Frequency shifting for solitons based on transformations in the Fourier domain and applications. (English) Zbl 1481.35357

Summary: We develop the theoretical procedures for shifting the frequency of a single soliton and of a sequence of solitons of the nonlinear Schrödinger equation. The procedures are based on simple transformations of the soliton pattern in the Fourier domain and on the shape-preserving property of solitons. These theoretical frequency shifting procedures are verified by numerical simulations with the nonlinear Schrödinger equation using the split-step Fourier method. In order to demonstrate the use of the frequency shifting procedures, two important applications are presented: (1) stabilization of the propagation of solitons in waveguides with frequency dependent linear gain-loss; (2) induction of repeated soliton collisions in waveguides with weak cubic loss. The results of numerical simulations with the nonlinear Schrödinger model are in very good agreement with the theoretical predictions.


35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems


Full Text: DOI arXiv


[2] Tao, T., Nonlinear dispersive equations: local and global analysis, Proceedings of the CBMS Regional Conference Series in Mathematics, 106 (2006), AMS · Zbl 1106.35001
[3] 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
[4] Kivshar, Y. S.; Malomed, B. A., Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys., 61, 763 (1989)
[7] Horton, W.; Ichikawa, Y. H., Chaos and Structure in Nonlinear Plasmas (1996), World Scientific: World Scientific Singapore
[8] Hasegawa, A.; Tappert, F., Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. part I. anomalous dispersion; part II. normal dispersion, Appl. Phys. Lett., 23, 142-144 (1973)
[9] Yoo, S. J.B., Wavelength conversion technologies for WDM network applications, J. Light. Technol., 14, 955-966 (1996)
[10] Ip, E.; Lau, A. P.T.; Barros, D. J.F.; Kahn, J. M., Coherent detection in optical fiber systems, Opt. Express, 16, 753 (2008)
[11] Mitschke, F. M.; Mollenauer, L. F., Discovery of the soliton self-frequency shift, Opt. Lett., 11, 659-661 (1986)
[12] Chung, Y.; Peleg, A., Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed raman response, Nonlinearity, 18, 1555-1574 (2005) · Zbl 1116.78015
[13] Peleg, A.; Chakraborty, D., Enhancement of transmission quality in soliton-based optical waveguide systems by frequency dependent linear gain – loss and the raman self-frequency shift, Phys. Rev. A, 98, 013853 (2018)
[14] Chakraborty, D.; Peleg, A.; Nguyen, Q. M., Stabilizing soliton-based multichannel transmission with frequency dependent linear gain-loss, Opt. Commun., 371, 252-262 (2016)
[15] Peleg, A.; Nguyen, Q. M.; Huynh, T. T., Stable scalable control of soliton propagation in broadband nonlinear optical waveguides, Eur. Phys. J. D, 71, 30 (2017)
[16] Husko, C.; Combrié, S.; Colman, P.; Zheng, J.; De Rossi, A.; Wong, C. W., Soliton dynamics in the multiphoton plasma regime, Sci. Rep., 3, 1100 (2013)
[17] Peleg, A.; Nguyen, Q. M.; Chung, Y., Crosstalk dynamics of optical solitons in a broadband kerr nonlinear system with weak cubic loss, Phys. Rev. A, 82, 053830 (2010)
[18] Nguyen, Q. M.; Peleg, A.; Tran, T. P., Robust transmission stabilization and dynamic switching in broadband hybrid waveguide systems with nonlinear gain and loss, Phys. Rev. A, 91, 013839 (2015)
[19] Kaup, D. J.; Math, J., Closure of the squared Zakharov-Shabat eigenstates, Anal. Appl., 54, 849-864 (1976) · Zbl 0333.34020
[20] Kaup, D. J., A perturbation expansion for the Zakharov-Shabat inverse scattering transform, SIAM J. Appl. Math., 31, 121-133 (1976) · Zbl 0334.47006
[23] Trefethen, L. N., Spectral Methods in MATLAB (2000), SIAM: SIAM Philadelphia · Zbl 0953.68643
[24] Yang, J., Nonlinear Waves in Integrable and Nonintegrable Systems (2010), SIAM: SIAM Philadelphia · Zbl 1234.35006
[25] Lakoba, T. I., Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation, Numer. Methods Partial Differ. Equ., 28, 641-669 (2012) · Zbl 1242.65208
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.