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Stability of Bragg grating solitons in a semilinear dual-core system with cubic–quintic nonlinearity

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Abstract

The existence and stability of quiescent Bragg grating solitons in a dual-core fiber, where one core contains a Bragg grating with cubic–quintic nonlinearity, and the other is a linear are studied. The model admits two disjoint bandgaps when the relative group velocity in the linear core, c, is zero: one in the upper half and the other in the lower half of the system’s linear spectrum. In the general case (i.e., \(c\ne 0\)), a central gap (which is a genuine gap) is formed, while the lower and upper gaps overlap with one branch of continuous spectrum, and therefore, they are not genuine bandgaps. For quiescent solitons, exact analytical solutions are found in implicit form for \(c=0\). For nonzero c, soliton solutions are obtained numerically. The system supports two disjoint families (referred to as Type 1 and Type 2) of zero-velocity soliton solutions, separated by a border. Both Type 1 and Type 2 soliton solutions exist throughout the upper and lower gaps but not in the central gap. The stability of both soliton families is investigated by means of systematic numerical simulations. It is found that Type 2 solitons are always unstable and are destroyed upon propagation. On the other hand, unstable Type 1 solitons may either decay into radiation or radiate some energy and evolve into a moving Type 1 soliton. Also, in the case of Type 1 solitons, we have identified stable regions in the plane of quintic nonlinearity and frequency. The influence of coupling coefficient and the relative group velocity in the linear core on the stability of solitons are analyzed.

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References

  1. Kashyap, R.: Fiber Bragg Gratings. Academic Press, Boston (2010)

    Google Scholar 

  2. Loh, W.H., Laming, R.I., Robinson, N., Cavaciuti, A., Vaninetti, F., Anderson, C.J., Zervas, M.N., Cole, M.J.: Dispersion compensation over distances in excess of 500 km for 10 Gb/s systems using chirped fiber gratings. IEEE Photon. Technol. Lett. 8, 944–946 (1996)

    Article  Google Scholar 

  3. Litchinitser, N.M., Eggleton, B.J., Patterson, D.B.: Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression. J. Lightwave Technol. 15, 1303–1313 (1997)

    Article  Google Scholar 

  4. Sankey, N.D., Prelewitz, D.F., Brown, T.G.: All-optical switching in a nonlinear periodic-waveguide structure. Appl. Phys. Lett. 60, 1427–1429 (1992)

    Article  Google Scholar 

  5. LaRochelle, S., Hibino, Y., Mizrahi, V., Stegeman, G.I.: All-optical switching of grating transmission using cross-phase modulation in optical fibres. Electron. Lett. 26, 1459–1460 (1990)

    Article  Google Scholar 

  6. Winful, H.G.: Pulse compression in optical fiber filters. Appl. Phys. Lett. 46, 527–529 (1985)

    Article  Google Scholar 

  7. Winful, H.G., Marburger, J.H., Garmire, E.: Theory of bistability in nonlinear distributed feedback structures. Appl. Phys. Lett. 35, 379–381 (1979)

    Article  Google Scholar 

  8. Radic, S., George, N., Agrawal, G.P.: Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures. J. Opt. Soc. Am. B 12, 671–680 (1995)

    Article  Google Scholar 

  9. Russell, P.S.J.: Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures. J. Mod. Opt. 38, 1599–1619 (1991)

    Article  Google Scholar 

  10. de Sterke, C.M., Sipe, J.E.: Gap solitons. Prog. Opt. 33, 203–260 (1994)

    Article  Google Scholar 

  11. Aceves, A.B., Wabnitz, S.: Self-induced transparency solitons in nonlinear refractive periodic media. Phys. Lett. A 141, 37–42 (1989)

    Article  Google Scholar 

  12. Christadoulides, D.N., Joseph, R.I.: Slow Bragg solitons in nonlinear periodic structures. Phys. Rev. Lett. 62, 1746–1749 (1989)

    Article  Google Scholar 

  13. Sipe, J.E., Winful, H.G.: Nonlinear Schrödinger solitons in a periodic structure. Opt. Lett. 13, 132–133 (1988)

    Article  Google Scholar 

  14. Barashenkov, I.V., Pelinovsky, D.E., Zemlyanaya, E.V.: Vibrations and oscillatory instabilities of gap solitons. Phys. Rev. Lett. 80, 5117–5120 (1998)

    Article  Google Scholar 

  15. Mak, W.C.K., Malomed, B.A., Chu, P.L.: Formation of a standing-light pulse through collision of gap soliton. Phys. Rev. E 68, 026609 (2003)

    Article  Google Scholar 

  16. Neill, D.R., Atai, J.: Collision dynamics of gap solitons in Kerr media. Phys. Lett. A 353, 416–421 (2006)

    Article  Google Scholar 

  17. Eggleton, B.J., de Sterke, C.M., Slusher, R.E.: Nonlinear pulse propagation in Bragg gratings. J. Opt. Soc. Am. B 14, 2980–2993 (1997)

    Article  Google Scholar 

  18. Benjamin, J.E., Slusher, R.E., de Sterke, C.M., Krug, P.A., Sipe, J.E.: Bragg grating solitons. Phys. Rev. Lett. 76, 1627–1630 (1996)

    Article  Google Scholar 

  19. Taverner, D., Broderick, N.G.R., Richardson, D.J., Laming, R.I., Ibsen, M.: Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating. Opt. Lett. 23, 328–330 (1998)

    Article  Google Scholar 

  20. de Sterke, C.M., Eggleton, B.J., Krug, P.A.: High-intensity pulse propagation in uniform gratings and grating superstructures. J. Lightwave Technol. 15, 1494–1502 (1997)

    Article  Google Scholar 

  21. Mok, J.T., de Sterke, C.M., Littler, I.C.M., Eggleton, B.J.: Dispersionless slow light using gap solitons. Nat. Phys. 2, 775–780 (2006)

    Article  Google Scholar 

  22. Conti, C., Trillo, S., Assanto, G.: Doubly resonant Bragg simultons via second-harmonic generation. Phys. Rev. Lett. 78, 2341–2344 (1997)

    Article  Google Scholar 

  23. He, H., Drummond, P.D.: Ideal soliton environment using parametric band gaps. Phys. Rev. Lett. 78, 4311–4315 (1997)

    Article  Google Scholar 

  24. Mak, W.C.K., Malomed, B.A., Chu, P.L.: Solitary waves in asymmetric coupled waveguides with quadratic nonlinearity. Opt. Commun. 154, 145–151 (1998)

    Article  Google Scholar 

  25. Atai, J.: Interaction of Bragg grating solitons in a cubic-quintic medium. J. Opt. B: Quantum Semiclass. 6, S177–S181 (2004)

    Article  Google Scholar 

  26. Atai, J., Malomed, B.A.: Families of Bragg-grating solitons in a cubic-quintic medium. Phys. Lett. A 284, 247–252 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dasanayaka, S., Atai, J.: Stability and collisions of moving Bragg grating solitons in a cubic-quintic nonlinear medium. J. Opt. Soc. Am. B 30, 396–404 (2013)

    Article  MATH  Google Scholar 

  28. Dasanayaka, S., Atai, J.: Interactions of Bragg grating solitons in a cubic-quintic nonlinear medium with dispersive reflectivity. Phys. Rev. E 84, 026613 (2011)

    Article  MATH  Google Scholar 

  29. Dasanayaka, S., Atai, J.: Stability of Bragg grating solitons in a cubic-quintic nonlinear medium with dispersive reflectivity. Phys. Lett. A 375, 225–229 (2010)

    Article  MATH  Google Scholar 

  30. Mak, W.C.K., Chu, P.L., Malomed, B.A.: Solitary waves in coupled nonlinear waveguides with Bragg gratings. J. Opt. Soc. Am. B 15, 1685–1692 (1998)

    Article  MathSciNet  Google Scholar 

  31. Atai, J., Malomed, B.A.: Solitary waves in systems with separated Bragg grating and nonlinearity. Phys. Rev. E 64, 066617 (2001)

    Article  Google Scholar 

  32. Atai, J., Malomed, B.A.: Bragg-grating solitons in a semilinear dual-core system. Phys. Rev. E 62, 8713–8718 (2000)

    Article  MathSciNet  Google Scholar 

  33. Atai, J., Malomed, B.A.: Gap solitons in Bragg gratings with dispersive reflectivity. Phys. Lett. A 342, 404–412 (2005)

    Article  MATH  Google Scholar 

  34. Neil, D.R., Atai, J., Malomed, B.A.: Gap solitons in a hollow optical fiber in the normal dispersion regime. Phys. Lett. A 367, 73–82 (2007)

    Article  Google Scholar 

  35. Skryabin, D.V.: Coupled core-surface solitons in photonic crystal fibers. Opt. Expr. 12, 4841–4846 (2004)

    Article  Google Scholar 

  36. Atai, J., Malomed, B.A., Merhasin, I.M.: Stability and collisions of gap solitons in a model of a hollow optical fiber. Opt. Commun. 265, 342–348 (2006)

    Article  Google Scholar 

  37. Atai, J., Malomed, B.A.: Spatial solitons in a medium composed of self-focusing and self-defocusing layers. Phys. Lett. A 298, 140–148 (2002)

    Article  Google Scholar 

  38. Gorbach, A.V., Malomed, B.A., Skryabin, D.V.: Gap polariton solitons. Phys. Lett. A 373, 3024–3027 (2009)

    Article  MATH  Google Scholar 

  39. Trillo, S., Wabnitz, S.: Nonlinear nonreciprocity in a coherent mismatched directional coupler. Appl. Phys. Lett. 49, 752–754 (1986)

    Article  Google Scholar 

  40. Fraga, W.B., Menezes, J.W.M., da Silva, M.G., Sobrinho, C.S., Sombra, A.S.B.: All optical logic gates based on an asymmetric nonlinear directional coupler. Opt. Commun. 262, 32–37 (2006)

    Article  Google Scholar 

  41. Romagnoli, M., Trillo, S., Wabnitz, S.: Soliton switching in nonlinear couplers. Opt. Quantum Electron. 24, S1237–S1267 (1992)

    Article  Google Scholar 

  42. Soto-Crespo, J.M., Akhmediev, N.N.: Stability of the soliton states in a nonlinear fiber coupler. Phys. Rev. E 48, 4710–4715 (1993)

    Article  Google Scholar 

  43. Akhmediev, N.N., Ankiewicz, A.A.: Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett. 70, 2395–2398 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mak, W.C.K., Chu, P.L., Malomed, B.A.: Asymmetric solitons in coupled second-harmonic-generating waveguides. Phys. Rev. E 57, 1092–1103 (1998)

    Article  Google Scholar 

  45. Atai, J., Chen, Y.: Nonlinear couplers composed of different nonlinear cores. J. Appl. Phys. 59, 24–27 (1992)

    Article  Google Scholar 

  46. Atai, J., Chen, Y.: Nonlinear mismatches between two cores of saturable nonlinear couplers. IEEE J. Quantum Electron. 29, 242–249 (1993)

    Article  Google Scholar 

  47. Bertolotti, M., Monaco, M., Sibilia, C.: Role of the asymmetry in a third-order nonlinear directional coupler. Opt. Commun. 116, 405–410 (1995)

    Article  Google Scholar 

  48. Wang, L., Zhang, J., Chong, L., Li, M., Qi, F.H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys. Rev. E 93, 062217 (2016)

    Article  Google Scholar 

  49. Wang, L., Zhang, J., Wang, Z.Q., Chong, L., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)

    Article  Google Scholar 

  50. Lü, X., Ma, W., Yu, J., Chaudry, M.K.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)

    Article  MathSciNet  Google Scholar 

  51. Lü, X., Lin, F.: Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. Numer. Simul. 32, 241–261 (2016)

    Article  MathSciNet  Google Scholar 

  52. Lü, X., Ma, W., Chaudry, M.K.: A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg-de Vries-like model. Appl. Math. Lett. 50, 37–42 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lü, X.: Madelung fluid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn. 81, 239–247 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Lü, X., Lin, F., Qi, F.: Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions. Appl. Math. Model. 39, 3221–3226 (2015)

    Article  MathSciNet  Google Scholar 

  55. Lü, X., Ma, W., Yu, J., Lin, F., Chaudry, M.K.: Envelope bright- and dark-soliton solutions for the Gerdjikov-Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)

    Article  MathSciNet  Google Scholar 

  56. Kong, L., Dai, C.: Some discussions about variable separation of nonlinear models using Riccati equation expansion method. Nonlinear Dyn. 81, 1553–1561 (2015)

    Article  MathSciNet  Google Scholar 

  57. Dai, C., Wang, Y.: Controllable combined Peregrine soliton and Kuznetsov-Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715–721 (2015)

    Article  MathSciNet  Google Scholar 

  58. Dai, C., Xu, Y.: Exact solutions for a Wick-type stochastic reaction Duffing equation. Appl. Math. Model. 39, 7420–7426 (2015)

    Article  MathSciNet  Google Scholar 

  59. Dai, C., Wang, Y., Liu, J.: Spatiotemporal Hermite-Gaussian solitons of a (3+1)-dimensional partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  60. Dai, C., Wang, Y.: Spatiotemporal localizations in (3+1)-dimensional PT-symmetric and strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 2453–2459 (2016)

    Article  MathSciNet  Google Scholar 

  61. Wang, Y., Dai, C.: Caution with respect to “new” variable separation solutions and their corresponding localized structures. Appl. Math. Model. 40, 3475–3482 (2016)

    Article  MathSciNet  Google Scholar 

  62. Chowdhury, S.A.M.S., Atai, J.: Stability of Bragg grating solitons in a semilinear dual core system with dispersive reflectivity. IEEE J. Quantum Electron. 50, 458–465 (2014)

    Article  Google Scholar 

  63. Boudebs, G., Cherukulappurath, S., Leblond, H., Troles, J., Smektala, F., Sanchez, F.: Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses. Opt. Commun. 219, 427–433 (2003)

    Article  Google Scholar 

  64. Zhan, C., Zhang, D., Zhu, D., Wang, D., Li, Y., Li, D., Lu, Z., Zhao, L., Nie, Y.: Third- and fifth-order optical nonlinearities in a new stilbazolium derivative. J. Opt. Soc. Am. B 19, 369–375 (2002)

    Article  Google Scholar 

  65. Lawrence, B.L., Cha, M., Torruellas, W.E., Stegeman, G.I., Etemad, S., Baker, G., Kajzar, F.: Measurement of the complex nonlinear refractive-index of single-crystal P-toluene sulfonate at 1064-nm. Appl. Phys. Lett. 64, 2773–2775 (1994)

    Article  Google Scholar 

  66. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, San Diego (2013)

    MATH  Google Scholar 

  67. Maimistov, A.I., Malomed, B.A., Desyatnikov, A.: A potential of incoherent attraction between multidimensional solitons. Phys. Lett. A 254, 179–184 (1999)

    Article  Google Scholar 

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  1. School of Electrical and Information Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

    Md. Jahirul Islam & Javid Atai

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Jahirul Islam, M., Atai, J. Stability of Bragg grating solitons in a semilinear dual-core system with cubic–quintic nonlinearity. Nonlinear Dyn 87, 1693–1701 (2017). https://doi.org/10.1007/s11071-016-3145-y

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