Skip to main content
SpringerLink
Log in

The Direct Scattering Problem for theZakharov–Shabat System

  • Chapter
Book cover Integrable Hamiltonian Hierarchies

Part of the book series: Lecture Notes in Physics ((LNP,volume 748))

  • 1344 Accesses

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V. E. Zakharov and A. B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 34:62–69, 1972.

    ADS  MathSciNet  Google Scholar 

  2. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur. The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math., 53:249–315, 1974.

    MathSciNet  Google Scholar 

  3. H. Segur and M. J. Ablowitz. Solitons and the Inverse Scattering Transform. Society for Industrial & Applied Mathematics, Philadelphia, PA 1981.

    MATH  Google Scholar 

  4. M. J. Ablowitz, A. D. Trubatch, and B. Prinari. Discrete and Continuous Nonlinear Schrodinger Systems. Cambridge University Press, Cambridge, 2003.

    Book  Google Scholar 

  5. L. D. Faddeev and L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  6. V. A. Marchenko. Sturm-Liouville operators and applications. Birkhäuser, Basel, 1987.

    Google Scholar 

  7. B. M. Levitan. Inverse Sturm-Liouville Problems. VSP Architecture, Zeist, 1987.

    MATH  Google Scholar 

  8. F. Guerra and R. Marra. Origin of the quantum observable operator algebra in the frame of stochastic mechanics. Phys. Rev. D, 28(8):1916–1921, 1983.

    Article  ADS  MathSciNet  Google Scholar 

  9. F. Guerra and R. Marra. Discrete stochastic variational principles and quantum mechanics. Phys. Rev. D, 29(8):1647–1655, 1984.

    Article  ADS  MathSciNet  Google Scholar 

  10. F. Guerra and L. M. Morato. Quantization of dynamical systems and stochastic control theory. Phys. Rev. D, 27(8):1774–1786, 1983.

    Article  ADS  MathSciNet  Google Scholar 

  11. F. Calogero, editor. Nonlinear Evolution Equations Solvable by the Spectral Transform, volume 26 of Res. Notes in Math. Pitman, London, 1978.

    Google Scholar 

  12. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. I. Pitaevskii. Theory of Solitons: The Inverse Scattering Method. Plenum, New York, 1984.

    Google Scholar 

  13. M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering, volume 149 of London Mathematical Society Lecture Notes Series. ,n.Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  14. I. D. Iliev, E. Kh. Christov, and K. P. Kirchev. Spectral Methods in Soliton Equations, volume 73 of Pitman Monographs and Surveys in Pure and Applied Mathematics. John Wiley & Sons, New York, 1991.

    Google Scholar 

  15. M. A. Naimark. Linear Differential Operators. Nauka, Moskow, 1969.

    Google Scholar 

  16. N. Dunford and J. T. Schwartz. Linear Operators. Part 1, 2, 3. Wiley Interscience Publications, New York, 1971.

    Google Scholar 

  17. A. B. Shabat. Inverse-scattering problem for a system of differential equations. Funct. Anal. Appl., 9(3):244–247, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. B. Shabat. An inverse scattering problem. Diff. Equ., 15(10):1299–1307, 1979.

    MathSciNet  Google Scholar 

  19. A. V. Mikhailov. Reduction in integrable systems. The reduction group. JETP Lett., 32:174, 1980.

    ADS  Google Scholar 

  20. P. J. Caudrey. The inverse problem for the third order equation u xxx+ q(x) ux+ r(x)u=-iζ3 u. Phys. Lett. A, 79(4):264–268, 1980.

    Article  ADS  MathSciNet  Google Scholar 

  21. P. J. Caudrey. The inverse problem for a general N×N spectral equation. Physica D: Nonl. Phen., 6(1):51–66, 1982.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. R. Beals and R. R. Coifman. Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math., 37:39–90, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  23. R. Beals and R. R. Coifman. Inverse scattering and evolution equations. Commun. Pure Appl. Math., 38(1):29–42, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. Beals and R. R. Coifman. The D-bar approach to inverse scattering and nonlinear evolutions. Physica D, 18(1-3):242–249, 1986.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. R. Beals and R. R. Coifman. Scattering and inverse scattering for first-order systems: II. Inverse Probl., 3(4):577–593, 1987.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. R. Beals and R. R. Coifman. Linear spectral problems, non-linear equations and the overline partial-method. Inverse Probl., 5(2):87–130, 1989.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. V. S. Gerdjikov and A. B. Yanovski. Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system. J. Math. Phys., 35:3687–3721, 1994.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. V. S. Gerdjikov. On the spectral theory of the integro-ifferential operator generating nonlinear evolution equations. Lett. Math. Phys., 6:315–324, 1982.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. L. D. Faddeev. Properties of the S-matrix of the one-dimensional Schrödinger equation. Amer. Math. Soc. Transl. (Ser. 2), 65:139–166, 1967.

    MATH  Google Scholar 

  30. L. D. Faddeev. Inverse problem of quantum scattering theory. II. J. Math. Sci., 5(3):334–396, 1976. In “Contemporary Mathematical Problems”, English translation from: VINITI, 3, 93–180 (1974).

    Google Scholar 

  31. I. S. Frolov. Inverse scattering problem for a Dirac system on the whole axis. Soviet Math. Dokl, 13:1468–1472, 1972.

    MATH  Google Scholar 

  32. J. Satsuma and N. Yajima. Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media. Prog. Theor. Phys. Suppl., 55:284, 1974.

    Article  ADS  MathSciNet  Google Scholar 

  33. A. C. Newell. The general structure of integrable evolution equations. Proc. R. Soc. Lond. A, Math. Phys. Sci., 365(1722):283–311, 1979.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. E. C. Titchmarsch. Eigenfunctions Expansions Associated with Second Order Differential Equations. Part I. Clarendon Press, Oxford, 1983.

    Google Scholar 

  35. T. Kawata. Inverse scattering transform of the higher order eigenvalue problem. J. Phys. Soc. Japan, 57(2):422–435, 1988.

    Article  ADS  MathSciNet  Google Scholar 

  36. X. Zhou. Direct and inverse scattering transforms with arbitrary spectral singularities. Commun. Pure Appl. Math., 42:895–938, 1989.

    Article  MATH  Google Scholar 

  37. A. C. Scott, F. Y. F. Chu, and D. W. McLaughlin. The soliton: A new concept in applied science. Proc. IEEE, 61(10):1443–1483, 1973.

    Article  MathSciNet  ADS  Google Scholar 

  38. P. G. Drazin and R. S. Johnson. Solitons: An Introduction. Cambridge texts in Applied Mathematics. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  39. V. S. Gerdjikov and M. I. Ivanov. A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Russ. Bulg. J. Phys. 10, 130–143, 1983.

    MathSciNet  Google Scholar 

  40. V. S. Gerdjikov and M. I. Ivanov. The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. Phys., 10:130–143, 1983.

    MathSciNet  Google Scholar 

  41. I. T. Gadjiev, V. S. Gerdjikov, and M. I. Ivanov. Hamiltonian structures of the nonlinear evolution equations related to the polynomial bundle. Notes of LOMI Sci., 120:55–68, 1982.

    Google Scholar 

  42. L. A. Bordag and A. B. Yanovski. Algorithmic construction of O(3) chiral field equation hierarchy and the Landau–Lifshitz equation hierarchy via polynomial bundle. J. Phys. A: Math. Gen., 29(17):5575–5590, 1996.

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Bulgarian Academy of Sciences, Inst. Nuclear Research & Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Germany

    V.S. Gerdjikov

  2. Dipto. Fisica Ist. Nazionale di Fisica Nucleare, Università Salerno, 84081 Baronissi, Italy

    G. Vilasi

  3. Dept. Mathematics & Applied Mathematics, University of Cape Town, Rondebosch, 7701 South Africa

    A.B. Yanovski

Authors
  1. V.S. Gerdjikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Vilasi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A.B. Yanovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Inst. Nuclear Research & Nuclear Energy, Tsarigradsko Chaussee 72, 1784 Sofia, Germany

    V.S. Gerdjikov

  2. Dipto. Fisica Ist. Nazionale di Fisica Nucleare, Università Salerno, 84081 Baronissi, Italy

    G. Vilasi

  3. Dept. Mathematics & Applied Mathematics, University of Cape Town, Rondebosch, 7701 South Africa

    A.B. Yanovski

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gerdjikov, V., Vilasi, G., Yanovski, A. (2008). The Direct Scattering Problem for theZakharov–Shabat System. In: Gerdjikov, V., Vilasi, G., Yanovski, A. (eds) Integrable Hamiltonian Hierarchies. Lecture Notes in Physics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77054-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation