Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation

Abstract

A (2 + 1)-dimensional variable-coefficient partially nonlocal nonlinear Schrödinger equation is considered, and analytical Peregrine solution (PS) and combined Akhmediev breather (AB) are presented from a reduced transformation and the Darboux transformation method. Based on these analytical solutions, the excitation control for three-dimensional PS and combined AB is investigated by comparing values between the maximum value of the effective propagation distance \(\zeta _{\max }\) and the crest position \(\zeta _{0}\) (for the first-order PS and second-order PS) or the crest position \(\zeta _n(n=1,2,3)\) (for the combined AB). If \(\zeta _{\max }<\zeta _{0}\), \(\zeta _{\max }=\zeta _{0}\) and \(\zeta _{\max }>\zeta _{0}\), the anterior excitation, crest excitation, and tail excitation of the first-order PS and second-order PS can be studied, respectively. Similarly, if \(\zeta _{\max }<\zeta _{n}\), \(\zeta _{\max }=\zeta _{n}\) and \(\zeta _{\max }>\zeta _{n}\), the anterior excitation, crest excitation, and tail excitation of the first first-order PS, the second-order PS and the second first-order PS in the combined AB can be discussed, respectively. The \(k+1\)-layer excited structures for the first-order PS, second-order PS, and the combined AB can be constructed for the fixed value of the Hermite polynomial parameter k.

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References

  1. 1.

    Chen, J.C., Ma, Z.Y., Hu, Y.H.: Nonlocal symmetry, nonlocal symmetry, Darboux transformation and soliton–cnoidal wave interaction solution for the shallow water wave equation. J. Math. Anal. Appl. 460, 987–1003 (2018)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Wang, Y.Y., Zhang, P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonlinear Dyn. 83, 1331–1339 (2016)

    MathSciNet  Google Scholar 

  3. 3.

    Guo, M., Fu, C., Zhang, Y., Liu, J., Yang, H.: Study of Ion-Acoustic solitary waves in a magnetized plasma using the three-dimensional time-space fractional Schamel-KdV equation. Complexity 2018, UNSP 6852548 (2018). https://doi.org/10.1155/2018/6852548

    MATH  Google Scholar 

  4. 4.

    Tao, M., Zhang, N., Gao, D., Yang, H.: Symmetry analysis for three-dimensional dissipation Rossby waves. Adv. Differ. Equ. 2018, 300 (2018). https://doi.org/10.1186/s13662-018-1768-7

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Zhang, Y., Dong, H., Zhang, X., Yang, H.: Rational solutions and lump solutions to the generalized(3 + 1)-dimensional shallow water-like equation. Comput. Math. Appl. 73, 246–252 (2017)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Ding, D.J., Jin, D.Q., Dai, C.Q.: Analytical solutions of differential-difference sine-Gordon equation. Therm. Sci. 21, 1701–1705 (2017)

    Google Scholar 

  7. 7.

    Dai, C.Q., Wang, Y.Y., Biswas, A.: Dynamics of dispersive long waves in fluids. Ocean Eng. 81, 77–88 (2014)

    Google Scholar 

  8. 8.

    Zhang, X.E., Chen, Y., Zhang, Y.: Breather, lump and X soliton solutions to nonlocal KP equation. Comput. Math. Appl. 74, 2341–2347 (2017)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Ma, Z.Y., Chen, J.C., Fei, J.X.: Lump and line soliton pairs to a (2 + 1)-dimensional integrable Kadomtsev–Petviashvili equation. Comput. Math. Appl. 76, 1130–1138 (2018)

    MathSciNet  Google Scholar 

  10. 10.

    Ma, W.X., Yong, X.L., Zhang, H.Q.: Diversity of interaction solutions to the (2 + 1)-dimensional Ito equation. Comput. Math. Appl. 75, 289–295 (2018)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Zhu, S.D., Song, J.F.: Residual symmetries, nth Bäcklund transformation and interaction solutions for (2 + 1)-dimensional generalized Broer–Kaup equations. Appl. Math. Lett. 83, 33–39 (2018)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Zhang, N., Xia, T., Jin, Q.: N-Fold Darboux transformation of the discrete Ragnisco–Tu system. Adv. Differ. Equ. 2018, 302 (2018). https://doi.org/10.1186/s13662-018-1751-3

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kong, L.Q., Liu, J., Jin, D.Q., Ding, D.J., Dai, C.Q.: Soliton dynamics in the three-spine \(\alpha \)-helical protein with inhomogeneous effect. Nonlinear Dyn. 87, 83–92 (2017)

    Google Scholar 

  14. 14.

    Fu, C., Lu, C., Yang, H.W.: Time-space fractional (2 + 1) dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions. Adv. Differ. Equ. 2018, 56 (2018). https://doi.org/10.1186/s13662-018-1512-3

    Article  MATH  Google Scholar 

  15. 15.

    Yuan, Q., Chen, C., Yang, H.: Existence of positive solutions for a Schrodinger–Poisson system with bounded potential and weighted functions in R3. Bound. Value Probl. 2017, 151 (2017). https://doi.org/10.1186/s13661-017-0886-6

    Article  MATH  Google Scholar 

  16. 16.

    Dai, C.Q., Wang, Y.Y., Fan, Y., Yu, D.G.: Reconstruction of stability for Gaussian spatial solitons in quintic-septimal nonlinear materials under PT-symmetric potentials. Nonlinear Dyn. 92, 1351–1358 (2018)

    Google Scholar 

  17. 17.

    Zhang, X.H., Liu, L.S., Wu, Y.H., Cui, Y.J.: Entire blow-up solutions for a quasilinear p-Laplacian Schrodinger equation with a non-square diffusion term. Appl. Math. Lett. 74, 85–93 (2017)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Wang, Y.Y., Chen, L., Dai, C.Q., Zheng, J., Fan, Y.: Exact vector multipole and vortex solitons in the media with spatially modulated cubic-quintic nonlinearity. Nonlinear Dyn. 90, 1269–1275 (2017)

    MathSciNet  Google Scholar 

  19. 19.

    Dai, C.Q., Wang, D.S., Wang, L.L., Zhang, J.F., Liu, W.M.: Quasi-two-dimensional Bose–Einstein condensates with spatially modulated cubic-quintic nonlinearities. Ann. Phys. 326, 2356–2368 (2011)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Wang, X.H., Guo, Q.: The propagation properties of the elliptic Gaussian beam in strongly nonlocal nonlinear media. Acta Phys. Sin. 54, 3183–3188 (2005). (in Chinese)

    Google Scholar 

  21. 21.

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

    MathSciNet  Google Scholar 

  22. 22.

    Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 92, 1261–1269 (2018)

    Google Scholar 

  23. 23.

    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 

  24. 24.

    Yan, Z.Y.: Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation. J. Math. Anal. Appl. 380, 689–696 (2011)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite-Gaussian solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)

    MATH  Google Scholar 

  26. 26.

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

    MATH  Google Scholar 

  27. 27.

    Broad, W.J.: Rogue giants at sea. The New York Times, 11 July 2006

  28. 28.

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

    Google Scholar 

  29. 29.

    Wang, Y.Y., Li, J.T., Dai, C.Q., Chen, X.F., Zhang, J.F.: Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution. Phys. Lett. A 377, 2097–2104 (2013)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Zhang, J.F., Dai, C.Q.: Control of nonautonomous matter rogue waves. Acta Phys. Sin. 65, 050501 (2016)

    Google Scholar 

  31. 31.

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

    MATH  Google Scholar 

  32. 32.

    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. 87, 67–73 (2017)

    Google Scholar 

  33. 33.

    Zhu, Y., Qin, W., Li, J.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Recurrence behavior for controllable excitation of rogue waves in a two-dimensional PT-symmetric coupler. Nonlinear Dyn. 88, 1883–1889 (2017)

    Google Scholar 

  34. 34.

    Dai, C.Q., Wang, X.G., Zhang, J.F.: Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Ann. Phys. 326, 645–656 (2011)

    MATH  Google Scholar 

  35. 35.

    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)

    MathSciNet  Google Scholar 

  36. 36.

    Zhang, J.F., Lou, J.H.: Line optical rogue waves and transmission controlling in inhomogeneous nonlinear waveguides. Acta Opt. Sin. 33, 0919001 (2013). (in Chinese)

    Google Scholar 

  37. 37.

    Wu, L., Li, L., Zhang, J.F.: Controllable generation and propagation of asymptotic parabolic optical waves in graded-index waveguide amplifiers. Phys. Rev. A 78, 013838 (2008)

    Google Scholar 

  38. 38.

    Dai, C.Q., Zhu, S.Q., Wang, L.L., Zhang, J.F.: Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrodinger equation with distributed coefficients. Europhys. Lett. 92, 24005 (2010)

    Google Scholar 

  39. 39.

    Dai, C.Q., Wang, Y.Y., Tian, Q., Zhang, J.F.: The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schröinger equation. Ann. Phys. 327, 512–521 (2012)

    MATH  Google Scholar 

  40. 40.

    Maruno, K., Ohta, Y.: Localized solitons of a (2 + 1)-dimensional nonlocal nonlinear Schrödinger equation. Phys. Lett. A 372, 4446–4450 (2008)

    MATH  Google Scholar 

  41. 41.

    Dai, C.Q., Zhang, J.F.: Exact spatial similaritons and rogons in 2D graded-index waveguides. Opt. Lett. 35, 2651–2653 (2010)

    Google Scholar 

  42. 42.

    Zhu, H.P., Chen, L., Chen, H.Y.: Hermite Gaussian vortex solitons of a (3 + 1)-dimensional partially nonlocal nonlinear Schrödinger equation with variable coefficients. Nonlinear Dyn. 85, 1913–1918 (2016)

    Google Scholar 

  43. 43.

    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Circular rogue wave clusters. Phys. Rev. E 84, 056611 (2011)

    MATH  Google Scholar 

  44. 44.

    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Second-order nonlinear Schrodinger equation breather solutions in the degenerate and rogue wave limits. Phys. Rev. E 85, 066601 (2012)

    Google Scholar 

  45. 45.

    Reeves-Hall, P.C., Taylor, J.R.: Wavelength and duration tunable sub-picosecond source using adiabatic Raman compression. Electron. Lett. 37, 417–418 (2001)

    Google Scholar 

  46. 46.

    Reeves-Hall, P.C., Lewis, S.A.E., Chernikov, S.V., Taylor, J.R.: Picosecond soliton pulse-duration-selectable source based on adiabatic compression in Raman amplifier. Electron. Lett. 36, 622–624 (2000)

    Google Scholar 

  47. 47.

    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)

    Google Scholar 

  48. 48.

    Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients. Phys. Rev. Lett. 90, 113902 (2003)

    Google Scholar 

  49. 49.

    Yang, R.C., Hao, R.Y., Li, L., Shi, X.J., Li, Z.H., Zhou, G.S.: Exact gray multi-soliton solutions for nonlinear Schrodinger equation with variable coefficients. Opt. Commun. 253, 177–185 (2005)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11775185) and the higher school visiting scholar development project (Grant No. FX2013103).

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Correspondence to Yi-Xiang Chen.

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Chen, Y., Xu, F. & Hu, Y. Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn 95, 1957–1964 (2019). https://doi.org/10.1007/s11071-018-4670-7

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Keywords

  • (2 + 1)-Dimensional variable-coefficient nonlinear
  • Partially nonlocal nonlinearity
  • Excitation control
  • Peregrine solution
  • Combined breather