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Baseband modulation instability, rogue waves and state transitions in a deformed Fokas–Lenells equation

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Abstract

We study a deformed Fokas–Lenells equation which is related to the integrable derivative nonlinear Schrödinger hierarchy with higher-order nonholonomic constraint. The baseband modulation instability as an origin of rogue waves is displayed. The explicit rogue wave solutions are obtained via the Darboux transformation. Typical rogue wave patterns such as the standard rogue wave, dark rogue wave and twisted rogue wave pair in three different components of the deformed Fokas–Lenells equation are presented. Besides, the state transitions between rogue waves and solitons are analytically found when the modulation instability growth rate tends to zero in the zero-frequency perturbation region. The explicit soliton solutions under the special parameter condition are given. The anti-dark and W-shaped solitons in their respective components are shown.

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References

  1. Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Lenells, J., Fokas, A.S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity 22, 11 (2008)

    Article  MATH  Google Scholar 

  4. Lenells, J., Fokas, A.S.: An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons. Inverse Probl. 25, 115006 (2009)

    Article  MATH  Google Scholar 

  5. Lenells, J.: Dressing for a novel integrable generalization of the nonlinear Schrödinger equation. J. Nonlinear Sci. 20, 709 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Matsuno, Y.: A direct method of solution for the Fokas–Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. J. Phys. A Math. Theor. 45, 475202 (2012)

    Article  MATH  Google Scholar 

  7. Wright III, O.C.: Some homoclinic connections of a novel integrable generalized nonlinear Schrödinger equation. Nonlinearity 22, 2633 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. He, J.S., Xu, S.W., Porsezian, K.: Rogue waves of the Fokas–Lenells equation. J. Phys. Soc. Jpn. 81, 124007 (2012)

    Article  Google Scholar 

  9. Chen, S.H., Song, L.Y.: Peregrine solitons and algebraic soliton pairs in Kerr media considering spacetime correction. Phys. Lett. A 378, 1228 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Triki, H., Wazwaz, A.M.: Combined optical solitary waves of the Fokas–Lenells equation. Wave Random Complex 27, 587 (2017)

    Article  MathSciNet  Google Scholar 

  11. Guo, B.L., Ling, L.M.: Riemann–Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. J. Math. Phys. 53, 073506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, Y., Yang, J.W., Chow, K.W., Wu, C.F.: Solitons, breathers and rogue waves for the coupled Fokas–Lenells system via Darboux transformation. Nonlinear Anal. RWA 33, 237 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, S.H., Ye, Y., Soto-Crespo, J.M., Grelu, P., Baronio, F.: Peregrine solitons beyond the threefold limit and their two-soliton interactions. Phys. Rev. Lett. 121, 104101 (2018)

    Article  Google Scholar 

  14. Ling, L.M., Feng, B.F., Zhu, Z.N.: General soliton solutions to a coupled Fokas–Lenells equation. Nonlinear Anal. RWA 40, 185 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kupershmidt, B.A.: KdV6: an integrable system. Phys. Lett. A 372, 2634 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kundu, A.: Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics. Theor. Math. Phys. 167, 800 (2011)

    Article  MathSciNet  Google Scholar 

  17. Wang, X., Liu, C., Wang, L.: Rogue waves and W-shaped solitons in the multiple self-induced transparency system. Chaos 27, 093106 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ren, Y., Yang, Z.Y., Liu, C., Yang, W.L.: Different types of nonlinear localized and periodic waves in an erbium-doped fiber system. Phys. Lett. A 379, 2991 (2015)

    Article  Google Scholar 

  19. Wang, L., Liu, C., Wu, X., Wang, X., Sun, W.R.: Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation. Nonlinear Dyn. 94, 977 (2018)

    Article  Google Scholar 

  20. Wang, L., Wu, X., Zhang, H.Y.: Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects. Phys. Lett. A 382, 2650 (2018)

    Article  MathSciNet  Google Scholar 

  21. Ren, Y., Liu, C., Yang, Z.Y., Yang, W.L.: Polariton superregular breathers in a resonant erbium-doped fiber. Phys. Rev. E 98, 062223 (2018)

    Article  Google Scholar 

  22. Kundu, A.: Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schrödinger and the Lenells–Fokas equation. J. Math. Phys. 51, 022901 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054 (2007)

    Article  Google Scholar 

  24. Dysthe, K., Krogstad, H.E., Muller, P.: Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009)

    Article  Google Scholar 

  26. Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675 (2009)

    Article  MATH  Google Scholar 

  27. Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B Appl. Math. 25, 16 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  29. He, J.S., Zhang, H.R., Wang, L.H., Porsezian, K., Fokas, A.S.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013)

    Article  Google Scholar 

  30. Zhang, G.Q., Yan, Z.Y., Wen, X.Y., Chen, Y.: Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. Phys. Rev. E 95, 042201 (2017)

    Article  MathSciNet  Google Scholar 

  31. Wei, J., Wang, X., Geng, X.G.: Periodic and rational solutions of the reduced Maxwell–Bloch equations. Commun. Nonlinear Sci. Numer. Simul. 59, 1 (2018)

    Article  MathSciNet  Google Scholar 

  32. Wang, X., Zhang, J.L., Wang, L.: Conservation laws, periodic and rational solutions for an extended modified Korteweg–de Vries equation. Nonlinear Dyn. 92, 1507 (2018)

    Article  Google Scholar 

  33. Li, P., Wang, L., Kong, L.Q., Wang, X., **e, Z.Y.: Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation. Appl. Math. Lett. 85, 110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, J.G., Zhang, Y.F.: Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation. Results Phys. 10, 94 (2018)

    Article  Google Scholar 

  35. Liu, J.G., Zhang, Y.F., Muhammad, I.: Resonant soliton and complexiton solutions for (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Comput. Math. Appl. 75, 3939 (2018)

    Article  MathSciNet  Google Scholar 

  36. Liu, J.G., Yang, X.J., Cheng, M.H., Feng, Y.Y., Wang, Y.D.: Abound rogue wave type solutions to the extended (3+ 1)-dimensional Jimbo–Miwa equation. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.03.034

  37. Chen, J.C., Zhu, S.D.: Residual symmetries and soliton–cnoidal wave interaction solutions for the negative-order Korteweg–de Vries equation. Appl. Math. Lett. 73, 136 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chen, J.C., Ma, Z.Y.: Consistent Riccati expansion solvability and soliton–cnoidal wave interaction solution of a (2 + 1)-dimensional Korteweg–de Vries equation. Appl. Math. Lett. 64, 87 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chen, J.C., Ma, Z.Y., Hu, Y.H.: Nonlocal symmetry, Darboux transformation and soliton–cnoidal wave interaction solution for the shallow water wave equation. J. Math. Anal. Appl. 460, 987 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81, 046602 (2010)

    Article  MathSciNet  Google Scholar 

  41. Tao, Y.S., He, J.S.: Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E 85, 026601 (2012)

    Article  Google Scholar 

  42. Bandelow, U., Akhmediev, N.: Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case. Phys. Lett. A 376, 1558 (2012)

    Article  MATH  Google Scholar 

  43. Chen, S.H.: Twisted rogue-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 023202 (2013)

    Article  Google Scholar 

  44. Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044102 (2012)

    Article  Google Scholar 

  45. Ling, L.M., Guo, B.L., Zhao, L.C.: High-order rogue waves in vector nonlinear Schrödinger equations. Phys. Rev. E 89, 041201 (2014)

    Article  Google Scholar 

  46. Chen, S.H., Song, L.Y.: Rogue waves in coupled Hirota systems. Phys. Rev. E 87, 032910 (2013)

    Article  Google Scholar 

  47. Wang, X., Liu, C., Wang, L.: Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations. J. Math. Anal. Appl. 449, 1534 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  48. Baronio, F., Conforti, M., Degasperis, A., Lombardo, S., Onorato, M., Wabnitz, S.: Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113, 034101 (2014)

    Article  Google Scholar 

  49. Chen, S., Baronio, F., Soto-Crespo, J.M., Grelu, P., Mihalache, D.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants (11705290, 11875126, 11671075, 11801133), the China Postdoctoral Science Foundation funded sixtieth and sixty-fourth batches (2016M602252, 2018M640678), the Young Scholar Foundation of ZUT (2018XQG16) and the Key Research Projects of Henan Higher Education Institutions (18A110038).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, 450001, Henan, China

    **n Wang & Jiao Wei

  2. College of Science, Zhongyuan University of Technology, Zhengzhou, 450007, Henan, China

    **n Wang & Jianlin Zhang

  3. School of Mathematics and Physics, North China Electric Power University, Bei**g, 102206, China

    Lei Wang

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  1. **n Wang
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Wang, X., Wei, J., Wang, L. et al. Baseband modulation instability, rogue waves and state transitions in a deformed Fokas–Lenells equation. Nonlinear Dyn 97, 343–353 (2019). https://doi.org/10.1007/s11071-019-04972-0

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