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Recurrence behavior for controllable excitation of rogue waves in a two-dimensional \(\varvec{\mathcal {PT}}\)-symmetric coupler

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Abstract

A (2 + 1)-dimensional variable-coefficient coupled nonlinear Schrödinger equation with different diffractions in parity-time symmetric coupler is studied, and exact solutions in the form of two-component Peregrine solution and rogue wave triplet are derived. Based on these solutions, by adjusting the relation between the maximal value \(Z_\mathrm{m}\) and the exciting location values \(Z_0\) for Peregrine solution and \(Z_1,Z_2\) for rogue wave triplet, recurrence behaviors for controllable excitation of Peregrine solution and rogue wave triplet including complete excitation, peak excitation, rear excitation and initial excitation are discussed in the exponential diffraction decreasing system. This phenomenon of recurrence for controllable excitation is owing to different values of diffractions in two transverse directions.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Zhou, Q., Ekici, M., Sonmezoglu, A., Mirzazadeh, M., Eslami, M.: Optical solitons with Biswas–Milovic equation by extended trial equation method. Nonlinear Dyn. 84, 1883–1900 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Mirzazadeh, M., Eslami, M., Zerrad, E., Mahmood, M.F., Biswas, A., Belic, M.: Optical solitons in nonlinear directional couplers by sine–cosine function method and Bernoulli’s equation approach. Nonlinear Dyn. 81, 1933–1949 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhou, Q., Liu, L., Liu, Y., Yu, H., Yao, P., Wei, C., Zhang, H.: Exact optical solitons in metamaterials with cubic-quintic nonlinearity and third-order dispersion. Nonlinear Dyn. 80, 1365–1371 (2015)

    Article  Google Scholar 

  6. Zhou, Q., Zhu, Q., Yu, H., **ong, X.: Optical solitons in media with time-modulated nonlinearities and spatiotemporal dispersion. Nonlinear Dyn. 80, 983–987 (2015)

    Article  MathSciNet  Google Scholar 

  7. Zhou, Q., Zhu, Q., Yu, H., Liu, Y., Wei, C., Yao, P., Bhrawy, A.H., Biswas, A.: Bright, dark and singular optical solitons in a cascaded system. Laser Phys. 25, 025402 (2015)

    Article  Google Scholar 

  8. Akhmediev, N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrodinger equation. Theor. Math. Phys. 69, 1089–1093 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  10. Ankiewicz, A., Kedziora, D.J., Akhmediev, N.: Rogue wave triplets. Phys. Lett. A 375, 2782–2786 (2011)

    Article  MATH  Google Scholar 

  11. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1056 (2007)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Li, J.T., Han, J.Z., Du, Y.D., Dai, C.Q.: Controllable behaviors of Peregrine soliton with two peaks in a birefringent fiber with higher-order effects. Nonlinear Dyn. 82, 1393–1398 (2015)

    Article  MathSciNet  Google Scholar 

  15. Dai, C.Q., Zhou, G.Q., Zhang, J.F.: Controllable optical rogue waves in the femtosecond regime. Phys. Rev. E 85, 016603 (2012)

    Article  Google Scholar 

  16. Li, J.T., Zhu, Y., Liu, Q.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Vector combined and crossing Kuznetsov-Ma solitons in PT-symmetric coupled waveguides. Nonlinear Dyn 85, 973–980 (2016)

    Article  MathSciNet  Google Scholar 

  17. Wang, Y.Y., Dai, C.Q., Zhou, G.Q., Fan, Y., Chen, L.: Rogue wave and combined breather with repeatedly excited behaviors in the dispersion/diffraction decreasing medium. Nonlinear Dyn (2016). doi:10.1007/s11071-016-3025-5

    Google Scholar 

  18. Chen, H.Y., Zhu, H.P.: Controllable behaviors of spatiotemporal breathers in a generalized variable-coefficient nonlinear Schrödinger model from arterial mechanics and optical fibers. Nonlinear Dyn. 81, 141–149 (2015)

    Article  Google Scholar 

  19. Dai, C.Q., Zhang, J.F.: Controllable dynamical behaviors for spatiotemporal bright solitons on continuous wave background. Nonlinear Dyn. 73, 2049–2057 (2013)

    Article  MathSciNet  Google Scholar 

  20. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT-symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)

    Article  Google Scholar 

  22. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)

    Article  MATH  Google Scholar 

  23. Dai, C.Q., Wang, Y.Y., Zhang, X.F.: Controllable Akhmediev breather and Kuznetsov-Ma soliton trains in PT-symmetric coupled waveguides. Opt. Express 22, 29862 (2014)

    Article  Google Scholar 

  24. Li, J.T., Zhang, X.T., Meng, M., Liu, Q.T., Wang, Y.Y., Dai, C.Q.: Control and management of the combined Peregrine soliton and Akhmediev breathers in PT-symmetric coupled waveguides. Nonlinear Dyn. 84, 473–479 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dai, C.Q., Wang, Y.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 

  26. Chen, Y.X., Jiang, Y.F., Xu, Z.X., Xu, F.Q.: Nonlinear tunnelling effect of combined Kuznetsov-Ma soliton in (3+1)-dimensional PT-symmetric inhomogeneous nonlinear couplers with gain and loss. Nonlinear Dyn. 82, 589–597 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  28. Abdullaeev, F.: Theory of solitons in inhomogeneous media. Wiley, New York (1994)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B25, 16 (1983)

    Article  MATH  Google Scholar 

  31. Serkin, V.N., Hasegawa, A.: Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion management. IEEE J. Sel. Top. Quant. Electron. 8, 418–431 (2002)

    Article  Google Scholar 

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

  33. Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Cheng, L.: Vector spatiotemporal localized structures in (3 + 1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11604395 and 11404289), the Aid Project for the Mainstay Young Teachers in Henan Provincial Institutions of Higher Education of China (2016GGJS-135), the High-level Talents Research and Startup Foundation Projects for Doctors of Zhoukou Normal University (zknu2014120), and the School-based Program of Zhoukou Normal University (zknuB1201605).

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Authors and Affiliations

  1. School of Physics and Telecommunications Engineering, Zhoukou Normal University, Zhoukou, 466001, People’s Republic of China

    Yu Zhu, Wei Qin, Ji-tao Li & **-zhong Han

  2. School of Sciences, Zhejiang A&F University, Lin’an, Zhejiang, 311300, People’s Republic of China

    Chao-qing Dai & Yue-yue Wang

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  1. Yu Zhu
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  2. Wei Qin
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  3. Ji-tao Li
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  4. **-zhong Han
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  6. Yue-yue Wang
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Correspondence to Ji-tao Li or **-zhong Han.

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Zhu, Y., Qin, W., Li, Jt. et al. Recurrence behavior for controllable excitation of rogue waves in a two-dimensional \(\varvec{\mathcal {PT}}\)-symmetric coupler. Nonlinear Dyn 88, 1883–1889 (2017). https://doi.org/10.1007/s11071-017-3350-3

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