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Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons

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Studies on the water waves contribute to the design of the related industries, such as the marine and offshore engineering, while the media with the negative refractive index can be applied as the carrier media in fiber optics. In consideration of the inhomogeneities of the media and nonuniformities of the boundaries in the real physical backgrounds, a quintic time-dependent-coefficient derivative nonlinear Schrödinger equation for certain hydrodynamic wave packets or medium with the negative refractive index is investigated in this paper. Bilinear forms and the N-soliton solutions with respect to the nonzero background, which are different from those in the existing studies, are derived under the certain constraints. Conditions for the dark/anti-dark/gray solitons are deduced due to the properties of the solitons derived via the asymptotic analysis. Effects of the dispersion coefficient \(\lambda (t)\), self-steepening coefficient \(\alpha (t)\), cubic nonlinearity \(\mu (t)\) and quintic nonlinearity \(\nu (t)\) on the interactions between the anti-dark and gray solitons under the certain condition are investigated. Interactions among the dark, anti-dark and gray solitons are discussed under two cases: when \({\alpha (t)}/{\lambda (t)}\) and \({\mu (t)}/{\lambda (t)}\) are the constants, whether the interaction is elastic or not depends on whether \(\lambda (t)\), \(\alpha (t)\) and \(\mu (t)\) are the constants or the functions of t; when \({\alpha (t)}/{\lambda (t)}\) and \({\mu (t)}/{\lambda (t)}\) are related to t, if the velocity of the soliton is a periodic function of t, the propagation of the corresponding soliton is periodic and the corresponding interaction is inelastic. Interactions among the three/four solitons are described to be elastic or inelastic based on the changes in the velocities and waveforms of the three/four solitons after the interactions.

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References

  1. Schneider, W., Yasuda, Y.: Stationary solitary waves in turbulent open-channel flow: analysis and experimental verification. J. Hydraul. Eng. 142, 04015035 (2016)

    Article  Google Scholar 

  2. Gao, X.Y.: Looking at a nonlinear inhomogeneous optical fiber through the generalized higher-order variable-coefficient Hirota equation. Appl. Math. Lett. 73, 143–149 (2017)

    Article  MathSciNet  Google Scholar 

  3. Gao, X.Y.: Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas. Appl. Math. Lett. 91, 165–172 (2019)

    Article  MathSciNet  Google Scholar 

  4. Li, Y.K., Wang, C.X., Liang, C.J., Li, J.D., Liu, W.A.: A simple early warning method for large internal solitary waves in the northern South China Sea. Appl. Ocean Res. 61, 167–174 (2016)

    Article  Google Scholar 

  5. Zhao, X.H., Tian, B., Guo, Y.J., Li, H.M.: Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer-Kaup system in water waves. Mod. Phys. Lett. B 32, 1750268 (2018)

    Article  MathSciNet  Google Scholar 

  6. Zhao, X.H., Tian, B., Xie, X.Y., Wu, X.Y., Sun, Y., Guo, Y.J.: Solitons, Backlund transformation and Lax pair for a (2+1)-dimensional Davey-Stewartson system on surface waves of finite depth. Wave. Random Complex 28, 356–366 (2018)

    Article  Google Scholar 

  7. Benitz, M.A., Lackner, M.A., Schmidt, D.P.: Hydrodynamics of offshore structures with specific focus on wind energy applications. Renew. Sustain. Energy Rev. 44, 692–716 (2015)

    Article  Google Scholar 

  8. Yuan, Y.Q., Tian, B., Liu, L., Wu, X.Y., Sun, Y.: Solitons for the (2+1)-dimensional Konopelchenko-Dubrovsky equations. J. Math. Anal. Appl. 460, 476–486 (2018)

    Article  MathSciNet  Google Scholar 

  9. Yuan, Y.Q., Tian, B., Chai, H.P., Wu, X.Y., Du, Z.: Vector semirational rogue waves for a coupled nonlinear Schrödinger system in a birefringent fiber. Appl. Math. Lett. 87, 50–56 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  11. Yin, H.M., Tian, B., Chai, J., Liu, L., Sun, Y.: Numerical solutions of a variable-coefficient nonlinear Schrödinger equation for an inhomogeneous optical fiber. Comput. Math. Appl. 76, 1827–1836 (2018)

    Article  MathSciNet  Google Scholar 

  12. Hu, Y.H., Zhu, Q.Y.: Dark and gray solitons of (2+1)-dimensional nonlocal nonlinear media with periodic response function. Nonlinear Dyn. 89, 225–233 (2017)

    Article  Google Scholar 

  13. Hu, C.C., Tian, B., Wu, X.Y., Du, Z., Zhao, X.H.: Lump wave-soliton and rogue wave-soliton interactions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in a fluid. Chin. J. Phys. 56, 2395–2403 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hu, C.C., Tian, B., Wu, X.Y., Yuan, Y.Q., Du, Z.: Mixed lump-kink and rogue wave-kink solutions for a (3 + 1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid mechanics. Eur. Phys. J. Plus 133, 40–47 (2018)

    Article  Google Scholar 

  15. Wang, M., Tian, B., Sun, Y., Yin, H.M.: Zhang, Z: Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave stripe and dark rogue wave solutions of a generalized Kadomtsev-Petviashvili equation in fluid mechanics. Chin. J. Phys. 60, 440–449 (2019)

    Article  Google Scholar 

  16. Wazwaz, A.M.: Abundant solutions of various physical features for the (2+1)-dimensional modified KdV–Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. 89, 1727–1732 (2017)

    Article  MathSciNet  Google Scholar 

  17. Du, Z., Tian, B., Chai, H.P., Yuan, Y.Q.: Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein. Commun. Nonlinear Sci. Numer. Simulat. 67, 49–59 (2019)

    Article  Google Scholar 

  18. Lu, X., Ma, W.X., Yu, J., Lin, F.H., Khalique, C.M.: Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)

    Article  MathSciNet  Google Scholar 

  19. Chen, S.S., Tian, B., Sun, Y., Zhang, C.R.: Generalized darboux transformations, rogue waves, and modulation instability for the coherently coupled nonlinear Schrödinger equations in nonlinear optics. Ann. Phys. 531, 1900011 (2019)

    Article  Google Scholar 

  20. Lan, Z.Z.: Dark solitonic interactions for the (3+1)-dimensional coupled nonlinear Schrödinger equations in nonlinear optical fibers. Opt. Laser Technol. 113, 462–466 (2019)

    Article  Google Scholar 

  21. Lan, Z.Z., Hu, W.Q., Guo, B.L.: General propagation lattice Boltzmann model for a variable-coefficient compound KdV-Burgers equation. Appl. Math. Model. 73, 695–714 (2019)

    Article  MathSciNet  Google Scholar 

  22. Zhang, C.R., Tian, B., Liu, L., Chai, H.P., Du, Z.: Vector breathers with the negatively coherent coupling in a weakly birefringent fiber. Wave Motion 84, 68–80 (2019)

    Article  MathSciNet  Google Scholar 

  23. Du, X.X., Tian, B., Wu, X.Y., Yin, H.M., Zhang, C.R.: Lie group analysis, analytic solutions and conservation laws of the (3 + 1)-dimensional Zakharov-Kuznetsov-Burgers equation in a collisionless magnetized electron-positron-ion plasma. Eur. Phys. J. Plus 133, 378–391 (2018)

    Article  Google Scholar 

  24. Lazarides, N., Tsironis, G.P.: Superconducting metamaterials. Phys. Rep. 752, 1–67 (2018)

    Article  MathSciNet  Google Scholar 

  25. Pazynin, L.A., Pazynin, V.L., Sliusarenko, H.O.: Negative refraction of plane electromagnetic waves in non-uniform double-negative media. Opt. Lett. 44, 1125–1128 (2019)

    Article  Google Scholar 

  26. Guo, R., Liu, Y.F., Hao, H.Q., Qi, F.H.: Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium. Nonlinear Dyn. 80, 1221–1230 (2015)

    Article  MathSciNet  Google Scholar 

  27. Golick, V.A., Kadygrob, D.V., Yampol’skii, V.A., Rakhmanov, A.L., Ivanov, B.A., Nori, Franco: Surface Josephson plasma waves in layered superconductors above the plasma frequency: evidence for a negative index of refraction. Phys. Rev. Lett. 104, 187003 (2010)

    Article  Google Scholar 

  28. Kivshar, Y.S., Shadrivov, I.V., Zharov, A.A., Ziolkowski, R.W.: Excitation of guided waves in layered structures with negative refraction. Opt. Express 13, 481–492 (2005)

    Article  Google Scholar 

  29. Marklund, M., Shukla, P.K., Stenflo, L.: Ultrashort solitons and kinetic effects in nonlinear metamaterials. Phys. Rev. E 73, 037601 (2006)

    Article  Google Scholar 

  30. Xu, S., Wang, L., Erdélyi, R., He, J.: Degeneracy in bright-dark solitons of the derivative nonlinear Schrödinger equation. Appl. Math. Lett. 87, 64–72 (2019)

    Article  MathSciNet  Google Scholar 

  31. Li, M., Tian, B., Liu, W.J., Zhang, H.Q., Meng, X.H., Xu, T.: Soliton-like solutions of a derivative nonlinear Schrödinger equation with variable coefficients in inhomogeneous optical fibers. Nonlinear Dyn. 62, 919–929 (2010)

    Article  Google Scholar 

  32. Xu, T., Chen, Y.: Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schrödinger equations. Nonlinear Dyn. 92, 2133–2142 (2018)

    Article  Google Scholar 

  33. Lü, X.: Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation. Chaos 23, 033137 (2013)

    Article  MathSciNet  Google Scholar 

  34. Jenkins, R., Liu, J., Perry, P., Sulem, C.: Soliton resolution for the derivative nonlinear Schrödinger equation. Commun. Math. Phys. 363, 1003–1049 (2018)

    Article  Google Scholar 

  35. Khare, A., Cooper, F., Dawson, J.F.: Exact solutions of a generalized variant of the derivative nonlinear Schrödinger equation in a Scarff II external potential and their stability properties. J. Phys. A 51, 445203 (2018)

    Article  MathSciNet  Google Scholar 

  36. Lü, X., Ma, W.X., Yu, J., Khalique, C.M.: 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 

  37. Triki, H., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Biswas, A., Belic, M.: Gray and black optical solitons with quintic nonlinearity. Optik 154, 354–359 (2018)

    Article  Google Scholar 

  38. Grecu, D., Grecu, A.T., Visinescu, A.: Madelung fluid description of a coupled system of derivative NLS equations. Rom. J. Phys. 57, 180–191 (2012)

    Google Scholar 

  39. Yu, W., Ekici, M., Mirzazadeh, M., Zhou, Q., Liu, W.J.: Periodic oscillations of dark solitons in nonlinear optics. Optik 165, 341–344 (2018)

    Article  Google Scholar 

  40. Li, M., Tian, B., Liu, W.J., Zhang, H.Q., Wang, P.: Dark and antidark solitons in the modified nonlinear Schrödinger equation accounting for the self-steepening effect. Phys. Rev. E 81, 046606 (2010)

    Article  Google Scholar 

  41. Zhang, Y.H., Guo, L.J., He, J.S., Zhou, Z.X.: Darboux transformation of the second-type derivative nonlinear Schrödinger equation. Lett. Math. Phys. 105, 853–891 (2015)

    Article  MathSciNet  Google Scholar 

  42. Triki, H., Wazwaz, A.M.: A new trial equation method for finding exact chirped soliton solutions of the quintic derivative nonlinear Schrödinger equation with variable coefficients. Wave. Random Complex 27, 153–162 (2017)

    Article  Google Scholar 

  43. Jia, T.T., Gao, Y.T., Feng, Y.J., Hu, L., Su, J.J., Li, L.Q., Ding, C.C.: On the quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics. Nonlinear Dyn. 96, 229–241 (2019)

    Article  Google Scholar 

  44. Rogers, C., Chow, K.W.: Localized pulses for the quintic derivative nonlinear Schrödinger equation on a continuous-wave background. Phys. Rev. E 86, 037601 (2012)

    Article  Google Scholar 

  45. Grimshaw, R.H.J., Annenkov, S.Y.: Water wave packets over variable depth. Stud. Appl. Math. 126, 409–427 (2011)

    Article  MathSciNet  Google Scholar 

  46. Ablowitz, M.J., Segur, H.: On the evolution of packets of water waves. J. Fluid Mech. 92, 691–715 (1979)

    Article  MathSciNet  Google Scholar 

  47. Fedele, R., Schamel, H.: Solitary waves in the Madelung’s fluid: connection between the nonlinear Schrödinger equation and the Korteweg–de Vries equation. Eur. Phys. J. B 27, 313–320 (2002)

    Article  Google Scholar 

  48. Slunyaev, A.V.: A high-order nonlinear envelope equation for gravity waves in finite-depth water. J. Exp. Theor. Phys. 101, 926–941 (2005)

    Article  Google Scholar 

  49. Benjamin, T.B., Feir, J.E.: The disintegration of wave trains on deep water. Part 1. J. Fluid Mech. 27, 417–430 (1967)

    Article  Google Scholar 

  50. Benilov, E.S., Flanagan, J.D., HOWLIN, C.P.: Evolution of packets of surface gravity waves over smooth topography. J. Fluid Mech. 533, 171–181 (2005)

    Article  MathSciNet  Google Scholar 

  51. Johnson, R.S.: On the modulation of water waves in the neighbourhood of \(kh\approx 1.363\). Proc. R. Soc. Lond. A 357, 131–141 (1977)

    Article  Google Scholar 

  52. Whitham, G.B.: Non-linear dispersion of water waves. J. Fluid Mech. 27, 399–412 (1967)

    Article  MathSciNet  Google Scholar 

  53. Veeresha, P., Prakasha, D.G.: Solution for fractional Zakharov–Kuznetsov equations by using two reliable techniques. Chin. J. Phys. 60, 313–330 (2019)

    Article  MathSciNet  Google Scholar 

  54. Veeresha, P., Prakasha, D.G.: New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos 29, 013119 (2019)

    Article  MathSciNet  Google Scholar 

  55. Veeresha, P., Prakasha, D.G.: A novel technique for (2+1)-dimensional time-fractional coupled Burgers equations. Math. Comput. Simul. 166, 324–345 (2019)

    Article  MathSciNet  Google Scholar 

  56. Song, N., Zhang, W., Yao, M.H.: Complex nonlinearities of rogue waves in generalized inhomogeneous higher-order nonlinear Schrödinger equation. Nonlinear Dyn. 82, 489–500 (2015)

    Article  Google Scholar 

  57. Song, N., Zhang, W., Wang, P., Xue, Y.K.: Rogue wave solutions and generalized Darboux transformation for an inhomogeneous fifth-order nonlinear Schrödinger equation. J. Funct. space 2017, 13 (2017)

    MATH  Google Scholar 

  58. Zhang, W., Wu, Q.L., Yao, M.H., Dowell, E.H.: Analysis on global and chaotic dynamics of nonlinear wave equations for truss core sandwich plate. Nonlinear Dyn. 94, 21–37 (2018)

    Article  Google Scholar 

  59. Zhang, W., Wu, Q.L., Ma, W.S.: Chaotic wave motions and chaotic dynamic responses of piezoelectric laminated composite rectangular thin plate under combined transverse and in-plane excitations. Int. J. Appl. Mech. 10, 1850114 (2018)

    Article  Google Scholar 

  60. Liu, W.H., Zhang, Y.F.: Optical soliton solutions, explicit power series solutions and linear stability analysis of the quintic derivative nonlinear Schrödinger equation. Opt. Quant. Electron. 51, 65–77 (2019)

    Article  Google Scholar 

  61. Fedele, R.: Envelope solitons versus solitons. Phys. Scr. 65, 502–508 (2002)

    Article  Google Scholar 

  62. Moses, J., Malomed, B.A., Wise, F.W.: Self-steepening of ultrashort optical pulses without self-phase-modulation. Phys. Rev. A 76, 021802 (2007)

    Article  Google Scholar 

  63. Emplit, P., Hamaide, J.P., Reinaud, F., Froehly, C., Bartelemy, A.: Picosecond steps and dark pulses through nonlinear single mode fibers. Opt. Commun. 62, 374–379 (1987)

    Article  Google Scholar 

  64. Il’ichev, A.T.: Envelope solitary waves and dark solitons at a water-ice interface. Proc. Steklov Inst. Math. 289, 152–166 (2015)

    Article  MathSciNet  Google Scholar 

  65. Kivshar, Y.S.: Nonlinear dynamics near the zero-dispersion point in optical fibers. Phys. Rev. A 43, 1677–1679 (1991)

    Article  Google Scholar 

  66. Hamaide, J.P., Emplit, P., Haelterman, M.: Dark-soliton jitter in amplified optical transmission systems. Opt. Lett. 16, 1578–1580 (1991)

    Article  Google Scholar 

  67. Kivshar, Y.S., Haelterman, M., Emplit, P., Hamaide, J.P.: Gordon–Haus effect on dark solitons. Opt. Lett. 19, 19–21 (1994)

    Article  Google Scholar 

  68. Hirota, R., Nagai, A., Nimmo, J.J.C., Gilson, C.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

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Acknowledgements

We express our sincere thanks to the each member of our discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017, and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

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  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Ting-Ting Jia, Yi-Tian Gao, Gao-Fu Deng & Lei Hu

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Jia, TT., Gao, YT., Deng, GF. et al. Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons. Nonlinear Dyn 98, 269–282 (2019). https://doi.org/10.1007/s11071-019-05188-y

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