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TMF, 1998, Volume 116, Number 1, Pages 3–53 (Mi tmf888)  

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

Towards an inverse scattering theory for two-dimensional nondecaying potentials

M. Boitia, F. Pempinellia, A. K. Pogrebkovb, B. Prinaria

a Lecce University
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The inverse scattering method is considered for the nonstationary Schrödinger equation with the potential $u(x_{1},x_{2})$ nondecaying in a finite number of directions in the $x$ 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.

DOI: https://doi.org/10.4213/tmf888

Full text: PDF file (463 kB)

English version:
Theoretical and Mathematical Physics, 1998, 116:1, 741–781

Bibliographic databases:

Received: 15.12.1997

Citation: M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Towards an inverse scattering theory for two-dimensional nondecaying potentials”, TMF, 116:1 (1998), 3–53; Theoret. and Math. Phys., 116:1 (1998), 741–781

Citation in format AMSBIB
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\jour Theoret. and Math. Phys.
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    This publication is cited in the following articles:
    1. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Bäcklund and Darboux Transformations for the Nonstationary Schrödinger Equation”, Proc. Steklov Inst. Math., 226 (1999), 42–62  mathnet  mathscinet  zmath
    2. Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “N-wave soliton solution on a generic background for KPI equation”, International Seminar Day on Diffraction, Proceedings, 1999, 167–175  crossref  mathscinet  isi  scopus  scopus  scopus
    3. Prinari, B, “On some nondecaying potentials and related Jost solutions for the heat conduction equation”, Inverse Problems, 16:3 (2000), 589  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    4. Fokas, AS, “On the integrability of linear and nonlinear partial differential equations”, Journal of Mathematical Physics, 41:6 (2000), 4188  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    5. Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “Some nondecaying potentials for the nonstationary Schrodinger equation”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After Needs '79, 2000, 33–41  crossref  mathscinet  zmath  isi
    6. Boiti M., Pempinelli F., Prinari B., Pogrebkov A.K., “Some nondecaying potentials for the heat conduction equation”, Proceedings of the Workshop on Nonlinearity, Integrability and All That: Twenty Years After Needs '79, 2000, 42–50  crossref  mathscinet  zmath  isi
    7. Boiti, M, “Towards an inverse scattering theory for non-decaying potentials of the heat equation”, Inverse Problems, 17:4 (2001), 937  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    8. Boiti, M, “Inverse scattering transform for the perturbed 1-soliton potential of the heat equation”, Physics Letters A, 285:5–6 (2001), 307  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    9. Boiti, M, “Extended resolvent and inverse scattering with an application to KPI”, Journal of Mathematical Physics, 44:8 (2003), 3309  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    10. O. M. Kiselev, “Asymptotics of solutions of higher-dimensional integrable equations and their perturbations”, Journal of Mathematical Sciences, 138:6 (2006), 6067–6230  mathnet  crossref  mathscinet  zmath  elib
    11. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrödinger Equation with a Bidimensionally Perturbed One-Dimensional Potential”, Proc. Steklov Inst. Math., 251 (2005), 6–48  mathnet  mathscinet  zmath
    12. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Spectral Theory of the Nonstationary Schrodinger Equation with a Two-Dimensionally Perturbed Arbitrary One-Dimensional Potential”, Theoret. and Math. Phys., 144:2 (2005), 1100–1116  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Boiti, M, “Scattering transform for nonstationary Schrodinger equation with bidimensionally perturbed N-soliton potential”, Journal of Mathematical Physics, 47:12 (2006), 123510  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    14. Boiti, M, “On the extended resolvent of the nonstationary Schrodinger operator for a Darboux transformed potential”, Journal of Physics A-Mathematical and General, 39:8 (2006), 1877  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus  scopus
    15. M. Boiti, F. Pempinelli, A. K. Pogrebkov, B. Prinari, “Building an extended resolvent of the heat operator via twisting transformations”, Theoret. and Math. Phys., 159:3 (2009), 721–733  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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