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Rogue waves and modulation instability in an extended Manakov system

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Abstract

We obtain the general nth-order rogue wave solutions of the vector cubic-quintic nonlinear Schrödinger equation with self-steepening, alias extended Manakov system, by means of a nonrecursive Darboux transformation scheme. We show that in such a two-component system, owing to the presence of the self-steepening effect, there would emerge an anomalous Peregrine soliton state on one wave component whose peak can grow three times higher than its background level, at the expense of a heavy falling-off on the other wave component. We also demonstrate other interesting rogue wave dynamics such as coexisting, doublet, quartet, and sextet rogue waves, which depend on the choice of structural parameters. In addition, the modulation instability responsible for the formation of rogue waves is discussed, revealing the broad-range existence of rogue waves, irrespective of what dispersion situations considered.

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References

  1. Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)

    MATH  Google Scholar 

  2. Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean. Springer, Berlin (2009)

    MATH  Google Scholar 

  3. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    MathSciNet  Google Scholar 

  4. Pisarchik, A.N., Jaimes-Reátegui, R., Sevilla-Escoboza, R., Huerta-Cuellar, G., Taki, M.: Rogue waves in a multistable system. Phys. Rev. Lett. 107, 274101 (2011)

    Google Scholar 

  5. Dudley, J.M., Dias, F., Erkintalo, M., Genty, G.: Instabilities, breathers and rogue waves in optics. Nat. Photon. 8, 755–764 (2014)

    Google Scholar 

  6. Wabnitz, S.: Nonlinear Guided Wave Optics: A Testbed for Extreme Waves. IOP Publishing, Bristol (2017)

    Google Scholar 

  7. Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)

    Google Scholar 

  8. Chabchoub, A., Hoffmann, N., Onorato, M., Akhmediev, N.: Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2, 011015 (2012)

    Google Scholar 

  9. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)

    Google Scholar 

  10. Lecaplain, C., Grelu, P., Soto-Crespo, J.M., Akhmediev, N.: Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser. Phys. Rev. Lett. 108, 233901 (2012)

    Google Scholar 

  11. Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)

    Google Scholar 

  12. Tsai, Y.Y., Tsai, J.Y., Lin, I.: Generation of acoustic rogue waves in dusty plasmas through three-dimensional particle focusing by distorted waveforms. Nat. Phys. 12, 573–577 (2016)

    Google Scholar 

  13. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)

    Google Scholar 

  14. Charalampidis, E.G., Cuevas-Maraver, J., Frantzeskakis, D.J., Kevrekidis, P.G.: Rogue waves in ultracold bosonic seas. Rom. Rep. Phys. 70, 504 (2018)

    Google Scholar 

  15. Malomed, B.A., Mihalache, D.: Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)

    Google Scholar 

  16. Yan, Z.Y.: Financial rogue waves. Commun. Theor. Phys. 54, 947–949 (2010)

    MATH  Google Scholar 

  17. Leonetti, M., Conti, C.: Observation of three dimensional optical rogue waves through obstacles. Appl. Phys. Lett. 106, 254103 (2015)

    Google Scholar 

  18. Dudley, J.M., Genty, G., Eggleton, B.J.: Harnessing and control of optical rogue waves in supercontinuum generation. Opt. Express 16, 3644–3651 (2008)

    Google Scholar 

  19. Birkholz, S., Nibbering, E.T.J., Brée, C., Skupin, S., Demircan, A., Genty, G., Steinmeyer, G.: Spatiotemporal rogue events in optical multiple filamentation. Phys. Rev. Lett. 111, 243903 (2013)

    Google Scholar 

  20. Walczak, P., Randoux, S., Suret, P.: Optical rogue waves in integrable turbulence. Phys. Rev. Lett. 114, 143903 (2015)

    MathSciNet  Google Scholar 

  21. Soto-Crespo, J.M., Devine, N., Akhmediev, N.: Integrable turbulence and rogue waves: breathers or solitons? Phys. Rev. Lett. 116, 103901 (2016)

    Google Scholar 

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

    MATH  Google Scholar 

  23. Onorato, M., Osborne, A.R., Serio, M.: Modulational instability in crossing sea states: a possible mechanism for the formation of freak waves. Phys. Rev. Lett. 96, 014503 (2006)

    Google Scholar 

  24. Degasperis, A., Lombardo, S., Sommacal, M.: Integrability and linear stability of nonlinear waves. J. Nonlinear Sci. 28, 1251–1291 (2018)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. Matveev, V.B., Salle, M.A.: Darboux Transformation and Solitons. Springer, Berlin (1991)

    MATH  Google Scholar 

  27. Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809 (1973)

    MathSciNet  MATH  Google Scholar 

  28. Chan, H.N., Malomed, B.A., Chow, K.W., Ding, E.: Rogue waves for a system of coupled derivative nonlinear Schrödinger equations. Phys. Rev. E 93, 012217 (2016)

    MathSciNet  Google Scholar 

  29. Osborne, A.R.: Nonlinear Ocean Waves and the Inverse Scattering Transform. Elsevier, Amsterdam (2010)

    MATH  Google Scholar 

  30. Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Series B Appl. Math. 25, 16–43 (1983)

    MathSciNet  MATH  Google Scholar 

  31. Shrira, V.I., Geogjaev, V.V.: What makes the Peregrine soliton so special as a prototype of freak waves? J. Eng. Math. 67, 11–22 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Chen, S., Baronio, F., Soto-Crespo, J.M., Liu, Y., Grelu, P.: Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations. Phys. Rev. E 93, 062202 (2016)

    MathSciNet  Google Scholar 

  33. Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)

    Google Scholar 

  34. Frisquet, B., Kibler, B., Morin, P., Baronio, F., Conforti, M., Millot, G., Wabnitz, S.: Optical dark rogue wave. Sci. Rep. 6, 20785 (2016)

    Google Scholar 

  35. Liu, W., Zhang, Y., He, J.: Rogue wave on a periodic background for Kaup–Newell equation. Rom. Rep. Phys. 70, 106 (2018)

    Google Scholar 

  36. Lan, Z.Z., Su, J.J.: Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system. Nonlinear Dyn. 96, 2535–2546 (2019)

    Google Scholar 

  37. Chen, S., Soto-Crespo, J.M., Baronio, F., Grelu, P., Mihalache, D.: Rogue-wave bullets in a composite (2+1)D nonlinear medium. Opt. Express 24, 15251–15260 (2016)

    Google Scholar 

  38. Dai, C.Q., Liu, J., Fan, Y., Yu, D.G.: Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Nonlinear Dyn. 88, 1373–1383 (2017)

    Google Scholar 

  39. Chen, S., 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)

    Google Scholar 

  40. Chen, S., Pan, C., Grelu, P., Baronio, F., Akhmediev, N.: Fundamental Peregrine solitons of ultrastrong amplitude enhancement through self-steepening in vector nonlinear systems. Phys. Rev. Lett. 124, 113901 (2020)

    MathSciNet  Google Scholar 

  41. Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716–1740 (2012)

    MathSciNet  MATH  Google Scholar 

  42. 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)

    Google Scholar 

  43. Yan, Z.: Two-dimensional vector rogue wave excitations and controlling parameters in the two-component Gross-Pitaevskii equations with varying potentials. Nonlinear Dyn. 79, 2515–2529 (2015)

    MathSciNet  Google Scholar 

  44. Chen, S., Zhou, Y., Bu, L., Baronio, F., Soto-Crespo, J.M., Mihalache, D.: Super chirped rogue waves in optical fibers. Opt. Express 27, 11370–11384 (2019)

    Google Scholar 

  45. Chabchoub, A., Akhmediev, N.: Observation of rogue wave triplets in water waves. Phys. Lett. A 377, 2590–2593 (2013)

    MATH  Google Scholar 

  46. Chen, S., Cai, X.M., Grelu, P., Soto-Crespo, J.M., Wabnitz, S., Baronio, F.: Complementary optical rogue waves in parametric three-wave mixing. Opt. Express 24, 5886–5895 (2016)

    Google Scholar 

  47. Zhang, G., Yan, Z., Wen, X.Y.: Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schrödinger equations. Proc. R. Soc. A 473, 20170243 (2017)

    MATH  Google Scholar 

  48. Cao, Y., He, J., Mihalache, D.: Families of exact solutions of a new extended (2+1)-dimensional Boussinesq equation. Nonlinear Dyn. 91, 2593–2605 (2018)

    Google Scholar 

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

    Google Scholar 

  50. Chen, S., Mihalache, D.: Vector rogue waves in the Manakov system: diversity and compossibility. J. Phys. A Math. Theor. 48, 215202 (2015)

    MathSciNet  MATH  Google Scholar 

  51. Ye, Y., Zhou, Y., Chen, S., Baronio, F., Grelu, P.: General rogue wave solutions of the coupled Fokas–Lenells equations and non-recursive Darboux transformation. Proc. R. Soc. A 475, 20180806 (2019)

    MathSciNet  Google Scholar 

  52. Xu, S., He, J.: The rogue wave and breather solution of the Gerdjikov–Ivanov equation. J. Math. Phys. 53, 063507 (2012)

    MathSciNet  MATH  Google Scholar 

  53. Zhang, Y., Cheng, Y., He, J.: Riemann-Hilbert method and N-soliton for two-component Gerdjikov–Ivanov equation. J. Nonlinear Math. Phys. 24, 210–223 (2017)

    MathSciNet  MATH  Google Scholar 

  54. Yildirim, Y.: Optical solitons to Gerdjikov–Ivanov equation in birefringent fibers with trial equation integration architecture. Optik 182, 349–355 (2019)

    Google Scholar 

  55. Wu, J.: Integrability aspects and multi-soliton solutions of a new coupled Gerdjikov–Ivanov derivative nonlinear Schrödinger equation. Nonlinear Dyn. 96, 789–800 (2019)

    MATH  Google Scholar 

  56. 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)

    Google Scholar 

  57. Kundu, A.: Explicit auto-Backlund relation through gauge transformation. J. Phys. A Math. Gen. 20, 1107–1114 (1987)

    MATH  Google Scholar 

  58. Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. QE–23, 510–524 (1987)

    Google Scholar 

  59. Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. Academic, San Diego (2007)

    MATH  Google Scholar 

  60. Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic, San Diego (2003)

    Google Scholar 

  61. Grelu, P.: Nonlinear Optical Cavity Dynamics: from Microresonators to Fiber Lasers. Wiley-VCH, Weinheim (2016)

    Google Scholar 

  62. Mihalache, D., Mazilu, D., Crasovan, L.C., Towers, I., Buryak, A.V., Malomed, B.A., Torner, L., Torres, J.P., Lederer, F.: Stable spinning optical solitons in three dimensions. Phys. Rev. Lett. 88, 073902 (2002)

    Google Scholar 

  63. Chen, S., Liu, Y., Mysyrowicz, A.: Unusual stability of a one-parameter family of dissipative solitons due to spectral filtering and nonlinearity saturation. Phys. Rev. A 81, 061806(R) (2010)

    Google Scholar 

  64. Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh. Eksp. Teor. Fiz. 65, 505–516 (1973)

    Google Scholar 

  65. 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)

    Google Scholar 

  66. Zhong, W.P., Belić, M., Malomed, B.A.: Rogue waves in a two-component Manakov system with variable coefficients and an external potential. Phys. Rev. E 92, 053201 (2015)

    MathSciNet  Google Scholar 

  67. Rogers, C., Schief, W.K., Malomed, B.: On modulated coupled systems. canonical reduction via reciprocal transformations. Commun. Nonlinear Sci. Numer. Simulat. 83, 105091 (2020)

    MathSciNet  MATH  Google Scholar 

  68. Frisquet, B., Kibler, B., Fatome, J., Morin, P., Baronio, F., Conforti, M., Millot, G., Wabnitz, S.: Polarization modulation instability in a Manakov fiber system. Phys. Rev. A 92, 053854 (2015)

    Google Scholar 

  69. Kang, J.U., Stegeman, G.I., Aitchison, J.S., Akhmediev, N.: Observation of Manakov spatial solitons in AlGaAs planar waveguides. Phys. Rev. Lett. 76, 3699–3702 (1996)

    Google Scholar 

  70. Bludov, Y.V., Driben, R., Konotop, V.V., Malomed, B.A.: Instabilities, solitons and rogue waves in PT-coupled nonlinear waveguides. J. Opt. 15, 064010 (2013)

    Google Scholar 

  71. Chen, S., Soto-Crespo, J.M., Grelu, P.: Coexisting rogue waves within the (2+1)-component long-wave-short-wave resonance. Phys. Rev. E 90, 033203 (2014)

    Google Scholar 

  72. 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)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants No. 11474051 and No. 11974075) and by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1872).

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Authors and Affiliations

  1. School of Physics and Quantum Information Research Center, Southeast University, Nanjing, 211189, China

    Yanlin Ye, Lili Bu, Changchang Pan & Shihua Chen

  2. China National Accreditation Service for Conformity Assessment (CNAS), Beijing, 100062, China

    Jia Liu

  3. Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, 077125, Magurele-Bucharest, Romania

    Dumitru Mihalache

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A Appendix

A Appendix

The formulas of the functions \(d_{0,1,2}\), \(f_{0,1,2}\), and \(g_{0,1,2}\) in Eqs. (40) and (41) are given by

$$\begin{aligned} d_{0}= & {} -\frac{\xi ^3}{6}+\frac{\mathrm {i}\sqrt{3}\xi ^2}{3\delta } +\frac{\mathrm {i}z\xi }{2}-\frac{\xi }{6\delta ^2}-\frac{\sqrt{3}z}{6\delta }, \\ d_{1}= & {} \rho _1\left( -\frac{\xi ^2}{2}+\frac{\mathrm {i}z}{2}+ \frac{\mathrm {i}\sqrt{3}\phi _1\xi }{3\delta } +\frac{\gamma a_{1}^2}{3\delta ^3}\right) -\frac{\gamma }{2\delta ^2\lambda _0}-\frac{1}{3\phi _1\delta ^3},\\ d_{2}= & {} \rho _2\left( -\frac{\xi ^2}{2}+\frac{\mathrm {i}z}{2}+\frac{\mathrm {i}\sqrt{3}\phi _2\xi }{3\delta } -\frac{\gamma a_{2}^2}{3\delta ^3}\right) -\frac{\gamma }{2\delta ^2\lambda _0}+\frac{1}{3\phi _2\delta ^3},\\ f_{0}= & {} \frac{\xi ^4}{24}-\frac{\mathrm {i}\sqrt{3}\xi ^3}{6\delta }-\frac{\mathrm {i}z\xi ^2}{4} +\frac{\mathrm {i}2\sqrt{3}\xi }{27\delta ^3}-\frac{z^2}{8}+\frac{\mathrm {i}z}{2\delta ^2},\\ f_{1}= & {} \rho _1\left( \frac{\xi ^3}{6}-\frac{\xi ^2}{2\phi _1\delta }-\frac{\gamma a_{1}^2\xi }{2\phi _1\delta ^3}-\frac{\mathrm {i}z\xi }{2}+\frac{\mathrm {i}z\phi _1}{2\delta } +\frac{\mathrm {i}2\sqrt{3}}{27\delta ^3} +\frac{\gamma }{2\delta ^2\lambda _0}\right) \\&+\frac{\xi }{2\delta ^2}\left( \frac{\gamma }{\lambda _0} -\frac{\phi _1}{\delta }\right) ,\\ f_{2}= & {} \rho _2\left( \frac{\xi ^3}{6}+\frac{\xi ^2}{2\phi _2\delta }+\frac{\gamma a_{2}^2\xi }{2\phi _2\delta ^3}-\frac{\mathrm {i}z\xi }{2}-\frac{\mathrm {i}z\phi _2}{2\delta } +\frac{\mathrm {i}2\sqrt{3}}{27\delta ^3} +\frac{\gamma }{2\delta ^2\lambda _0}\right) \\&+\frac{\xi }{2\delta ^2}\left( \frac{\gamma }{\lambda _0} +\frac{\phi _2}{\delta }\right) ,\\ g_{0}= & {} \frac{\xi ^5}{120}-\frac{\mathrm {i}\sqrt{3}\xi ^4}{18\delta }-\frac{\mathrm {i}\xi ^3z}{12} -\frac{\xi ^3}{12\delta ^2}-\frac{\sqrt{3}\xi ^2 z}{12\delta } +\frac{\mathrm {i}\sqrt{3}\xi ^2}{54\delta ^3} -\frac{\xi z^2}{8}\\&+\frac{\mathrm {i}\xi z}{4\delta ^2}+\frac{\xi }{27\delta ^4}-\frac{\mathrm {i}\sqrt{3}z^2}{12\delta } +\frac{5\sqrt{3}z}{27\delta ^3},\\ g_{1}= & {} \rho _1\left[ \frac{\xi ^4}{24}+\frac{\xi ^3(5\phi _1-4)}{18\delta } +\frac{\sqrt{3}}{6\phi _1\delta }\left( \xi z-\frac{2\sqrt{3}}{27\delta ^3} +\frac{\mathrm {i}\gamma }{\delta ^2\lambda _0}\right) -\frac{\mathrm {i}\xi ^2 z}{4}\right. \\&\left. -\frac{z^2}{8} -\frac{\gamma a_{1}^2}{3\delta ^3}\left( \frac{\mathrm {i}\sqrt{3}\xi ^2}{2}+\frac{7\xi \phi _1}{18\delta } -\frac{10\xi }{9\delta }+\frac{3\xi \gamma }{2\phi _1\lambda _0} -\sqrt{3}z\right) \right] \\&+\left( \frac{\gamma }{2\lambda _0}+\frac{\mathrm {i}\sqrt{3}}{3\phi _1\delta }\right) \frac{\xi ^2}{2\delta ^2} +\left( \frac{\mathrm {i}\sqrt{3}\gamma }{3\phi _1\lambda _0} +\frac{20\phi _1-13}{27\delta }\right) \frac{\xi }{2\delta ^3}\\&-\left( \frac{\sqrt{3}}{3\delta \phi _1} +\frac{\mathrm {i}\gamma }{4\lambda _0}\right) \frac{z}{\delta ^2},\\ g_{2}= & {} \rho _2\left[ \frac{\xi ^4}{24}-\frac{\xi ^3(5\phi _2-4)}{18\delta } +\frac{\sqrt{3}}{6\phi _2\delta }\left( \xi z-\frac{2\sqrt{3}}{27\delta ^3} +\frac{\mathrm {i}\gamma }{\delta ^2\lambda _0}\right) -\frac{\mathrm {i}\xi ^2 z}{4}\right. \\&\left. -\frac{z^2}{8} -\frac{\gamma a_{2}^2}{3\delta ^3}\left( \frac{\mathrm {i}\sqrt{3}\xi ^2}{2}+\frac{7\xi \phi _2}{18\delta } -\frac{10\xi }{9\delta }-\frac{3\xi \gamma }{2\phi _2\lambda _0} -\sqrt{3}z\right) \right] \\&+\left( \frac{\gamma }{2\lambda _0}+\frac{\mathrm {i}\sqrt{3}}{3\phi _2\delta }\right) \frac{\xi ^2}{2\delta ^2} +\left( \frac{\mathrm {i}\sqrt{3}\gamma }{3\phi _2\lambda _0} +\frac{20\phi _2-13}{27\delta }\right) \frac{\xi }{2\delta ^3}\\&-\left( \frac{\sqrt{3}}{3\delta \phi _2} +\frac{\mathrm {i}\gamma }{4\lambda _0}\right) \frac{z}{\delta ^2}. \end{aligned}$$

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Ye, Y., Liu, J., Bu, L. et al. Rogue waves and modulation instability in an extended Manakov system. Nonlinear Dyn 102, 1801–1812 (2020). https://doi.org/10.1007/s11071-020-06029-z

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