Abstract
We study the dynamics of snakelike similaritons to the nonlinear Schrödinger equation in combined harmonic-lattice potentials with a varying source term. Using the self-similar and Möbius transformations, we construct a large number of analytical solutions which generate many types of novel snakelike similaritons under some constraint conditions, such as the W- and bell-shaped breathing similaritons and the elliptic function solutions. Then, four specific examples of physical interest are considered in details to show the dynamical behaviors of the similaritons. Especially, the compression effect of snakelike similaritons is demonstrated in dispersion-decreasing media with Bessel modulated nonlinearity. We find that the structures of snakelike similaritons can be controlled by tuning the coefficient parameters, the combined harmonic-lattice potentials, and the source term. The stability of the solutions has also been performed by numerical simulations. Our results may have potential applications in the dual-core fiber waveguides and Bose–Einstein condensates in the presence of an inhomogeneous source.
Similar content being viewed by others
References
Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502 (2000)
Wu, L., Zhang, J.F., Finot, C., Li, L.: Propagation of dark similaritons on the compact parabolic background in dispersion-managed optical fibers. Opt. Express 17, 8278–8286 (2009)
Xu, S., Petrovi, N.Z., Belić, M.R., Deng, W.: Exact solutions for the quintic nonlinear Schrödinger equation with time and space. Nonlinear Dyn. 84, 251–259 (2016)
Ozeki, Y., Inoue, T.: Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression. Opt. Lett. 31, 1606–1608 (2006)
Dai, C.Q., Wang, Y., Yan, C.: Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun. 283, 1489–1494 (2010)
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Hidden symmetry reductions and the Ablowitz–Kaup–Newell–Segur hierarchies for nonautonomous solitons. In: Porsezian, K., Ganapathy, R. (eds.) Odyssey of Light in Nonlinear Optical Fibers: Theory and Applications, pp. 145–187. CRC Press, Taylor Francis, Boca Raton (2015)
Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, New York (1995)
Chen, Y.X., Xu, F.Q., Hu, Y.L.: Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 1957–1964 (2019)
Wu, H.Y., Jiang, L.H.: Excitation management of (2+1)-dimensional breathers for a coupled partially nonlocal nonlinear Schrödinger equation with variable coefficients. Nonlinear Dyn. 95, 3401–3409 (2019)
Yang, C., Liu, W., Zhou, Q., Mihalache, D., Malomed, B.A.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)
Ponomarenko, S.A., Agrawal, G.P.: Do solitonlike self-similar waves exist in nonlinear optical media? Phys. Rev. Lett. 97, 013901 (2006)
Serkin, V.N., Chapela, V.M., Percino, J., Belyaeva, T.L.: Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides. Opt. Commun. 192, 237–244 (2001)
Malomed, B.A., Skinner, I.M., Chu, P.L., Peng, G.D.: Symmetric and asymmetric solitons in twin-core nonlinear optical fibers. Phys. Rev. E 53, 4084 (1996)
Li, L., Zhao, X., Xu, Z.: Dark solitons on an intense parabolic background in nonlinear waveguides. Phys. Rev. A 78, 063833 (2008)
Zhao, X., Li, L., Xu, Z.: Dark-soliton stripes on a paraboloidal background in a bulk nonlinear medium. Phys. Rev. A 79, 043827 (2009)
Wu, L., Zhang, J.F., Li, L., Tian, Q., Porsezian, K.: Similaritons in nonlinear optical systems. Opt. Express 16, 6352–6360 (2008)
Xu, S., Petrović, N.Z., Belić, M.R., Hu, Z.L.: Light bullet supported by parity-time symmetric potential with power-law nonlinearity. Nonlinear Dyn. 84, 1877–1882 (2016)
Malomed, B.A., Mihalache, D.: Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)
Xu, S., Petrović, N., Belić, M.R.: Exact solutions of the (2+1)-dimensional quintic nonlinear Schrödinger equation with variable coefficients. Nonlinear Dyn. 80, 583–589 (2015)
Xu, S., Xue, L., Belić, M.R., He, J.R.: Spatiotemporal soliton clusters in strongly nonlocal media with variable potential coefficients. Nonlinear Dyn. 87, 827–834 (2017)
Dai, C.Q., Zhang, J.F.: Controllable dynamical behaviors for spatiotemporal bright solitons on continuous wave background. Nonlinear Dyn. 73, 2049–2057 (2013)
Mani Rajan, M.S.: Dynamics of optical soliton in a tapered erbium-doped fiber under periodic distributed amplification system. Nonlinear Dyn. 85, 599–606 (2016)
Subramanian, K., Alagesan, T., Mahalingam, A., Mani Rajan, M.S.: Propagation properties of optical soliton in an erbium-doped tapered parabolic index nonlinear fiber: soliton control. Nonlinear Dyn. 87, 1575–1587 (2017)
Yang, Z.Y., Zhao, L.C., Zhang, T., Li, Y.H., Yue, R.H.: Snakelike nonautonomous solitons in a graded-index grating waveguide. Phys. Rev. A 81, 043826 (2010)
Dai, C.Q., Zhu, H.P.: Superposed Kuznetsov–Ma solitons in a two-dimensional graded-index grating waveguide. J. Opt. Soc. Am. B 30, 3291–3297 (2013)
Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. Lett. 90, 113902 (2003)
Chernikov, S.V., Richardson, D.J., Dianov, E.M., Payn, D.N.: Soliton pulse compression in dispersion-decreasing fiber. Opt. Lett. 18, 476–478 (1993)
Moores, J.D.: Nonlinear compression of chirped solitary waves with and without phase modulation. Opt. Lett. 21, 555–557 (1996)
Biswas, A., Milovic, D.: Mathematical Theory of Dispersion-Managed Optical Solitons. Springer, Berlin (2010)
Serkin, V.N., Hasegawa, A.: Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion and nonlinear management. IEEE J. Select. Top. Quant. Electron. 8, 418–431 (2002)
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons. J. Mod. Opt. 57, 456–1472 (2010)
Yang, Q., Zhang, J.F.: Bose-Einstein solitons in time-dependent linear potential. Opt. Commun. 258, 35–42 (2006)
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, 164102 (2008)
Anderson, B.P., Kasevich, M.A.: Macroscopic quantum interference from atomic tunnel arrays. Science 282, 1686 (1998)
Bronski, J.C., Carr, L.D., Deconinck, B., Kutz, J.N.: Bose–Einstein condensates in standing waves: the cubic nonlinear Schrödinger equation with a periodic potential. Phys. Rev. Lett. 86, 1402 (2001)
Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179 (2006)
Blakie, P.B., Bezett, A., Buonsante, P.: Degenerate Fermi gas in a combined harmonic-lattice potential. Phys. Rev. A 75, 063609 (2007)
Hassan, A.S.: Critical points of a quasi two-dimensional harmonically trapped Bose gas. Physica B 405, 1040–1044 (2010)
Liu, M., Shum, P.: Generalized coupled nonlinear equations for the analysis of asymmetric two-core fiber coupler. Opt. Express 11, 116–119 (2003)
Raju, T.S., Panigrahi, P.K., Porsezian, K.: Nonlinear compression of solitary waves in asymmetric twin-core fibers. Phys. Rev. E 71, 026608 (2005)
Raju, T.S., Panigrahi, P.K.: Self-similar propagation in a graded-index nonlinear-fiber amplifier with an external source. Phys. Rev. A 81, 043820 (2010)
He, J.R., Yi, L.: Exact optical self-similar solutions in a tapered graded-index nonlinear-fiber amplifier with an external source. Opt. Commun. 320, 129–137 (2014)
Raju, T.S.: Dynamics of self-similar waves in asymmetric twin-core fibers with Airy–Bessel modulated nonlinearity. Opt. Commun. 346, 74–79 (2015)
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, 053508 (2015)
Yan, Z., Zhang, X.F., Liu, W.M.: Nonautonomous matter waves in a waveguide. Phys. Rev. A 84, 023627 (2011)
Raju, T.S.: Spatiotemporal optical similaritons in dual-core waveguide with an external source. Commun. Nonlinear. Sci. Numer. Simulat. 45, 75–80 (2017)
Bhat, N.A.R., Sipe, J.E.: Optical pulse propagation in nonlinear photonic crystals. Phys. Rev. E 64, 056604 (2001)
He, J.R., Xu, S., Xue, L.: Snakelike similaritons in tapered grating dual-core waveguide amplifiers. Phys. Scr. 94, 105216 (2019)
Eiermann, B., Treutlein, P., Anker, T., Albiez, M., Taglieber, M., Marzlin, K.P., Oberthaler, M.K.: Dispersion management for atomic matter waves. Phys. Rev. Lett. 91, 060402 (2003)
Kartashov, Y.V., Konotop, V.V., Abdullaev, FKh: Gap solitons in a spin-orbit-coupled Bose–Einstein condensate. Phys. Rev. Lett. 111, 060402 (2013)
Paul, T., Richter, K., Schlagheck, P.: Nonlinear resonant transport of Bose–Einstein condensates. Phys. Rev. Lett. 94, 020404 (2005)
Raju, T.S., Kumar, C.N., Panigrahi, P.K.: On exact solitary wave solutions of the nonlinear Schrödinger equation with a source. J. Phys. A Math. Gen. 38, L271 (2005)
Malomed, B.A.: Soliton Management in Periodic Systems. Springer, New York (2006)
Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant No. 11847103 and the Field Grade Project of HuBei University of Science and Technology under Grant No. 2016-19XB006.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
He, JR., Deng, WW. & Xue, L. Snakelike similaritons in combined harmonic-lattice potentials with a varying source. Nonlinear Dyn 100, 1599–1609 (2020). https://doi.org/10.1007/s11071-020-05584-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05584-9