Highly controllable stabilization and switching of multiple colliding soliton sequences with generic Ginzburg-Landau gain-loss. (English) Zbl 07868614
Summary: We investigate propagation of \(\boldsymbol{J}\) soliton sequences in a nonlinear optical waveguide array with generic weak Ginzburg-Landau (GL) gain-loss and nearest-neighbor (NN) interaction. The propagation is described by a system of \(\boldsymbol{J}\) perturbed coupled nonlinear Schrödinger (NLS) equations. The NN interaction property leads to the elimination of collisional three-pulse interaction effects, which prevented the observation of stable multisequence soliton propagation with \(\boldsymbol{J} > 2\) sequences in the presence of generic GL gain-loss in all previous studies. We show that the dynamics of soliton amplitudes can be described by a generalized \(\boldsymbol{J}\)-dimensional Lotka-Volterra (LV) model. Stability and bifurcation analysis for the equilibrium points of the LV model, which is augmented by an application of the Lyapunov function method, is used to develop setups that lead to robust and scalable transmission stabilization and switching for a general \(\boldsymbol{J}\) value. The predictions of the LV model are confirmed by extensive numerical simulations with the perturbed coupled-NLS model with \(\boldsymbol{J} = 3\), 4, and 5 soliton sequences. Furthermore, soliton stability and the agreement between the LV model’s predictions and the simulations are not reduced with increasing value of \(\boldsymbol{J}\). Therefore, our study provides the first demonstration of robust control of multiple colliding sequences of NLS solitons in the presence of generic weak GL gain-loss with an arbitrary number of sequences. Due to the robustness and scalability of the results, they can have important applications in stabilization and switching of broadband soliton-based optical waveguide transmission.
MSC:
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 35Q51 | Soliton equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q56 | Ginzburg-Landau equations |
| 35C08 | Soliton solutions |
Keywords:
soliton; nonlinear Schrödinger equation; multisequence propagation; Lotka-Volterra model; nonlinear gain and loss; soliton stabilizationReferences:
| [1] | Kivshar, Y. S.; Malomed, B. A., Rev. Mod. Phys., 61, 763, 1989 |
| [2] | Asano, N.; Taniuti, T.; Yajima, N., J. Math. Phys., 10, 2020, 1969 |
| [3] | Horton, W.; Ichikawa, Y. H., Chaos and Structure in Nonlinear Plasmas, 1996, World Scientific: World Scientific Singapore |
| [4] | Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons: The Inverse Scattering Method, 1984, Plenum: Plenum New York · Zbl 0598.35002 |
| [5] | Newell, A. C., Solitons in Mathematics and Physics, 1985, SIAM: SIAM Philadelphia · Zbl 0565.35003 |
| [6] | Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Rev. Mod. Phys., 71, 463, 1999 |
| [7] | Carretero-González, R.; Frantzeskakis, D. J.; Kevrekidis, P. G., Nonlinearity, 21, R139, 2008 · Zbl 1216.82023 |
| [8] | Agrawal, G. P., Nonlinear Fiber Optics, 2019, Academic: Academic San Diego, CA |
| [9] | Hasegawa, A.; Kodama, Y., Solitons in Optical Communications, 1995, Clarendon: Clarendon Oxford · Zbl 0840.35092 |
| [10] | Iannone, E.; Matera, F.; Mecozzi, A.; Settembre, M., Nonlinear Optical Communication Networks, 1998, Wiley: Wiley New York |
| [11] | Mollenauer, L. F.; Gordon, J. P., Solitons in Optical Fibers: Fundamentals and Applications, 2006, Academic: Academic San Diego, CA |
| [12] | Agrawal, G. P., Applications of Nonlinear Fiber Optics, 2020, Academic: Academic San Diego, CA |
| [13] | Hasegawa, A., Front. Phys., 10, Article 1044845 pp., 2022 |
| [14] | Forghieri, F.; Tkach, R. W.; Chraplyvy, A. R., (Kaminow, I. P.; Koch, T. L., Optical Fiber Telecommunications, Vol. III, 1997, Academic: Academic San Diego, CA), (Chapter 8) |
| [15] | Essiambre, R.-J.; Kramer, G.; Winzer, P. J.; Foschini, G. J.; Goebel, B., J. Lightwave Technol., 28, 662, 2010 |
| [16] | Nguyen, Q. M.; Peleg, A., Opt. Commun., 283, 3500, 2010 |
| [17] | Peleg, A.; Nguyen, Q. M.; Chung, Y., Phys. Rev. A, 82, Article 053830 pp., 2010 |
| [18] | Peleg, A.; Chung, Y., Phys. Rev. A, 85, Article 063828 pp., 2012 |
| [19] | Chakraborty, D.; Peleg, A.; Jung, J.-H., Phys. Rev. A, 88, Article 023845 pp., 2013 |
| [20] | Nguyen, Q. M.; Peleg, A.; Tran, T. P., Phys. Rev. A, 91, Article 013839 pp., 2015 |
| [21] | Chakraborty, D.; Peleg, A.; Nguyen, Q. M., Opt. Commun., 371, 252, 2016 |
| [22] | Peleg, A.; Nguyen, Q. M.; Tran, T. P., Opt. Commun., 380, 41, 2016 |
| [23] | Peleg, A.; Nguyen, Q. M.; Huynh, T. T., Eur. Phys. J. D, 71, 30, 2017 |
| [24] | Peleg, A.; Chakraborty, D., Commun. Nonlinear Sci. Numer. Simul., 63, 145, 2018 · Zbl 1510.78051 |
| [25] | Peleg, A.; Nguyen, Q. M.; Glenn, P., Phys. Rev. E, 89, Article 043201 pp., 2014 |
| [26] | van Saarloos, W.; Hohenberg, P. C., Phys. D, 56, 303, 1992 · Zbl 0763.35088 |
| [27] | Aranson, I. S.; Kramer, L., Rev. Mod. Phys., 74, 99, 2002 · Zbl 1205.35299 |
| [28] | Newell, A. C.; Passot, T.; Lega, J., Annu. Rev. Fluid Mech., 25, 399, 1993 |
| [29] | Stewartson, K.; Stuart, J. T., J. Fluid Mech., 48, 529, 1971 · Zbl 0222.76045 |
| [30] | Malomed, B. A.; Nepomnyashchy, A. A., Phys. Rev. A, 42, 6009, 1990 |
| [31] | Moores, J. D., Opt. Commun., 96, 65, 1993 |
| [32] | Akhmediev, N. N.; Afanasjev, V. V.; Soto-Crespo, J. M., Phys. Rev. E, 53, 1190, 1996 |
| [33] | Kutz, J. N., SIAM Rev., 48, 629, 2006 · Zbl 1121.35130 |
| [34] | Renninger, W. H.; Chong, A.; Wise, F. W., Phys. Rev. A, 77, Article 023814 pp., 2008 |
| [35] | Kuramoto, Y.; Tsuzuki, T., Progr. Theoret. Phys., 54, 687, 1975 |
| [36] | Yochelis, A.; Hagberg, A.; Meron, E.; Lin, A. L.; Swinney, H. L., SIAM J. Appl. Dyn. Syst., 1, 236, 2002 · Zbl 1004.35064 |
| [37] | (Islam, M. N., Raman Amplifiers for Telecommunications 1: Physical Principles, 2004, Springer: Springer New York) |
| [38] | (Headley, C.; Agrawal, G. P., Raman Amplification in Fiber Optical Communication Systems, 2005, Elsevier: Elsevier San Diego, CA) |
| [39] | Hayat, A.; Nevet, A.; Orenstein, M., Phys. Rev. Lett., 102, Article 183002 pp., 2009 |
| [40] | Hayat, A.; Nevet, A.; Ginzburg, P.; Orenstein, M., Semicond. Sci. Technol., 26, Article 083001 pp., 2011 |
| [41] | Reichert, M.; Smirl, A. L.; Salamo, G.; Hagan, D. J.; Van Stryland, E. W., Phys. Rev. Lett., 117, Article 073602 pp., 2016 |
| [42] | Penzkofer, A.; Falkenstein, W., Opt. Commun., 16, 247, 1976 |
| [43] | He, G. S.; Tan, L.-S.; Zheng, Q.; Prasad, P. N., Chem. Rev., 108, 1245, 2008 |
| [44] | Fan, H. H.; He, Y. J.; Dong, J. W.; Chen, B. C.; Wang, H. Z.; Tian, Y. P.; Reid, M. F., Appl. Phys. Lett., 96, Article 021109 pp., 2010 |
| [45] | Husko, C.; Combrié, S.; Colman, P.; Zheng, J.; De Rossi, A.; Wong, C. W., Sci. Rep., 3, 1100, 2013 |
| [46] | Peleg, A.; Chakraborty, D., Phys. D, 406, Article 132397 pp., 2020 · Zbl 1490.81085 |
| [47] | Mollenauer, L. F.; Mamyshev, P. V., IEEE J. Quantum Electron., 34, 2089, 1998 |
| [48] | Nakazawa, M., IEEE J. Sel. Top. Quantum Electron., 6, 1332, 2000 |
| [49] | Peleg, A.; Nguyen, Q. M.; Huynh, T. T., Eur. Phys. J. D, 71, 315, 2017 |
| [50] | Kaup, D. J., Phys. Rev. A, 44, 4582, 1991 |
| [51] | Herstein, I. A., Topics in Algebra, 251, 1975, Xerox Corporation: Xerox Corporation Lexington, MA · Zbl 1230.00004 |
| [52] | Lyapunov, A. M., The General Problem of the Stability of Motion, 1992, Taylor and Francis: Taylor and Francis London · Zbl 0786.70001 |
| [53] | Hirsch, M. W.; Smale, S., Differential Equations, Dynamical Systems, and Linear Algebra, 1974, Academic: Academic New York · Zbl 0309.34001 |
| [54] | Meiss, J. D., Differential Dynamical Systems, 2007, SIAM: SIAM Philadelphia · Zbl 1144.34001 |
| [55] | Yang, J., Nonlinear Waves in Integrable and Nonintegrable Systems, 2010, SIAM: SIAM Philadelphia · Zbl 1234.35006 |
| [56] | Kuznetsov, E. A.; Mikhailov, A. V.; Shimokhin, I. A., Phys. D, 87, 201, 1995 · Zbl 1194.78030 |
| [57] | Pelinovsky, D. E.; Kivshar, Y. S.; Afanasjev, V. V., Phys. D, 116, 121, 1998 · Zbl 0934.35175 |
| [58] | Peleg, A.; Chung, Y., J. Phys. A, 36, 10039, 2003 |
| [59] | Nguyen, Q. M.; Peleg, A., J. Opt. Soc. Am. B, 27, 1985, 2010 |
| [60] | Chertkov, M.; Chung, Y.; Dyachenko, A.; Gabitov, I.; Kolokolov, I.; Lebedev, V., Phys. Rev. E, 67, Article 036615 pp., 2003 |
| [61] | Kaup, D. J., J. Math. Anal. Appl., 54, 849, 1976 · Zbl 0333.34020 |
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