. Boiti, F. Pempinelli, and A. Pogrebkov,J. Math. Phys.,35, 4683 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  11. M. Boiti, J. Léon, and F. Pempinelli,Phys. Lett. A,141, 96 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  12. M. Boiti, F. Pempinelli, A. K. Progrebkov, and M. C. Polivanov,Inverse Problems,8, 331 (1992).

    Article  ADS  MathSciNet  Google Scholar 

  13. S. V. Manakov, V. E. Zakharov, L. A. Bordag, A. R. Its, and V. B. Matveev,Phys. Lett. A,63, 205 (1977).

    Article  ADS  Google Scholar 

  14. B. A. Dubrovin, T. M. Malanyuk, I. M. Krichever, and V. G. Makhankov,Sov. J. Part. Nucl.,19, 252 (1988).

    MathSciNet  Google Scholar 

  15. A. S. Fokas and V. E. Zakharov,J. Nonlinear Sci.,2, 109 (1992).

    Article  MathSciNet  Google Scholar 

  16. J. M. Ghidaglia and J. C. Saut,Nonlinearity,3, 475 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Boiti, J. Léon, L. Martina, and F. Pempinelli,Phys. Lett. A,132, 432 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  18. M. Boiti, L. Martina, and F. Pempinelli,Chaos. Solitons, and Fractals,5, 2377 (1995).

    Article  MathSciNet  Google Scholar 

  19. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov, “Resolvent approach for the nonstationary Schrödinger equation. Standard case of rapidly decreasing potential,” in:Nonlinear Evolution Equations and Dynamical Systems (M. Boiti, L. Martina, and F. Pempinelli, eds.), World Scientific, Singapore (1992), p. 97.

    Google Scholar 

  20. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,Theor. Math. Phys.,93, 1200 (1992).

    Article  MathSciNet  Google Scholar 

  21. M. Boiti, F. Pempinelli, and A. Pogrebkov,Theor. Math. Phys.,99, 511 (1994).

    Article  MathSciNet  Google Scholar 

  22. M. Boiti, F. Pempinelli, and A. Pogrebkov, “Spectral theory of solitons on a generic background for the KPI equation,” in:Nonlinear Physics. Theory and Experiment (E. Alfinito, M. Boiti, L. Martina, and F. Pempinelli, eds.), World Scientific, Singapore (1996), p. 37.

    Google Scholar 

  23. M. Boiti, F. Pempinelli, and A. Pogrebkov,Inverse Problems,13, L7 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  24. V. B. Matveev and M. A. Salle,Darboux Transformations and Solitons, Springer, Berlin (1991).

    MATH  Google Scholar 

  25. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov,Inverse Problems,7, 43 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  26. M. Boiti, F. Pempinelli, and A. Porgrebkov,Physica D,87, 123 (1995).

    Article  ADS  MathSciNet  Google Scholar 

  27. A. S. Fokas and A. K. Pogrebkov, unpublished.

  28. V. D. Lipovsky,Funct. Anal. Appl.,20, 282 (1986).

    Article  Google Scholar 

  29. Xin Zhou,Commun. Math. Phys.,128, 551 (1990).

    Article  ADS  Google Scholar 

  30. M. Boiti, B. Konopelchenko, and F. Pempinelli,Inverse Problems,1, 33 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  31. I. M. Gelfand and G. E. Shilov,Generalized Functions, Vol. 1,Properties and Operations, Acad. Press, New York (1964).

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Fisica dell'Università and Sezione INFN, Italy

    M. Boiti, F. Pempinelli & B. Prinari

  2. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    A. K. Pogrebkov

Authors
  1. M. Boiti
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  2. F. Pempinelli
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  3. A. K. Pogrebkov
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  4. B. Prinari
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Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 116. No. 1, pp. 3–53, July, 1998.

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Boiti, M., Pempinelli, F., Pogrebkov, A.K. et al. Towards an inverse scattering theory for two-dimensional nondecaying potentials. Theor Math Phys 116, 741–781 (1998). https://doi.org/10.1007/BF02557122

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  • DOI: https://doi.org/10.1007/BF02557122

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