Abstract
In this work we study numerically the self-frequency shift of the equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and intrapulse Raman scattering (IRS). It has been established that the stationary value of the central frequency of solutions, which appears as a result of the collective action of these physical effects, is a linearly increasing function of the Raman parameter γ for the solution of Tsoy and Akhmediev (Phys Lett A 343:17–422, 2005), Tsoy et al. (Phys Rev E 73:036621, 2006), and the quadratic function of γ for the parameters used in Uzunov et al. (Phys Rev E 90:042906, 2014). We have found a complete suppression of the self-frequency shift of the equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and IRS below and at the point of the Poincare–Andronov–Hopf bifurcation (PAHB) (Uzunov et al. in Phys Rev E 90:042906, 2014). We have numerically observed stable pairs and sequences of equidistant equilibrium solutions and pulsating solutions propagating in the presence of linear and nonlinear gain, spectral filtering, and IRS below and at the point of the PAHB. The pairs and equidistant sequences of pulsating solutions at the point of bifurcation require larger initial separation and exist at smaller distances of propagation than the pairs and equidistant sequences of the equilibrium solutions below the point of bifurcation.
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References
Afanasiev, V.V., Serkin, V.N., Vysloukh, V.A.: Amplification and compression of femtosecond optical solitons in active fibers. Sov. Lightwave Commun. 2, 35–38 (1992)
Afanasjev, V.V.: Soliton singularity in the system with nonlinear gain. Opt. Lett. 20, 704–706 (1995)
Afanasjev, V.V., Akhmediev, N.N.: Soliton interaction and bound states in amplified-damped fiber systems. Opt. Lett. 20, 1970–1972 (1995)
Afanasjev, V.V., Akhmediev, N.N.: Soliton interaction in nonequilibrium dynamical systems. Phys. Rev. E 53, 6471–6475 (1996)
Agrawal, G.P.: Nonlinear fiber optics, 3rd edn. Academic Press, San Diego (2001)
Akhmediev, N.N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)
Akhmediev, N.N., Ankiewicz, A., Soto-Crespo, J.M.: Multisoliton solutions of the complex Ginzburg–Landau equation. Phys. Rev. Lett. 79, 4047–4051 (1997)
Akhmediev, N., Soto-Crespo, J.M., Town, G.: Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg–Landau equation approach. Phys. Rev. E 63, 056602 (2001)
Akhmediev, N., Soto-Crespo, J.M., Grelu, P.: Vibrating and shaking soliton pairs in dissipative systems. Phys. Lett. A 364, 413–416 (2007)
Bélanger, P.-A., Gagnon, L., Pare, C.: Solitary pulses in an amplified nonlinear dispersive medium. Opt. Lett. 14, 943–945 (1989)
Blow, K.J., Doran, N.J., Wood, D.: Suppression of the soliton self-frequency shift by bandwidth-limited amplification. J. Opt. Soc. Am. B 5, 1301–1304 (1988)
Chang, W., Ankiewicz, A., Akhmediev, N.N., Soto-Crespo, J.M.: Creeping solitons in dissipative systems and their bifurcations. Phys. Rev. E 76, 016607 (2007)
Conte, R., Musette, M.: Exact solitons to the complex Ginzburg–Landau equation of non-linear optics. Pure Appl. Opt. 4, 315–320 (1995)
Conte, R., Musette, M.: Solitary waves of nonlinear nonintegrable equations. In: Akhmediev, N., Ankievicz, A. (eds.) Dissipative Solitons. Springer, Berlin (2005)
Cross, M.C., Hohenberg, P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 854–1112 (1993)
Gerdjikov, V.S., Uzunov, I.M., Evstatiev, E.G., Diankov, G.L.: The nonlinear Schrödinger and N-soliton interaction. Generalization of Karpman–Solov’ev approach. Phys. Rev. E 55, 6039–6060 (1997)
Gorshkov, K.A. Ph.D. thesis, Institute of Applied Physics, Gorky, (1981) unpublished
Gorshkov, K.A., Ostrovsky, L.A.: Interactions of solitons in nonintegrable systems: direct perturbation method and applications. Phys. D 3, 428–438 (1981)
Hasegawa, A., Kodama, Y.: Solitons in Optical Communications. Clarendon Press, Oxford (1995)
Haus, H.A., Fujimoto, J.G., Ippen, E.P.: Structures for additive pulse mode locking. J. Opt. Soc. Am. B 8, 2068–2076 (1991)
Heidt, A.: Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers. J. Lightwave Technology 27(18), 3984–3991 (2009)
Karpman, V.I., Solov’ev, V.V.: A perturbation theory for soliton systems. Phys. D 3, 142–164 (1981)
Kärtner, F.X., Au, J.A., Keller, U.: Mode-locking with slow and fast saturable absorbers—what‘s the difference. IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998)
Kodama, Y., Wabnitz, S.: Reduction and suppression of soliton interactions by bandpass filters. Opt. Lett. 18, 1311–1313 (1993)
Kodama, Y., Romagnoli, M., Wabnitz, S.: Soliton stability and interactions in fibre lasers. Electron. Lett. 28, 1981–1982 (1992)
Latas, S.C.V., Ferreira, M.F.S.: Soliton propagation in the presence of intrapulse Raman scattering and nonlinear gain. Opt. Commun. 251, 415–422 (2005)
Malomed, B.A.: Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation. Phys. Rev. A 44, 6954–6960 (1991)
Mancas, S.C., Choudhury, S.R.: A novel variational approach to pulsating soliitons in the cubic–quintic Ginzburg–Landau equation. Theor. Math. Phys. 152(2), 339–355 (2007)
Mancas, S.C., Choudhury, S.R.: Spatiotemporal structure of pulsating solitons in the cubic–quintic Ginzburg–Landau equation: a novel variational formulation. Chaos Solitons Fractals 40, 91–105 (2009)
Matsumoto, M., Ikeda, H., Uda, T., Hasegawa, A.: Stable soliton transmission in the system with nonlinear gain. J. Lightwave Technol. 13, 658–665 (1995)
Mitschke, F.M., Mollenauer, L.F.: Discovery of the soliton self frequency shift. Opt. Lett. 11, 659–661 (1986)
Nakazawa, M., Kurokawa, K., Kubota, H., Yamada, E.: Observation of the trapping of an optical soliton by adiabatic gain narrowing and its shape. Phys. Rev. Lett. 65, 1881–1884 (1990)
Pereira, N.R., Stenflo, L.: Nonlinear Schrodinger equation including growth and damping. Phys. Fluids 20, 1733–1743 (1977)
Serkin, V.N.: Colored envelope solitons in optical fibers. Sov. Tech. Phys. Lett. 13, 320–321 (1987)
Soto-Crespo, J.M., Akhmediev, N., Afanasjev, V.V.: Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Am. B 13, 1439–1449 (1996)
Soto-Crespo, J.M., Akhmediev, N., Afanasjev, V.V., Wabnitz, S.: Pulse solutions of the quintic complex Ginzburg–Landau equation in the case of normal dispersion. Phys. Rev. E 55, 4783–4796 (1997)
Soto-Crespo, J.M., Akhmediev, N., Ankiewicz, A.: Pulsating, creeping, and erupting solitons in dissipative systems. Phys. Rev. Lett. 85, 2937–2940 (2000)
Soto-Crespo, J.M., Grapinet, M., Grelu, P., Akhmediev, N.: Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser. Phys. Rev. E 70, 066612 (2004)
Tsoy, E., Akhmediev, N.: Bifurcations from stationary to pulsating solitons in the cubic quintic complex Ginzburg Landau equation. Phys. Lett. A 343, 417–422 (2005)
Tsoy, E., Ankiewicz, A., Akhmediev, N.: Dynamical models for dissipative localized waves of the complex Ginzburg Landau equation. Phys. Rev. E 73, 036621 (2006)
Uzunov, I.M.: Description of the suppression of the soliton self-frequency shift by bandwidth-limited amplification. Phys. Rev. E 82, 066603 (2010)
Uzunov, I.M., Arabadzhiev, T.N.: Suppression of the soliton self-frequency shift by BLA. Phys. Rev. E 84, 026607 (2011)
Uzunov, I.M., Arabadzhiev, T.N., Georgiev, Z.D.: Influence of higher-order effects on pulsating solutions, stationary solutions and moving fronts in the presence of linear and nonlinear gain/loss and spectral filtering. Opt. Fiber Technol. doi:10.1016/j.yofte.2015.04.003 (2015)
Uzunov, I.M., Muschall, R., Gölles, M., Lederer, F., Wabnitz, S.: Effect of nonlinear gain and filtering on soliton interactions. Opt. Commun. 118, 577–580 (1995)
Uzunov, I.M., Gerdjikov, V.S., Gölles, M., Lederer, F.: On the description of N-soliton interaction in optical fibers. Opt. Commun. 125, 237–242 (1996)
Uzunov, I.M., Georgiev, Z.D., Arabadzhiev, T.N.: Influence of intrapulse Raman scattering on the stationary pulses in the presence of linear and nonlinear gain as well as spectral filtering. Phys. Rev. E 90, 042906 (2014)
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Uzunov, I.M., Arabadzhev, T.N. & Georgiev, Z.D. Self-frequency shift and nonlinear interaction of equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and intrapulse Raman scattering. Opt Quant Electron 47, 2969–2981 (2015). https://doi.org/10.1007/s11082-015-0184-4
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DOI: https://doi.org/10.1007/s11082-015-0184-4