Skip to main content
SpringerLink
Log in

Solvability of the derivative nonlinear Schrödinger equation and the massive thirring model

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

Here we review some results of J.-H. Lee of theN×N Zakharov-Shabat system with a polynomial spectral parameter. We define a scattering transform following the set-up of Beals-Coifman [2]. In the 2×2 cases, we modify the Kaup-Newell and Kuznetsov-Mikhailov system to assure the normalization with respect to the spectral parameter. Then we are able to apply the technique of Zakharov-Shabat for the solitons of NLS to our cases. We obtain the long-time behavior of the equations which can be transformed into DNLS and MTM in laboratory coordinates respectively.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Beals andR. R. Coifman, Scattering, transformation spectrales, et equations d'evolution non lineaires Seminaire Goulaouic-Meyer-Schwartz. 1980–1981, exp. 22, Ecole Polytechnique, Palaiseau.

    Google Scholar 

  2. R. Beals andR. R. Coifman, Scattering and inverse scattering for first order system Comm. Pure Appl. Math. 1984. V. 37, P. 39–90.

    Google Scholar 

  3. R. Beals andR. R. Coifman, Inverse scattering and evolution equations Comm. Pure Appl. Math. 1985. V. 38, P. 29–42.

    Google Scholar 

  4. R. Beals andR. R. Coifman, Scattering and inverse scattering for first-order system: II Inverse Problem. 1987. V. 3. P. 577–593.

    Google Scholar 

  5. R. Beals andR. R. Coifman, Direct and Inverse Scattering on the line Math. Surveys and Mon<ographs. V. 28 A.M.S., Providence, 1988.

    Google Scholar 

  6. R. Beals andR. R. Coifman, Linear spectral Problems, non-linear equations and D-BAR-method Inverse Problems. 1989. V. 5. P. 87–130.

    Google Scholar 

  7. I. V. Barashenkov, B. S. Getmanov, and V. E. Kovtun, The unified approach to integrable relativistic equations: soliton solutions over non-vanishing backgrounds, in “Nonlinear Fields: Classical Random Semiclassical” edited by P. Garbaczewski and Z. Popowicz World Scientific, 1992.

  8. R. K. Bullough and P. J. Caudrey (eds), Solitons, Topics in Current Physics No. 17, Springer-Verlag, 1980.

  9. A. S. Fokas and V. E. Zakharov, Important Developments in Soliton Theory Springer-Verlag, 1993.

  10. A. P. Fordy, Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces J. Phys. A: Math. Gen. 1984. V. 17. P. 1235–45.

    Google Scholar 

  11. V. S. Gerzhikov et al., Quadratic Bundle and Nonlinear Equations Theoret. and Math. Phys. 1980. V. 44, No.3. P. 784–795.

    Google Scholar 

  12. D. J. Kaup andA. C. Newell, An exact solution for a derivative nonlinear Schrodinger equation J. Math. Phys. 19(4), April 1978. P. 789–801.

    Google Scholar 

  13. D. J. Kaup andA. C. Newell, On the Coleman correspondence and the solution of the Massive Thirring Model Lettere Al Nuovo Comento. 1977. V. 20, No.9. P. 325–331.

    Google Scholar 

  14. E. A. Kuznetsov andA. B. Mikhailov, On the complete integrability of the two-dimensional classical Thirring Model Thet. Math. Physics. 1977. V. 30, No. 3. P. 193–200.

    Google Scholar 

  15. J. H. Lee, Analytic properties of Zakharov-Shabat inverse scattering problem with polynomial spectral dependence of degree 1 in the potential Ph.D. dissertation, Yale University, May 1983.

  16. J. H. Lee, A Zakharov-Shabat inverse scattering problem and the associated evolution equations Chinese J. of Mathematics. 1984. V. 12, No.4. P. 223–233.

    Google Scholar 

  17. J. H. Lee, Analytic properties of a Zakharov-Shabat inverse scattering problem (I) Chinese J. of Mathematics. 1986. V. 14, No.4 Dec. P. 225–248.

    Google Scholar 

  18. J. H. Lee, Analytic properties of a Zakharov-Shabat inverse scattering problem (II) Chinese J. of Mathematics. 1987. V. 16, No.2. P. 81–110.

    Google Scholar 

  19. J. H. Lee, Global solvability of the derivative nonlinear Schrödinger equation Transactions of the Americal Mathematical Society. 1989. V. 314, No. 1. P. 107–118.

    Google Scholar 

  20. J. H. Lee, On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter Transactions of the A.M.S. 1989. V. 316, No. 1. P. 327–336.

    Google Scholar 

  21. J. H. Lee, AN×N Zakharov-Shabat System with a quadratic spectral parameter Proc. of the 5th workshop on nonlinear evolution equations and dynamical systems, Kolymbari, Crete, Greece, July 2–16, Springer-Verlag.

  22. J. H. Lee, AN×N Zakharov-Shabat system with a quadratic spectral parameter and Skew-Hermitian potentials Preprint.

  23. J. H. Lee, Inverse scattering transform and D-Bar method NSC Report, Taipei. 1989–90.

  24. J. H. Lee, Solvability of the massive Thirrng model preprint.

  25. J. H. Lee, AN×N Zakharov-Shabat system with a polynomial spectral parameter to appear in the proc. of 7th workshop on nonlinear evolution equations and dynamical systems, Gallipolli, Lecce, Italy. V. June 19–29, 1991.

  26. V. G. Makhankov, Soliton phenomenology Mathematics and its Applications (Soviet Series) 33, Kluwer Academic Publishers Group, Dordrecht, 1980.

    Google Scholar 

  27. C. T. Lin, Inverse scattering transofrm and the associated evolution equations Master's thesis, National Taiwan University, May 1992.

  28. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons Springer-Verlag 1991.

  29. W. Mio et al., Modified nonlinear Schrodinger equation for Alfven waves propagating along the magnetic field in cold plasmas J. Phys Soc. Japan. 1976. V. 41. P. 265–271.

    Google Scholar 

  30. E. Mjolhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field J. Plasma Physics. 1976. V. 16, part. 3. P. 321–334.

    Google Scholar 

  31. H. C. Morris andR. K. Dodd, The two component derivative nonlinear Schrodinger equation Physica Scripta. 1979. V. 20. P. 505–508.

    Google Scholar 

  32. R. Sasaki, Canonical structure of soliton equations, II. The Kaup-Newell system Physica V. 5D. P. 66–74.

  33. J. C. Shaw andH. C. Yen, Solving DNLS equation by Riemann problem technique Chinese J. of Math. 1990. V. 18. P. 283–293.

    Google Scholar 

  34. V. E. Zakharov andA. B. Shabat, A refined theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in non-linear media Soviet Physics JETP. 1972. V. 34. P. 62–69.

    Google Scholar 

  35. X. Zhou, Riemann-Hilbert problem and inverse scattering SIAM J. Math. Anal. 1989. V. 20. P. 966–986.

    Google Scholar 

  36. X. Zhou, Direct and inverse transform with arbitrary spectral singularities Comm. Pure Applied Math. 1989. V. XL II. P. 895–938.

    Google Scholar 

  37. X. Zhou, Inverse scattering transform for systems with rational spectral dependence. P. Preprint.

Download references

Authors
  1. Jyh-Hao Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Institute of Mathematics, Academia Sinica, Taipei 11529, Taiwan, R.O.C. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 322–328, May, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, JH. Solvability of the derivative nonlinear Schrödinger equation and the massive thirring model. Theor Math Phys 99, 617–621 (1994). https://doi.org/10.1007/BF01016148

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01016148

Keywords

Search

Navigation