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Soliton dynamics in an extended nonlinear Schrödinger equation with a spatial counterpart of the stimulated Raman scattering

Published online by Cambridge University Press:  30 July 2013

E. M. GROMOV  and
B. A. MALOMED
E. M. GROMOV
Affiliation:
National Research University Higher School of Economics, 25/12 Bolshaja Pecherskaja Ulitsa, Nizhny Novgorod 603155, Russia (egromov@hse.ru)
B. A. MALOMED
Affiliation:
Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Abstract

The dynamics of solitons is considered in the framework of the extended nonlinear Schrödinger equation (NLSE), which is derived from a system of Zakharov's type for the interaction between high-frequency (HF) and low-frequency (LF) waves, in which the LF field is subject to diffusive damping. The model may apply to the propagation of HF waves in plasmas. The resulting NLSE includes a pseudo-stimulated-Raman-scattering (PSRS) term, i.e. a spatial-domain counterpart of the SRS term, which is well known as an ingredient of the temporal-domain NLSE in optics. Also included is inhomogeneity of the spatial second-order diffraction (SOD). It is shown that the wavenumber downshift of solitons, caused by the PSRS, may be compensated by an upshift provided by the SOD whose coefficient is a linear function of the coordinate. An analytical solution for solitons is obtained in an approximate form. Analytical and numerical results agree well, including the predicted balance between the PSRS and the linearly inhomogeneous SOD.

Type
Papers
Information
Journal of Plasma Physics , Volume 79 , Special Issue 6: Special issue in memory of Professor Padma Kant Shukla 1950-2013 , December 2013 , pp. 1057 - 1062
Copyright
Copyright © Cambridge University Press 2013 

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References

Agrawal, G. P. 2001 Nonlinear Fiber Optics. San Diego, CA: Academic Press.Google Scholar
Andrianov, A., Muraviev, S., Kim, A. and Sysoliatin, A. 2007 DDF-based all-fiber optical source of femtosecond pulses smoothly tuned in the telecommunication range. Laser Phys. 17, 1296–1302.CrossRefGoogle Scholar
Biancalama, F., Skrybin, D. V. and Yulin, A. V. 2004 Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers. Phys. Rev. E 70, 011615.Google Scholar
Blit, R. and Malomed, B. A. 2012 Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides. Phys. Rev. A 86, 043841.CrossRefGoogle Scholar
Chernikov, S., Dianov, E., Richardson, D. and Payne, D. 1993 Soliton pulse compression in dispersion-decreasing fiber. Opt. Lett. 18, 476–478.CrossRefGoogle ScholarPubMed
Dauxois, T. and Peyrard, M. 2006 Physics of Solitons. Cambridge, UK: Cambridge University Press.Google Scholar
Dickey, L. A. 2005 Soliton Equation and Hamiltonian Systems. New York: World Scientific.Google Scholar
Essiambre, R.-J. and Agrawal, G. P. 1997a Timing jitter of ultrashort solitons in high-speed communication systems. I. General formulation and application to dispersion-decreasing fibers. J. Opt. Soc. Am. B 14, 314–322.CrossRefGoogle Scholar
Essiambre, R.-J. and Agrawal, G. P. 1997b Timing jitter of ultrashort solitons in high-speed communication systems. II. Control of jitter by periodic optical phase conjugation. J. Opt. Soc. Am. B 14, 323–330.CrossRefGoogle Scholar
Gordon, J. P. 1986 Theory of the soliton self–frequency shift. Opt. Lett. 11, 662–664.CrossRefGoogle ScholarPubMed
Gromov, E. M., Piskunova, L. V. and Tyutin, V. V. 1999 Dynamics of wave packets in the frame of third-order nonlinear Schrödinger equation. Phys. Lett. A 256, 153–158.CrossRefGoogle Scholar
Gromov, E. M. and Talanov, V. I. 1996 Nonlinear dynamics of short wave trains in dispersive media. Sov. Phys. JETP 83, 73–79.Google Scholar
Gromov, E. M. and Talanov, V. I. 2000 Short optical solitons in fibers. Chaos 10, 551–558.CrossRefGoogle ScholarPubMed
Hasegawa, A. and Tappert, F. 1973 Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: anomalous dispersion. Appl. Phys. Lett. 23, 142–144.CrossRefGoogle Scholar
Hong, B. and Lu, D. 2009 New Jacobi functions solitons for the higher-order nonlinear Schrödinger equation. Inter. J. Nonlin. Sci. 7, 360–367.Google Scholar
Infeld, E. and Rowlands, G. 2000 Nonlinear Waves, Solitons, and Chaos. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Karpman, V. I. 2004 The extended third-order nonlinear Schrödinger equation and Galilean transformation Eur. Phys. J. B 39, 341–350.CrossRefGoogle Scholar
Kivshar, Y. S. 1990 Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers. Phys. Rev. A 42, 1757–1761.CrossRefGoogle ScholarPubMed
Kivshar, Y. S. and Agrawal, G. P. 2003 Optical Solitons: From Fibers to Photonic Crystals. San Diego, CA: Academic Press.Google Scholar
Kivshar, Y. S. and Malomed, B. A. 1993 Raman-induced optical shocks in nonlinear fibers. Opt. Lett. 18, 485–487.CrossRefGoogle ScholarPubMed
Kodama, Y. J. 1985 Optical solitons in a monomode fiber. Stat. Phys. 39, 597–614.CrossRefGoogle Scholar
Kodama, Y. and Hasegawa, A. 1987 Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. 23, 510–524.CrossRefGoogle Scholar
Malomed, B. A. 2006 Soliton Management in Periodic Systems. New York: Springer.Google Scholar
Malomed, B. A. and Tasgal, R. S. 1998 Matching intrapulse self-frequency shift to sliding-frequency filters for transmission of narrow solitons. J. Opt. Soc. Am. B 15, 162–170.CrossRefGoogle Scholar
Marklund, M., Shukla, P. K. and Stenflo, L. 2006 Ultrashort solitons and kinetic effects in nonlinear metamaterials. Phys. Rev. E 73, 037601.Google ScholarPubMed
Mitschke, F. M. and Mollenauer, L. F. 1986 Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661.CrossRefGoogle ScholarPubMed
Obregon, M. A. and Stepanyants, Yu. A. 1998 Oblique magneto-acoustic solitons in a rotating plasma. Phys. Lett. A 249, 315–323.CrossRefGoogle Scholar
Oliviera, J. R. and Moura, M. A. 1998 Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation. Phys. Rev. E 57, 4751–4755.Google Scholar
Scalora, M., Syrchin, M., Akozbek, N., Poliakov, E. Y., D'Aguanno, G., Mattiucci, N., Bloemer, M. J. and Zheltikov, A. M. 2005 Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials. Phys. Rev. Lett. 95, 013902.CrossRefGoogle ScholarPubMed
Tsitsas, N. L., Rompotis, N., Kourakis, I., Kevrekidis, P. G. and Frantzeskakis, D. J. 2009 Higher-order effects and ultra-short solitons in left-handed metamaterials. Phys. Rev. Lett. E 79, 037601.Google Scholar
Wen, S. C., Wang, Y., Su, W., Xiang, Y., Fu, X. and Fan, D. 2006 Modulation instability in nonlinear negative-index material. Phys. Rev. E 73, 036617.Google ScholarPubMed
Yang, Y. 2001 Solitons in Field Theory and Nonlinear Analysis. New York: Springer.CrossRefGoogle Scholar
Zakharov, V. E. 1971 Hamiltonian formalism for hydrodynamic plasma model. Sov. Phys. JETP 33, 927–932.Google Scholar
Zakharov, V. E. 1974 The Hamiltonian formalism for waves in nonlinear media having dispersion. Radiophys. Quantum Electron. 17, 326–343.CrossRefGoogle Scholar
Zakharov, V. E. and Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69.Google Scholar
Zaspel, C. E. 1999 Optical solitary wave and shock solutions of the higher order nonlinear Schrödinger equation. Phys. Rev. Lett. 82, 723–726.CrossRefGoogle Scholar