Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 083, 9 pages      arXiv:0708.3506      https://doi.org/10.3842/SIGMA.2007.083
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Stability Analysis of Continuous Waves in Nonlocal Random Nonlinear Media

Maxim A. Molchan
B.I. Stepanov Institute of Physics, 68 Nezalezhnasci Ave., 220072 Minsk, Belarus

Received July 26, 2007, in final form August 15, 2007; Published online August 26, 2007

Abstract
On the basis of the competing cubic-quintic nonlinearity model, stability (instability) of continuous waves in nonlocal random non-Kerr nonlinear media is studied analytically and numerically. Fluctuating media parameters are modeled by the Gaussian white noise. It is shown that for different response functions of a medium nonlocality suppresses, as a rule, both the growth rate peak and bandwidth of instability caused by random parameters. At the same time, for a special form of the response functions there can be an ''anomalous'' subjection of nonlocality to the instability development which leads to further increase of the growth rate. Along with the second-order moments of the modulational amplitude, higher-order moments are taken into account.

Key words: nonlocality; competing nonlinearity; stochasticity.

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References

  1. Kivshar Yu.S., Luther-Davies B., Dark optical solitons: physics and applications, Phys. Rep. 298 (1998), 81-197.
  2. Mamyshev P.V., Bosshard C., Stegeman G.I., Generation of a periodic array of dark spatial solitons in the regime of effective amplification, J. Opt. Soc. Amer. B Opt. Phys. 11 (1994), 1254.
  3. Hasewaga A., Observation of self-trapping instability of a plasma cyclotron wave in a computer experiment, Phys. Rev. Lett. 24 (1970), 1165-1168.
  4. Whitham G.B., Nonlinear dispersive waves, Proc. R. Soc. London A 283 (1965), 238-261.
  5. Benjamin T.B., Feir J.E., The disintegration of wave trains on deep water, J. Fluid Mech. 27 (1967), 417-430.
  6. Wu B., Niu Q., Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices, Phys. Rev. A 64 (2001), 061603, 4 pages, cond-mat/0009455.
  7. Kevrekidis P.G., Frantzeskakis D.J., Pattern forming dynamical instabilities of Bose-Einstein condensates, Modern Phys. Lett. B 18 (2004), 173-202, cond-mat/0406657.
  8. Stegeman G.I., Segev M., Optical spatial solitons and their interactions: universality and diversity, Science 286 (1999), 1518-1522.
  9. Tai K., Hasegawa A., Tomita A., Observation of modulational instability in optical fibers, Phys. Rev. Lett. 56 (1986), 135-138.
  10. Kivshar Yu.S., Agraval G.P., Optical solitons: from fibers to photonic crystals, Academic Press, San Diego, 2003.
  11. Abdullaev F.Kh., Darmanyan S.A., Garnier J., Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, Progr. Opt. 44 (2002), 303-365.
  12. Peccianti M., Conti C., Alberici E., Assanto G., Spatially incoherent modulational instability in a nonlocal medium, Laser Phys. Lett. 2 (2005), 25-29, physics/0409139.
  13. Dreischuh A., Paulus G.G., Zacher F., Grasbon F., Walther H., Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium, Phys. Rev. E 60 (1999), 6111-6117.
  14. Suter D., Blasberg T., Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium, Phys. Rev. A 48 (1993), 4583-4587.
  15. Perez-Garcia V.M., Konotop V.V., Garcia-Ripoll J.J., Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions, Phys. Rev. E 62 (2000), 4300-4308.
  16. Turitsyn S.K., Spatial dispersion of nonlinearity and stability of multidimensional solitons, Teoret. Mat. Fiz. 64 (1985), 226-232 (English transl.: Theoret. and Math. Phys. 64 (1985), 797-801).
  17. Bang O., Krolikowski W., Wyller J., Rasmussen J.J., Collapse arrest and soliton stabilization in nonlocal nonlinear media, Phys. Rev. E 66 (2002), 046619, 5 pages, nlin.PS/0201036.
  18. Dreischuh A., Neshev D., Peterson D.E., Bang O., Krolikowski W., Observation of attraction between dark solitons, Phys. Rev. Lett. 96 (2006), 043901, 4 pages, physics/0504003.
  19. Krolikowski W., Bang O., Rasmussen J.J., Wyller J., Modulational instability in nonlocal nonlinear Kerr media, Phys. Rev. E 64 (2001), 016612, 8 pages, nlin.PS/0105049.
  20. Krolikowski W., Bang O., Nikolov N.I., Neshev D., Wyller J., Rasmussen J.J., Edmundson D., Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media, J. Opt. B Quantum Semiclass. Opt. 6 (2004), S288-S294, nlin.PS/0402040.
  21. Roussignol P., Ricard D., Lukasik J., Flytzanis C., New results on optical phase conjugation in semiconductor-doped glasses, J. Opt. Soc. Amer. B Opt. Phys. 4 (1987), 5.
  22. Lederer F., Biehlig W., Bright solitons and light bullets in semiconductor waveguides, Electron. Lett. 30 (1994), 1871-1872.
  23. Lawrence B., Cha M., Torruellas W.E., Stegeman G.I., Etemad S., Baker G., Kajzar F., Measurement of the complex nonlinear refractive index of single crystal p-toluene sulfonate at 1064 nm, Appl. Phys. Lett. 64 (1994), 2773-2775.
  24. Doktorov E.V., Molchan M.A., Modulational instability in nonlocal Kerr-type media with random parameters, Phys. Rev. A 75 (2007), 053819, 6 pages, arXiv:0704.1093.
  25. Wyller J., Krolikowski W., Bang O., Rasmussen J.J., Generic features of modulational instability in nonlocal Kerr media, Phys. Rev. E 66 (2002), 066615, 13 pages, nlin.PS/0105049.
  26. Novikov E.A., Functionals and the random force method in turbulence theory, Zh. Exp. Teor. Fiz. 47 (1964), 1919-1926 (English transl.: Sov. Phys. JETP 20 (1964), 1290-1295).

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