Skip to main content
SpringerLink
Log in

Inverse Spectral Problem Related to the N-wave Equation

  • Conference paper
Differential Operators and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 117))

Abstract

A skewselfadjoint linear system of differential equations that is auxiliary for the well known N-wave equation is considered. Direct and inverse spectral problems for this system are formulated and solved in terms of the generalized Weyl functions.

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

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. M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform,SIAM Stud. Appl. Math. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1981.

    Book  MATH  Google Scholar 

  2. T. Ya. Azizov, Dissipative operators in Hilbert space with indefinite metrics, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 639–662.

    MathSciNet  Google Scholar 

  3. R. Beals and R.R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), 39–90.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Beals, P. Deift and C. Tomei, Direct and inverse scattering on the line, Math. Surveys and Monographs 28, AMS, 1988.

    MATH  Google Scholar 

  5. R. Beals, P. Deift and X. Zhou, The inverse scattering transform on the line. In: A.S. Fokas and V.E. Zakharov (Eds.), Important Developments in Soliton Theory. Springer Ser. Nonlinear Dynamics, Springer, Berlin, 1993, pp. 7–32.

    Google Scholar 

  6. R.P. Boas, Entire functions. Academic Press, New York, 1954.

    MATH  Google Scholar 

  7. P. Deift and X. Zhou, Direct and inverse scattering on the line with arbitrary singularities, Comm. Pure Appl. Math. 44 (1991), 485–533.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space. Amer. Math. Soc. Transl. 24, AMS, Providence, R.I., 1970.

    Google Scholar 

  9. I. Gohberg, M.A. Kaashoek and A L Sakhnovich, Pseudo-canonical systems with rational Weyl functions: explicit formulas and applications, J. Diff. Eqs. 146 (1998), 375–398.

    Article  MathSciNet  MATH  Google Scholar 

  10. M.G. Krein, Continuous analogues of propositions on polynomials orthogonal on the unit circle, Dokl. Akad. Nauk SSSR 105 (1955), 637–640.

    MathSciNet  Google Scholar 

  11. M.G. Krein, Topics in differential and integral equations and operator theory. OT 7, Birkhäuser Verlag, 1983.

    Google Scholar 

  12. Z.L. Leibenzon, The inverse problem of spectral analysis for higher order ordinary differential operators, Trudy Moskov. Mat. Obšč. 15 (1966), 70–144.

    MathSciNet  Google Scholar 

  13. M.M. Malamud, Spectral analysis of Volterra operators and inverse problems for systems of ordinary operators. SFB 288, Preprint no. 269, Berlin, 1997.

    Google Scholar 

  14. V.P. Potapov, The multiplicative structure of J -contractive matrix functions,Amer. Math. Soc. Transl. 15 (1960), 131–243.

    MathSciNet  MATH  Google Scholar 

  15. R. Paley and N. Wiener, Fourier transforms in the complex domain. Amer. Math. Soc., NY, 1934.

    MATH  Google Scholar 

  16. A.L. Sakhnovich, A nonlinear Schrödinger equation on the semiaxis and a related inverse problem, Ukrain. Math. J. 42:3 (1990), 316–323.

    Article  MathSciNet  MATH  Google Scholar 

  17. A.L. Sakhnovich, The N -wave problem on the semiaxis, Russ. Math. Surveys 46:4 (1991), 198–200.

    Article  MathSciNet  Google Scholar 

  18. A.L. Sakhnovich, The N -wave problem on the semiaxis,in: 16 All-Union school on the operator theory in functional spaces (lecture materials), Nydzni Novgorod (1992), 95–114.

    Google Scholar 

  19. A.L. Sakhnovich, Spectral theory for the systems of differential equations and applications. Thesis for secondary doctorship, Kiev, Institute of Mathematics, 1992.

    Google Scholar 

  20. L.A. Sakhnovich, Factorisation problems and operator identities, Russian Math. Surveys 41 (1986), 1–64.

    Article  MathSciNet  MATH  Google Scholar 

  21. L.A. Sakhnovich, Interpolation theory and its applications. Kluwer, Dordrecht, 1997.

    Book  MATH  Google Scholar 

  22. L.A. Sakhnovich, Spectral theory of canonical differential systems, method of operator identities. OT, Birkhäuser Verlag, to appear.

    Google Scholar 

  23. V.A. Yurko, An inverse problem for differential operators on the halfaxis,Sov. Math. Iz. VUZ 35 (1991), no. 12, 67–74.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Branch of Hydroacoustics, Marine Institute of Hydrophysics Academy of Sciences of Ukraine, Preobradzenskaya 3, 270100, Odessa, Ukraine

    Alexander Sakhnovich

Authors
  1. Alexander Sakhnovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, University of Odessa, 270026, Odessa, Ukraine

    V. M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    I. Gohberg

  3. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    M. Gorbachuk  & V. Gorbachuk  & 

  4. Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    M. A. Kaashoek

  5. Department of Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/1411, 1040, Vienna, Austria

    H. Langer

  6. Institute of Mathematics, Economics and Mechanics, Odessa State University, 270057, Dvoryanskaya str. 2, Odessa, Ukraine

    G. Popov

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this paper

Cite this paper

Sakhnovich, A. (2000). Inverse Spectral Problem Related to the N-wave Equation. In: Adamyan, V.M., et al. Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 117. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8403-7_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8403-7_24

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9552-1

  • Online ISBN: 978-3-0348-8403-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation