Abstract
In this paper we study the modulation of localized solutions by an inhomogeneous saturable nonlinear medium. Throughout an appropriate ansatz we convert the inhomogeneous saturable nonlinear Schrödinger equation in a homogeneous one. Then, via a variational approach we construct localized solutions of the autonomous equation and we present some modulation patterns of this localized structures. We have checked the stability of such solutions through numerically simulations.
Similar content being viewed by others
References
Agrawal, G.P.: Nonlinear Fiber Optics. Optics and Photonics. Elsevier Science (2013).
Ainslie, B.J., Girdlestone, H.P., Cotter, D.: Semiconductor-doped fibre waveguides exhibiting picosecond optical nonlinearity. Electron. Lett. 23, 405–406 (1987)
Arroyo Meza, L.E., de Souza Dutra, A., Hott, M.B.: Wide localized solitons in systems with time- and space-modulated nonlinearities. Phys. Rev. E 86(2), 026605 (2012). https://doi.org/10.1103/PhysRevE.86.026605
Arroyo Meza, L.E., de Souza Dutra, A., Hott, M.B.: Wide vector solitons in systems with time- and space-modulated nonlinearities. Phys. Rev. E 88(5), 053202 (2013). https://doi.org/10.1103/PhysRevE.88.053202
Avelar, A.T., Bazeia, D., Cardoso, W.B.: Solitons with cubic and quintic nonlinearities modulated in space and time. Phys. Rev. E 79(2), 025602 (2009). https://doi.org/10.1103/PhysRevE.79.025602
Avelar, A.T., Bazeia, D., Cardoso, W.B.: Modulation of breathers in the three-dimensional nonlinear Gross–Pitaevskii equation. Phys. Rev. E 82(5), 057601 (2010). https://doi.org/10.1103/PhysRevE.82.057601
Belmonte-Beitia, J., Calvo, G.F.: Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials. Phys. Lett. A 373(4), 448–453 (2009). https://doi.org/10.1016/j.physleta.2008.11.056
Belmonte-Beitia, J., Pérez-García, V.M., Vekslerchik, V., Konotop, V.V.: Localized nonlinear waves in systems with time- and space-modulated nonlinearities. Phys. Rev. Lett. 100(16), 164102 (2008). https://doi.org/10.1103/PhysRevLett.100.164102
Belobo Belobo, D., Ben-Bolie, G.H., Kofane, T.C.: Dynamics of matter-wave condensates with time-dependent two- and three-body interactions trapped by a linear potential in the presence of atom gain or loss. Phys. Rev. E 89(4), 042913 (2014). https://doi.org/10.1103/PhysRevE.89.042913
Calaça, L., Avelar, A.T., Bazeia, D., Cardoso, W.B.: Modulation of localized solutions for the Schrödinger equation with logarithm nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 19(9), 2928–2934 (2014). https://doi.org/10.1016/j.cnsns.2014.02.002
Cardoso, W., Avelar, A., Bazeia, D.: Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearities modulated in space and time. Nonlinear Anal. Real World Appl. 11(5), 4269–4274 (2010a). https://doi.org/10.1016/j.nonrwa.2010.05.013
Cardoso, W., Avelar, A., Bazeia, D.: Modulation of breathers in cigar-shaped Bose–Einstein condensates. Phys. Lett. A 374(26), 2640–2645 (2010b). https://doi.org/10.1016/j.physleta.2010.04.050
Cardoso, W.B., Avelar, A.T., Bazeia, D.: Modulation of localized solutions in a system of two coupled nonlinear Schrödinger equations. Phys. Rev. E 86(2), 027601 (2012). https://doi.org/10.1103/PhysRevE.86.027601
Cardoso, W.B., Avelar, A.T., Bazeia, D., Hussein, M.S.: Solitons of two-component Bose–Einstein condensates modulated in space and time. Phys. Lett. A 374(23), 2356–2360 (2010). https://doi.org/10.1016/j.physleta.2010.03.065
Cardoso, W.B., Couto, H.L.C., Avelar, A.T., Bazeia, D.: Modulation of localized solutions in quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients. Commun. Nonlinear Sci. Numer. Simul. 48, 474–483 (2017). https://doi.org/10.1016/j.cnsns.2017.01.012
Coutaz, J.L., Kull, M.: Saturation of the nonlinear index of refraction in semiconductor-doped glass. J. Opt. Soc. Am. B 8(1), 95–98 (1991). https://doi.org/10.1364/JOSAB.8.000095.
Dai, C., Zhu, S., Zhang, J.: Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation. Opt. Commun. 283(19), 3784–3791 (2010). https://doi.org/10.1016/j.optcom.2010.05.027
Dai, C.Q., Wang, Y.Y., Wang, X.G.: Ultrashort self-similar solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients in the inhomogeneous fiber. J. Phys. A Math. Theor. 44(15), 155203 (2011). https://doi.org/10.1088/1751-8113/44/15/155203
Denschlag, J.: Generating solitons by phase engineering of a Bose–Einstein condensate. Science 287(5450), 97–101 (2000). https://doi.org/10.1126/science.287.5450.97
De Kumar, K., Goyal, A., Raju, T.S., Kumar, C., Panigrahi, P.K.: Riccati parameterized self-similar waves in two-dimensional graded-index waveguide. Opt. Commun. 341, 15–21 (2015). https://doi.org/10.1016/j.optcom.2014.11.101
Dmitriev, S.V., Kevrekidis, P.G., Kivshar, Y.S.: Radiationless energy exchange in three-soliton collisions. Phys. Rev. E 78(4), 046604 (2008). https://doi.org/10.1103/PhysRevE.78.046604
Dmitriev, S.V., Kivshar, Y.S., Shigenari, T.: Fractal structures and multiparticle effects in soliton scattering. Phys. Rev. E 64(5), 056613 (2001). https://doi.org/10.1103/PhysRevE.64.056613
Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation in photonic crystal fiber. Rev. Mod. Phys. 78(4), 1135–1184 (2006). https://doi.org/10.1103/RevModPhys.78.1135
He, J.d., Zhang, J.f.: Self-similar optical pulses tunneling in cubic-quintic nonlinear media with distributed coefficients. J. Phys. A Math. Theor. 44(20), 205203 (2011). https://doi.org/10.1088/1751-8113/44/20/205203
He, Jd, Zhang, Jf, Zhang, My, Dai, Cq: Analytical nonautonomous soliton solutions for the cubic-quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun. 285(5), 755–760 (2012). https://doi.org/10.1016/j.optcom.2011.10.087
He, J.R., Li, H.M.: Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with different external potentials. Phys. Rev. E 83(6), 066607 (2011). https://doi.org/10.1103/PhysRevE.83.066607
He, J.R., Yi, L.: Formations of n-order two-soliton bound states in Bose–Einstein condensates with spatiotemporally modulated nonlinearities. Phys. Lett. A 378(16–17), 1085–1090 (2014). https://doi.org/10.1016/j.physleta.2014.01.050
He, J.R., Yi, L., Li, H.M.: Localized nonlinear waves in combined time-dependent magnetic-optical potentials with spatiotemporally modulated nonlinearities. Phys. Lett. A 377(34–36), 2034–2040 (2013). https://doi.org/10.1016/j.physleta.2013.06.025
Heng-Nong, X., Miao, Z.: Matter-wave solitons in two-component Bose—Einstein condensates with tunable interactions and time varying potential. Commun. Theor. Phys. 56(6), 1035–1040 (2011). https://doi.org/10.1088/0253-6102/56/6/11
Hirota, R., Suzuki, K.: Theoretical and experimental studies of lattice solitons in nonlinear lumped networks. P. IEEE 61(10), 1483–1491 (1973). https://doi.org/10.1109/PROC.1973.9297
Hukriede, J., Runde, D., Kip, D.: Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides. J. Phys. D Appl. Phys. 36(3), R1–R16 (2003). https://doi.org/10.1088/0022-3727/36/3/201
Il’ichev, N.N., Kir’yanov, A.V., Shapkin, V.P., Nasibov, S.A., Mosaleva, S.Y.: Nonlinear change in refractive index of Co2+:ZnSe at short-pulse single-beam 1.54-\(\mu\)m Z-scan probing. Appl. Phys. B 81(1), 83–91 (2005). https://doi.org/10.1007/s00340-005-1860-z
Inouye, S., Andrews, M.R., Stenger, J., Miesner, H.J., Stamper-Kurn, D.M., Ketterle, W.: Observation of Feshbach resonances in a Bose–Einstein condensate. Nature 392(6672), 151–154 (1998). https://doi.org/10.1038/32354
Jin, H.Q., Dai, W., Tong, A., Cai, Z.B., Liang, J.C., He, J.R.: Dynamics of analytical three-dimensional matter-wave solutions in Bose–Einstein condensates with multi-body interactions. Phys. Lett. A 378(14–15), 1017–1021 (2014). https://doi.org/10.1016/j.physleta.2014.01.055
Kivshar, Y.S., Agrawal, G.: Optical Solitons: From Fibers to Photonic Crystals. Elsevier Science (2003).
Kumar, A.: Bistability and hysteresis of solitons in inhomogeneously doped fibers with saturating nonlinearity. Phys. Rev. E 58(4), 5021–5024 (1998). https://doi.org/10.1103/PhysRevE.58.5021
Li, J., Zong, F.D., Song, C.S., Wang, Y., Li, F.B.: Dynamics of analytical three-dimensional solutions in Bose–Einstein condensates with time-dependent gain and potential. Phys. Rev. E 85(3), 036607 (2012). https://doi.org/10.1103/PhysRevE.85.036607
Loomba, S., Pal, R., Kumar, C.N.: Bright solitons of the nonautonomous cubic-quintic nonlinear Schrödinger equation with sign-reversal nonlinearity. Phys. Rev. A 92(3), 033811 (2015). https://doi.org/10.1103/PhysRevA.92.033811
Malomed, B.A.: Soliton Management in Periodic Systems. Springer US (2006).
Meza, L.E.A., Dutra, AdS, Hott, M.B., Roy, P.: Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation. Phys. Rev. E 91(1), 013205 (2015). https://doi.org/10.1103/PhysRevE.91.013205
Mikeska, H.J., Steiner, M.: Solitary excitations in one-dimensional magnets. Adv. Phys. 40(3), 191–356 (1991). https://doi.org/10.1080/00018739100101492
Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. Lett. 45(13), 1095–1098 (1980). https://doi.org/10.1103/PhysRevLett.45.1095
Nath, A., Roy, U.: A unified model for an external trap in a cigar-shaped Bose–Einstein condensate. J. Phys. A Math. Theor. 47(41), 415301 (2014). https://doi.org/10.1088/1751-8113/47/41/415301
Peng, G.D., Xiong, Z., Chu, P.L.: Photosensitivity and gratings in dye-doped polymer optical fibers. Opt. Fiber Technol. 5(2), 242–251 (1999). https://doi.org/10.1006/ofte.1998.0298
Peyrard, M.: Nonlinear dynamics and statistical physics of DNA. Nonlinearity 17(2), R1–R40 (2004). https://doi.org/10.1088/0951-7715/17/2/R01
Serkin, V.N., Belyaeva, T.L.: High-energy optical Schrödinger solitons. JETP Lett. 74(12), 573–577 (2001). https://doi.org/10.1134/1.1455063
Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85(21), 4502–4505 (2000a). https://doi.org/10.1103/PhysRevLett.85.4502
Serkin, V.N., Hasegawa, A.: Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain. J. Exp. Theor. Phys. Lett. 72(2), 89–92 (2000b). https://doi.org/10.1134/1.1312019
Serkin, V.N., Hasegawa, A.: Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion. IEEE J. Sel. Top. Quant. 8, 418–431 (2002). https://doi.org/10.1109/JSTQE.2002.1016344
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98(7), 074102 (2007). https://doi.org/10.1103/PhysRevLett.98.074102
Soloman Raju, T.: Dynamics of self-similar waves in asymmetric twin-core fibers with Airy–Bessel modulated nonlinearity. Opt. Commun. 346, 74–79 (2015). https://doi.org/10.1016/j.optcom.2015.02.025
Teixeira, R.M., Cardoso, W.B.: Fractal scattering of Gaussian solitons in directional couplers with logarithmic nonlinearities. Phys. Lett. A 380(35), 2738–2749 (2016). https://doi.org/10.1016/j.physleta.2016.06.041
Theis, M., Thalhammer, G., Winkler, K., Hellwig, M., Ruff, G., Grimm, R., Denschlag, J.H.: Tuning the scattering length with an optically induced feshbach resonance. Phys. Rev. Lett. 93(12), 123001 (2004). https://doi.org/10.1103/PhysRevLett.93.123001
Wang, D.S., Hu, X.H., Hu, J., Liu, W.M.: Quantized quasi-two-dimensional Bose–Einstein condensates with spatially modulated nonlinearity. Phys. Rev. A 81(2), 025604 (2010). https://doi.org/10.1103/PhysRevA.81.025604
Wang, D.S., Hu, X.H., Liu, W.M.: Localized nonlinear matter waves in two-component Bose–Einstein condensates with time- and space-modulated nonlinearities. Phys. Rev. A 82(2), 023612 (2010). https://doi.org/10.1103/PhysRevA.82.023612
Yan, Z.: Novel wave structures in the two-dimensional cubic-quintic nonlinear Schrödinger equation with space-modulated potential and nonlinearities. Nonlinear Dyn. 82(1–2), 119–129 (2015). https://doi.org/10.1007/s11071-015-2143-9
Yan, Z., Hang, C.: Analytical three-dimensional bright solitons and soliton pairs in Bose–Einstein condensates with time-space modulation. Phys. Rev. A 80(6), 063626 (2009). https://doi.org/10.1103/PhysRevA.80.063626
Yan, Z., Konotop, V.V.: Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities. Phys. Rev. E 80(3), 036607 (2009). https://doi.org/10.1103/PhysRevE.80.036607
Yang, J.: Nonlinear waves in integrable and nonintegrable systems. Soc. Ind. Appl. Math. (2010). https://doi.org/10.1137/1.9780898719680
Yang, Y., Yan, Z., Mihalache, D.: Controlling temporal solitary waves in the generalized inhomogeneous coupled nonlinear Schrödinger equations with varying source terms. J. Math. Phys. 56(5), 053508 (2015). https://doi.org/10.1063/1.4921641
Yomba, E.: Traveling-waves and solitons in a generalized time-variable coefficients nonlinear Schrödinger equation with higher-order terms. Phys. Lett. A 377(3–4), 167–175 (2013). https://doi.org/10.1016/j.physleta.2012.11.049
Yomba, E., Zakeri, G.A.: Solitons in a generalized space- and time-variable coefficients nonlinear Schrödinger equation with higher-order terms. Phys. Lett. A 377(42), 2995–3004 (2013). https://doi.org/10.1016/j.physleta.2013.09.011
Zhang, J.F., Tian, Q., Wang, Y.Y., Dai, C.Q., Wu, L.: Self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients. Phys. Rev. A 81(2), 023832 (2010). https://doi.org/10.1103/PhysRevA.81.023832
Zhong, W.P., Belić, M.R., Assanto, G.: Localized nonlinear wavepackets with radial–azimuthal modulated nonlinearity and an external potential. Phys. Scr. 84(5), 055001 (2011). https://doi.org/10.1088/0031-8949/84/05/055001
Zhong, W.P., Belić, M.R., Huang, T.: Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity. J. Opt. Soc. Am. B 30(5), 1276–1283 (2013). https://doi.org/10.1364/JOSAB.30.001276.
Zhu, Y., Haberman, R., Yang, J.: Universal map for fractal structures in weak interactions of solitary waves. Phys. Rev. Lett. 100(14), 143901 (2008). https://doi.org/10.1103/PhysRevLett.100.143901
Acknowledgements
We thank the CNPq (Grant #458889/2014-8), FAPEG, and Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ), Brazilian agencies, for the partial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Calaça, L., Cardoso, W.B. Modulation of localized solutions in an inhomogeneous saturable nonlinear Schrödinger equation. Opt Quant Electron 49, 379 (2017). https://doi.org/10.1007/s11082-017-1214-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-017-1214-1