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Vector breathers, rogue and breather-rogue waves for a coupled mixed derivative nonlinear Schrödinger system in an optical fiber

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Abstract

In this paper, a coupled mixed derivative nonlinear Schrödinger system, which describes the short pulses in the femtosecond or picosecond regime of a birefringent optical fiber, is investigated. Based on the known Nth-order breather solutions, we derive the first-order breathers and investigate their properties, e.g., velocities and peak amplitudes, where N is a positive integer. Then, two kinds of the second-order breathers are presented. We construct the Nth-order semirational solutions, with only one spectral parameter involved. Based on the obtained Nth-order semirational solutions, we analytically investigate and graphically illustrate the vector degenerate breathers, rogue and breather-rogue waves. We discuss how certain parameters, e.g., the nonlinear coefficients, affect the shapes of the degenerate breathers, rogue and breather-rogue waves.

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Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. Kuznetsov-Ma breather exhibits waves localized in time and periodic in space [57].

  2. Spatiotemporal breather exhibits waves periodic in both space and time [57].

References

  1. Yin, H.M., Tian, B., Zhao, X.C.: Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system. Appl. Math. Comput. 368, 124768 (2020)

    MathSciNet  MATH  Google Scholar 

  2. Du, Z., Tian, B., Chai, H.P., et al.: Dark-bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber. Appl. Math. Lett. 102, 106110 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yang, D.Y., Tian, B., Wang, M., Zhao, X., Shan, W.R., Jiang, Y.: Lax pair, Darboux transformation, breathers and rogue waves of an N-coupled nonautonomous nonlinear Schrödinger system for an optical fiber or plasma. Nonlinear Dyn. 107, 2657–2666 (2022)

    Article  Google Scholar 

  4. Li, B.Q., Ma, Y.L.: N-order rogue waves and their novel colliding dynamics for a transient stimulated Raman scattering system arising from nonlinear optics. Nonlinear Dyn. 101, 2449–2461 (2020)

    Article  Google Scholar 

  5. Mirzazadeh, M., Arnous, A.H., Mahmood, M.F., et al.: Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach. Nonlinear Dyn. 81, 277–282 (2015)

    Article  MATH  Google Scholar 

  6. Yang, C., Liu, W., Zhou, Q., et al.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Article  Google Scholar 

  7. 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  MATH  Google Scholar 

  8. Dong, M.J., Tian, L.X., Wei, J.D., et al.: Some localized wave solutions for the coupled Gerdjikov-Ivanov equation. Appl. Math. Lett. 122, 107483 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lü, X., Hua, Y.F., Chen, S.J., et al.: Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws. Commun. Nonlinear Sci. Numer. Simul. 95, 105612 (2021)

    Article  MATH  Google Scholar 

  10. Lü, X., Chen, S.J.: New general interaction solutions to the KPI equation via an optional decoupling condition approach. Commun. Nonlinear Sci. Numer. Simul. 103, 105939 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, S.J., Lü, X., Li, M.G., et al.: Derivation and simulation of the M-lump solutions to two (2+1)-dimensional nonlinear equations. Phys. Scr. 96, 095201 (2021)

    Article  Google Scholar 

  12. Chen, S.J., Lü, X., Tang, X.F.: Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 95, 105628 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, X., Zhang, H., Liu, W.: The dynamic characteristics of pure-quartic solitons and soliton molecules. Appl. Math. Model. 102, 305–312 (2022)

    Article  MathSciNet  Google Scholar 

  14. Yan, Y.Y., Liu, W.J.: Soliton rectangular pulses and bound states in a dissipative system modeled by the variable-coefficients complex cubic-quintic Ginzburg-Landau equation. Chin. Phys. Lett. 38, 094201 (2021)

    Article  Google Scholar 

  15. Wang, T.Y., Zhou, Q., Liu, W.J.: Soliton fusion and fission for the high-order coupled nonlinear Schrödinger system in fiber lasers. Chin. Phys. B 31, 020501 (2022)

    Article  Google Scholar 

  16. Yang, C., Liu, W., Zhou, Q., et al.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Article  Google Scholar 

  17. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Considering the shallow water of a wide channel or an open sea through a generalized (2+1)-dimensional dispersive long-wave system. Qual. Theory Dyn. Syst. 21, 104 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gao, X.Y., Guo, Y.J., Shan, W.R.: Auto-Bäcklund transformation, similarity reductions and solitons of an extended (2+1)-dimensional coupled Burgers system in fluid mechanics. Qual. Theory Dyn. Syst. 21, 60 (2022)

  19. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.: Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Nonlinear Dyn. 108, 1599–1616 (2022)

    Article  Google Scholar 

  20. Gao, X.T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, F.Y., Gao, Y.T.: Lie group analysis for a higher-order Boussinesq-Burgers system. Appl. Math. Lett. 132, 108094 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: N-fold Darboux transformation and solitonic interactions for the Kraenkel-Manna-Merle system in a saturated ferromagnetic material. Nonlinear Dyn. (2022) in press, https://doi.org/10.1007/s11071-022-07959-6

  23. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C.: N-fold generalized Darboux transformation and soliton interactions for a three-wave resonant interaction system in a weakly nonlinear dispersive medium. Chaos Solitons Fract. 165, 112786 (2022)

    Article  MathSciNet  Google Scholar 

  24. Yang, D.Y., Tian, B., Hu, C.C., Liu, S.H., Shan, W.R., Jiang, Y.: Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber. Wave. Random Complex (2022) in press, https://doi.org/10.1080/17455030.2021.1983237

  25. Zhou, T.Y., Tian, B.: Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Appl. Math. Lett. 133, 108280 (2022)

    Article  MATH  Google Scholar 

  26. Cheng, C.D., Tian, B., Zhang, C.R., Zhao, X.: Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg-de Vries equation in a fluid. Nonlinear Dyn. 105, 2525–2538 (2021)

    Article  Google Scholar 

  27. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Li, L.Q.: Modified generalized Darboux transformation, degenerate and bound-state solitons for a Laksmanan-Porsezian-Daniel equation in a ferromagnetic spin chain. Chaos Solitons Fract. 162, 112399 (2022)

  28. Shen, Y., Tian, B., Liu, S.H., Zhou, T.Y.: Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Nonlinear Dyn. 108, 2447–2460 (2022)

    Article  Google Scholar 

  29. Cheng, C.D., Tian, B., Ma, Y.X., Zhou, T.Y., Shen, Y.: Pfaffian, breather and hybrid solutions for a (2+1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. Phys. Fluids 34, 115132 (2022)

    Article  Google Scholar 

  30. Zhou, T.Y., Tian, B., Chen, Y.Q., Shen, Y.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2+1)-dimensional generalized Burgers system with the variable coefficients in a fluid. Nonlinear Dyn. 108, 2417–2428 (2022)

    Article  Google Scholar 

  31. Gao, X.Y., Guo, Y.J., Shan, W.R.: Reflecting upon some electromagnetic waves in a ferromagnetic film via a variable-coefficient modified Kadomtsev-Petviashvili system. Appl. Math. Lett. 132, 108189 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, M., Tian, B., Zhou, T.Y.: Darboux transformation, generalized Darboux transformation and vector breathers for a matrix Lakshmanan-Porsezian-Daniel equation in a Heisenberg ferromagnetic spin chain. Chaos Solitons Fract. 152, 111411 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Tian, S.F.: Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation. Appl. Math. Lett. 100, 106056 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lan, Z.Z.: Rogue wave solutions for a coupled nonlinear Schrödinger equation in the birefringent optical fiber. Appl. Math. Lett. 98, 128–134 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, D.Y., Tian, B., Tian, H.Y., Wei, C.C., Shan, W.R., Y., Jiang: Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber. Chaos Solitons Fract. 156, 111719 (2022)

  36. Wang, X., Han, B.: Novel rogue waves and dynamics in the integrable pair-transition-coupled nonlinear Schrödinger equation. Appl. Math. Lett. 99, 105987 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang, J.J., Tian, S.F., Peng, W.Q., et al.: The N-coupled higher-order nonlinear Schrödinger equation: Riemann-Hilbert problem and multi-soliton solutions. Math. Meth. Appl. Sci. 43, 2458–2472 (2020)

    Article  MATH  Google Scholar 

  38. Yan, X.W., Tian, S.F., Dong, M.J., et al.: Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76, 179–186 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chen, S., Soto-Crespo, J.M., Grelu, P.: Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence. Opt. Express. 22, 27632–27642 (2014)

    Article  Google Scholar 

  40. Chen, S., Baronio, F., Soto-Crespo, J.M., et al.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, X.B., Han, B.: The three-component coupled nonlinear Schrödinger equation: Rogue waves on a multi-soliton background and dynamics. EPL 126, 15001 (2019)

    Article  Google Scholar 

  42. Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. J. Differ. Equ. 262, 506–558 (2017)

    Article  MATH  Google Scholar 

  43. Ji, T., Zhai, Y.: Soliton, breather and rogue wave solutions of the coupled Gerdjikov-Ivanov equation via Darboux transformation. Nonlinear Dyn. 101, 619–631 (2020)

    Article  Google Scholar 

  44. Li, B.Q., Ma, Y.L.: Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation. Appl. Math. Comput. 386, 125469 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Hu, B., Xia, T.: A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line. Math. Meth. Appl. Sci. 41, 5112–5123 (2018)

    Article  MATH  Google Scholar 

  46. Hisakado, M., Wadati, M.: Integrable multi-component hybrid nonlinear Schrd̈inger equations. J. Phys. Soc. Jpn. 64, 408–413 (1995)

    Article  MATH  Google Scholar 

  47. Yan, X.W.: Lax pair, Darboux-dressing transformation and localized waves of the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber. Appl. Math. Lett. 107, 106414 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  48. Janutka, A.: Collisions of optical ultra-short vector pulses. J. Phys. A Math. Theor. 41, 285204 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhang, H.Q., Tian, B., Lü, X., et al.: Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations. Phys. Lett. A. 373, 4315–4321 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Matsuno, Y.: The N-soliton solution of a two-component modified nonlinear Schrödinger equation. Phys. Lett. A 375, 3090–3094 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Li, M., Xiao, J.H., Jiang, Y., et al.: Bound-state dark/antidark solitons for the coupled mixed derivative nonlinear Schrd̈inger equations in optical fibers. Eur. Phys. J. D 66, 1–14 (2012)

    Article  Google Scholar 

  52. Li, M., Tian, B., Liu, W.J., et al.: Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers. Eur. Phys. J. D 59, 279–289 (2010)

    Article  Google Scholar 

  53. Song, N., Lei, Y., Cao, D.: Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrödinger equation. Acta Mech. Sin. 38, 1–7 (2022)

    Article  Google Scholar 

  54. Dong, M.J., Tian, L.X., Wei, J.D.: Novel rogue waves for a mixed coupled nonlinear Schrödinger equation on Darboux-Dressing transformation. East Asian J. Appl. Math. 12, 22–34 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  55. Hang, C., Wu, Q.L., Zhang, H.Q.: Breathers and double-pole solutions of coupled mixed derivative nonlinear Schrödinger equations from optical fibers. Mod. Phys. Lett. B 35, 2150373 (2021)

    Article  Google Scholar 

  56. Zhang, H.Q.: Darboux transformation and N-soliton solution for the coupled modified nonlinear Schrödinger equations. Z. Naturforsch. A 67, 711–722 (2012)

    Article  Google Scholar 

  57. Priya, N.V., Senthilvelan, M., Lakshmanan, M.: Akhmediev breathers, Ma solitons, and general breathers from rogue waves: A case study in the Manakov system. Phys. Rev. E 88, 022918 (2013)

    Article  Google Scholar 

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017.

Funding

This project was funded by the National Natural Science Foundation of China under Grant No. 11772017.

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Authors and Affiliations

  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

    Xi-Hu Wu, Yi-Tian Gao, Xin Yu, Liu-Qing Li & Cui-Cui Ding

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  1. Xi-Hu Wu
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  3. Xin Yu
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Wu, XH., Gao, YT., Yu, X. et al. Vector breathers, rogue and breather-rogue waves for a coupled mixed derivative nonlinear Schrödinger system in an optical fiber. Nonlinear Dyn 111, 5641–5653 (2023). https://doi.org/10.1007/s11071-022-08058-2

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