Abstract
A discrete Ablowitz–Ladik equation with variable coefficients, which can describe certain phenomena in an electrical/optical system, is analytically studied in this paper. Bright one- and two-soliton solutions are derived from the bilinear forms for such an equation. Soliton propagation and collision are graphically presented and analyzed with the choice of the functions \(\upsilon _{n}(t)\), \(\gamma (t)\) and \(\Lambda (t)\), which are respectively the space-time modulated inhomogeneous frequency shift, time-modulated effective gain/loss term and coefficient of the tunnel coupling between sites, where n and t are the lattice site and scaling time, respectively. With \(\upsilon _{n}(t)\), \(\gamma (t)\) and \(\Lambda (t)\) being the constants, the one soliton is shown to maintain its original amplitude and width during the propagation, and head-on collision between the two solitons is graphically illustrated with the amplitude of each soliton unchanging during the collision. Amplitudes and travelling directions of the one and two solitons are seen to be influenced by \(\gamma (t)\) and \(\Lambda (t)\), respectively. It is shown that \(\upsilon _{n}(t)\) does not affect the propagation and collision features of the solitons.
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References
Ablowitz, M.J., Ladik, J.: Nonlinear differential–difference equations. J. Math. Phys. 16, 598–603 (1975)
Akhmediev, N., Ankiewicz, A.: Modulation instability, Fermi–Pasta–Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz–Ladik equation. Phys. Rev. E 83, 046603 (2011)
Chow, K.W., Conte, R., Xu, N.: Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schrödinger (Ablowitz–Ladik) model. Phys. Lett. A 349, 422–429 (2006)
Doktorov, E.V., Matsuka, N.P., Rothos, V.M.: Dynamics of the Ablowitz–Ladik soliton train. Phys. Rev. E 69, 056607 (2004)
Gerdjikov, V.S., Baizakov, B.B., Salerno, M., Kostov, N.A.: Adiabatic N-soliton interactions of Bose–Einstein condensates in external potentials. Phys. Rev. E 73, 046606 (2006)
Hennig, D., Tsironis, G.P.: Wave transmission in nonlinear lattices. Phys. Rep. 307, 333–432 (1999)
Hirota, R.: The Direct Method in Soliton Theory, pp. 1–55. Cambridge University Press, Cambridge (2004)
Huang, Q.M., Gao, Y.T., Jia, S.L., Wang, Y.L., Deng, G.F.: Bilinear Backlund transformation, soliton and periodic wave solutions for a (3+1)-dimensional variable-coefficient generalized shallow water wave equation. Nonlinear Dyn. 87, 2529–2540 (2017)
Huang, W.H., Liu, Y.L.: Jacobi elliptic function solutions of the Ablowitz–Ladik discrete nonlinear Schrödinger system. Chaos Soliton. Fract. 40, 786–792 (2009)
Kominis, Y., Bountis, T., Hizanidis, K.: Breathers in a nonautonomous Toda lattice with pulsating coupling. Phys. Rev. E 81, 066601 (2010)
Lan, Z.Z., Gao, Y.T., Yang, J.W., Su, C.Q., Wang, Q.M.: Solitons, Backlund transformation and Lax pair for a (2+1)-dimensional B-type Kadomtsev-Petviashvili equation in the fluid/plasma mechanics. Mod. Phys. Lett. B 30, 1650265 (2016)
Lan, Z.Z., Gao, Y.T., Yang, J.W., Su, C.Q., Mao, B.Q.: Solitons, Backlund transformation and Lax pair for a (2+1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water of uniform depth. Commun. Nonlinear Sci. Numer. Simulat. 44, 360–372 (2017)
Marquié, P., Bilbault, J.M., Remoissenet, M.: Observation of nonlinear localized modes in an electrical lattice. Phys. Rev. E 51, 6217 (1995)
Su, J.J., Gao, Y.T.: Bilinear forms and solitons for a generalized sixth-order nonlinear Schrodinger equation in an optical fiber. Eur. Phys. J. P. 132, 53 (2017)
Su, J.J., Gao, Y.T., Jia, S.L.: Solitons for a generalized sixth-order variable-coefficient nonlinear Schrodinger equation for the attosecond pulses in an optical fiber. Commun. Nonlinear Sci. Numer. Simul. 50, 128–141 (2017)
Trombettoni, A., Smerzi, A.: Discrete solitons and breathers with dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353 (2001)
Wang, Y.F., Tian, B., Li, M., Wang, P., Jiang, Y.: Soliton dynamics of a discrete integrable Ablowitz–Ladik equation for some electrical and optical systems. Appl. Math. Lett. 35, 46–51 (2014)
Yang, J.W., Gao, Y.T., Feng, Y.J., Su, C.Q.: Solitons and dromion-like structures in an inhomogeneous optical fiber. Nonlinear Dyn. 87, 851–862 (2017)
Yu, F.J.: Nonautonomous discrete bright soliton solutions and interaction management for the Ablowitz–Ladik equation. Phys. Rev. E 91, 032914 (2015)
Acknowledgements
This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Beijing University of Posts and Telecommunications (BUPT) Excellent Ph.D. Students Foundation (No. CX2016308).
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Xie, XY., Tian, B., Chai, J. et al. Soliton collisions of a discrete Ablowitz–Ladik equation with variable coefficients for an electrical/optical system. Opt Quant Electron 49, 155 (2017). https://doi.org/10.1007/s11082-017-0978-7
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DOI: https://doi.org/10.1007/s11082-017-0978-7