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Exact and approximate symmetries for light propagation equations with higher order nonlinearity

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Abstract

For the first time exact analytical solutions to the eikonal equations in (1+1) dimensions with a refractive index being a saturated function of intensity are constructed. It is demonstrated that the solutions exhibit collapse; an explicit analytical expression for the self-focusing position, where the intensity tends to infinity, is found. Based on an approximated Lie symmetry group, solutions to the eikonal equations with arbitrary nonlinear refractive index are constructed. Comparison between exact and approximate solutions is presented. Approximate solutions to the nonlinear Schrödinger equation in (1 + 2) dimensions with arbitrary refractive index and initial intensity distribution are obtained. A particular case of refractive index consisting of Kerr refraction and multiphoton ionization is considered. It is demonstrated that the beam collapse can take place not only at the beam axis but also in an off-axis ring region around it. An analytical condition distinguishing these two cases is obtained and explicit formula for the self-focusing position is presented.

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References

  1. CRC Hanbook of Lie Group Analysis of Differential Equations, ed. N. H. Ibragimov (CRC Press Inc., Boca Raton, 1994–1996).

    Google Scholar 

  2. Y. R. Shen, The principles of nonlinear optics (JohnWiley, New York and Chichester, 1984).

    Google Scholar 

  3. S. A. Akhmanov, A. P. Sukhorikov, and R. V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Sov. Phys. Usp. 10, 609 (1968).

  4. R.W. Boyd, Nonlinear Optics (Academic Press, Amsterdam, Tokyo, 2003).

    Google Scholar 

  5. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Elsevier, Amsterdam, 2006).

    Google Scholar 

  6. A. Couairon and A. Mysyrowicz, Femtosecond filamentation in transparent media, Phys. Rep. 441, 47–189 (2007).

    Article  Google Scholar 

  7. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Ultrashort filaments of light in weakly-ionized, optically-transparent media, Rep.Prog. Phys. 70, 1633 (2007).

    Article  Google Scholar 

  8. S. A. Akhmanov, R. V. Khokhlov and A. P. Sukhorukov, On the self-focusing and self-chanelling of intense laser beams in nonlinear medium, Sov.Phys. JETP 23, 1025 (1966).

    Google Scholar 

  9. V. F. Kovalev and V. V. Pustovalov, Group and renormgroup symmetry of a simple model for nonlinear phenomena in optics, gas dynamics and plasma theory, Mathem. Comp.Modelling 25, 165 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  10. V. F. Kovalev, Renormgroup symmetries in problems of nonlinear geometrical optics, Theor.Math. Phys. 111, 686 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Gagnon and P. Winternitz, Exact solutions of the cubic and quintic nonlinear Schrödinger equation for a cylindrical geometry, Phys. Rev. A 39, 296 (1998).

    Article  MathSciNet  Google Scholar 

  12. L. Gagnon and P. Winternitz, Lie symmetries of a generalised nonlinear Schrödinger equation: I. The symmetry group and its subgroups, J. Phys.A:Math.Gen. 21, 1493 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  13. L. Gagnon and P. Winternitz, Lie symmetries of a generalised nonlinear Schrodinger equation. II. Exact solutions, J. Phys. A: Math. Gen. 22, 469 (1989).

    Article  MATH  MathSciNet  Google Scholar 

  14. V. I. Pulov, I. M. Uzunov, and E. J. Chacarov, Solutions and laws of conservation for coupled nonlinear Schrödinger equations: Lie group analysis, Phys. Rev. E 57, 3468 (1998).

    Article  MathSciNet  Google Scholar 

  15. V. F. Kovalev, Renonrmalization group analysis for singularities in the wave beam self-focusing problem, Theor. Math. Phys. 113, 719 (1990).

    MathSciNet  Google Scholar 

  16. V. F. Kovalev, V. Yu. Bychenkov, and V. T. Tikhonchuk, Renormalization-group approach to the problem of light-beam self-focusing, Phys. Rev. A 61, 033809 (1–10) (2000).

    Article  Google Scholar 

  17. L. L. Tatarinova and M. E. Garcia, Exact solutions of the eikonal equations describing self-focusing in highly nonlinear geometrical optics, Phys. Rev. A 78, 021806(R) (1–4) (2008).

    Google Scholar 

  18. D. V. Shirkov and V. F. Kovalev, The Bogoliubov renormalization group and solution symmetry in mathematical physics, Phys. Rep. 352, 219 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. F. Kovalev, Approximate transformation groups and renormgroup symmetries, Nonlinear Dynamics 22, 73 (2000).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Theoretical Physics, University of Kassel, Heinrich-Plett-Str. 40, 34132, Kassel, Germany

    M. E. Garcia & L. L. Tatarinova

  2. Institute for mathematical modelling RAS, Miusskaya Pl., 4-A, 125047, Moscow, Russia

    V. F. Kovalev

  3. Theoretical Physics, University of Fribourg, Chemin du Museé 3, 1700, Fribourg, Switzerland

    L. L. Tatarinova

Authors
  1. M. E. Garcia
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  2. V. F. Kovalev
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  3. L. L. Tatarinova
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Submitted by N.H. Ibragimov

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Garcia, M.E., Kovalev, V.F. & Tatarinova, L.L. Exact and approximate symmetries for light propagation equations with higher order nonlinearity. Lobachevskii J Math 31, 123–140 (2010). https://doi.org/10.1134/S1995080210020046

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  • DOI: https://doi.org/10.1134/S1995080210020046

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