Advertisement

Self-similar solutions of equations of the nonlinear Schrödinger type

  • V. G. Marikhin
  • A. B. Shabat
  • M. Boiti
  • F. Pempinelli
Nonlinear Physics

Abstract

A study is made of self-similar solutions of an entire family of one-dimensional integrable dynamic systems of the nonlinear Schrödinger equation type. This family is reduced to one of three canonical forms corresponding to a Toda chain, a Volterra chain, or to the Landau-Lifshitz model, which can also be reduced to three self-similar systems coupled by Miura transformations with the fourth Painleve equation. A commutative representation is constructed for this equation. A relationship is established between the poles of the rational solutions of the fourth Painleve equation and the steady-state distribution of the electric charges in a parabolic potential. A self-similar solution is constructed for the spin dynamics. An exact solution is obtained for the nonlinear Schrödinger equation with variable dispersion (optical soliton).

Keywords

Soliton Elementary Particle Electric Charge Canonical Form Equation Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Boiti and F. Pempinelli, Nuovo Cimento B 59, 40 (1980).MathSciNetGoogle Scholar
  2. 2.
    V. G. Marikhin and A. B. Shabat, Teor. Mat. Fiz. 118, 217 (1999).MathSciNetGoogle Scholar
  3. 3.
    A. V. Mikhailov and A. B. Shabat, Teor. Mat. Fiz. 62, 163 (1985).MathSciNetGoogle Scholar
  4. 4.
    Yu. N. Ovchinnikov, Pis’ma Zh. Éksp. Teor. Fiz. 69, 387 (1999) [JETP Lett. 69, 418 (1999)].MathSciNetGoogle Scholar
  5. 5.
    V. I. Gromak, Diff. Eqns. 23, 506 (1987).zbMATHMathSciNetGoogle Scholar
  6. 6.
    A. P. Veselov and A. B. Shabat, Funct. Anal. Appl. 27, 1 (1993).CrossRefMathSciNetGoogle Scholar
  7. 7.
    N. A. Lukashevich, Diff. Eqns. 3, 395 (1967).Google Scholar
  8. 8.
    V. E. Adler, Physica D 73, 335 (1994).CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    P. Estévez and P. Clarkson, E-print archives, solvint/9904002.Google Scholar
  10. 10.
    F. W. Nijhoff, G. R. W. Quispel, J. van der Linden, and H. W. Capel, Physica A 119, 101 (1983); G. R. W. Quispel and H. W. Capel, Phys. Lett. A 88, 371 (1982); Physica A 117, 76 (1983).CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    F. J. Bureau, Annali di Matematica (IV) 91, 163 (1972).zbMATHMathSciNetGoogle Scholar
  12. 12.
    S. Turitsyn, Phys. Rev. E 58, R1256 (1998).ADSGoogle Scholar

Copyright information

© MAIK "Nauka/Interperiodica" 2000

Authors and Affiliations

  • V. G. Marikhin
    • 1
  • A. B. Shabat
    • 1
  • M. Boiti
    • 2
  • F. Pempinelli
    • 2
  1. 1.Landau Institute of Theoretical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Dipartimento di Fisica dell’Università and SezioneLecceItaly

Personalised recommendations