Exact self-similar solution to a generalized nonlocal nonlinear Schrödinger model

Abstract

Abstract:

Exact self-similar solution of a generalized nonlinear Schrödinger equation with varying cubic-quintic nonlinearity, weakly nonlocality, gain and nonlinear gain was obtained. The stability of the solution was studied numerically. The results show that the self-similar solitary wave can exist and propagate in the media with or without both nonlocality and quintic nonlinearity, and that the stability of the self-similar solitary wave is drastically influenced by the degree of nonlocality and the cumulative diffraction under the condition that the phase parameter is far from .

Key words: nonlinear optics, self-similar solution, weakly nonlocal nonlinear Schrödinger model, nonlinear gain

CLC Number:

O437.5

ZHANG Shao-Wu, Yi-Lin. Exact self-similar solution to a generalized nonlocal nonlinear Schrödinger model[J]. J4, 2009, 26(4): 465-472.

References

[1] Barenblatt G I. Scaling, Self-Similarity, and Intermediate Asymptotics
[M]. Cambridge:Cambridge University Press, 1996.

[2] Olver P J. Applications of Lie Groups to Differential Equations
[M]. New York: Springer, 1986.

[3] Turitsyn S K. Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton
[J]. Phys. Rev. E, 1998, 58: R1256.

[4] Kruglov V I. Peacock A C, and Harvey J D. Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients
[J]. Phys. Rev. Lett., 2003, 90: 113902.

[5] Chen S, Yi L, Guo D, and Lu P. Self-similar evolutions of parabolic, HG, and hybrid optical pulses: Universality and diversity
[J]. Phys. Rev. E, 2005, 72: 016622.

[6] Chen S, Yi L. Chirped self-similar solutions of a generalized nonlinear Schrödinger equation model
[J]. Phys. Rev. E, 2005, 71: 016606.

[7] Raju T S, Panigrahi P K, and Porezian K. Self-similar propagation and compression of chirped self-similar waves in asymmetric twin-core fibers with nonlinear gain
[J]. Phys. Rev. E, 2005, 72: 046612.

[8] Wadati M. Bäcklund transformation for solutions of the modified Korteweg-de Vries equation
[J]. J. Phys. Soc. Japan., 1974, 36: 1498.

[9] Ablowitz M J and Clarkson P A. Nonlinear evolution equations and inverse scattering
[M]. Cambridge: Cambridge University Press, 1999.

[10] Dong Z, Yu X, and Wang L. Exact traveling wave solutions of the nonlinear coupled KdV equations
[J]. Chinese Journal of Quantum Electronics (量子电子学报), 2006, 23(3): 379 (in Chinese).

[11] Zheng B. New soliton solutions to 2+1 dimensional breaking soliton equation
[J]. Chinese Journal of Quantum Electronics (量子电子学报), 2006, 23(4): 451 (in Chinese).

[12] Zhang Shao and Yi Lin. Exact solutions of a generalized nonlinear Schrödinger equation
[J]. Phys. Rev. E, 2008, 78: 026602.

[13] Shih M-f, Segev M, and Salamo G. Three-dimensional spiraling of interacting spatial solitons
[J]. Phys. Rev. Lett., 1997, 78: 2551.

[14] Werner A G J, Hensler S, Stuhler J, and Pfau T. Bose-Einstein Condensation of Chromium
[J]. Phys. Rev. Lett., 2005, 94: 160401.

[15] Lindl J D. Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain
[J]. Phys. Plasmas, 1995, 2(11): 3933.

[16] Korabel N and Klages R. Fractal Structures of Normal and Anomalous Diffusion in Nonlinear Nonhyperbolic Dynamical Systems
[J]. Phys. Rev. Lett., 2002, 89: 214102.

[17] Fermann M E, Kruglov V I. Thomsen B C, et al. Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers
[J]. Phys. Rev. Lett., 2000, 84: 6010.

[18] Chang G, Winful H G, Galvanauskas A, and Norris T B. Self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients
[J]. Phys. Rev. E, 2005, 72: 016609.

[19] Peccianti M, Brzdakiewicz K A, and Assanto G. Nonlocal spatial soliton interactions in nematic liquid crystals
[J]. Opt. Lett., 2002, 27: 1460.

[20] Królikowski W and Bang O. Solitons in nonlocal nonlinear media: Exact solutions
[J]. Phys. Rev. E, 2000, 63: 016610.

[21] Conti C, Peccianti M, and Assanto G. Observation of optical spatial solitons in a highly nonlocal medium
[J]. Phys. Rev. Lett., 2004, 92: 113902.

[22] Zhong W and Yi L. Two-dimensional Laguerre-Gaussian soliton family in strongly nonlocal nonlinear media
[J]. Phys. Rev. A, 2007, 75: R061801.

[23] Mihalache D, Mazilu D, Lederer F, et al. Stable solitons of even and odd parities supported by competing nonlocal nonlinearities
[J]. Phys. Rev. E, 2006, 74: 066614.

[24] Królikowski W, Bang O, Rasmussen J J, and Wyller J. Modulational instability in nonlocal nonlinear Kerr media
[J]. Phys. Rev. E, 2001, 64: 016612.

[25] Agrawal G P. Nonlinear Fiber Optics. Boston: Academic Press, 2006: 4th ed., Chap.2.