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Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides

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Abstract

With the help of the similarity transformation connected the variable-coefficient nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions in two dimensional graded-index waveguides. Then, we investigate the nonlinear tunneling of controllable rogue waves when they pass through nonlinear barrier and nonlinear well. Our results indicate that the propagation behaviors of rogue waves, such as postpone, sustainment and restraint, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z m and the effective propagation distance corresponding to maximum amplitude of rogue waves Z 0. Postponed, sustained and restrained rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged and decreasing amplitudes by modulating the ratio of the amplitudes of rogue waves to barrier or well height.

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References

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

    Article  MATH  Google Scholar 

  2. Moslem, W.M., Shukla, P.K., Eliasson, B.: Surface plasma rouge waves. Europhys. Lett. 96, 25002 (2011)

    Article  Google Scholar 

  3. 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 

  4. Wen, L., Li, L., Li, Z.D., Zhang, X.F., Liu, W.M.: Matter rogue wave in Bose–Einstein condensates with attractive atomic interaction. Eur. Phys. J. D 64, 473 (2011)

    Article  Google Scholar 

  5. Yan, Z.Y.: Financial rogue waves appearing in the coupled nonlinear volatility and option pricing model. Phys. Lett. A 375, 4274 (2011)

    Article  MATH  Google Scholar 

  6. 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 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  8. Zhong, W.P., Belic, M.R., Xia, Y.Z.: Special soliton structures in the (2+1)-dimensional nonlinear Schrodinger equation with radially variable diffraction and nonlinearity coefficients. Phys. Rev. E 83, 036603 (2011)

    Article  Google Scholar 

  9. Zhong, W.P., Belic, M.R., Huang, T.W.: Two-dimensional accessible solitons in PT-symmetric potentials. Nonlinear Dyn. 70, 2027 (2012)

    Article  MathSciNet  Google Scholar 

  10. Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B, Quantum Semiclass. Opt. 7, R53 (2005)

    Article  Google Scholar 

  11. Wu, X.F., Hua, G.S., Ma, Z.Y.: Evolution of optical solitary waves in a generalized nonlinear Schrodinger equation with variable coefficients. Nonlinear Dyn. 70(3), 2259 (2012). doi:10.1007/s11071-012-0616-7

    Article  MathSciNet  Google Scholar 

  12. Dai, C.Q., Wang, Y.Y., Zhang, J.F.: Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrodinger equation. Opt. Lett. 35, 1437 (2010)

    Article  Google Scholar 

  13. 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 

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

    Article  Google Scholar 

  15. Erkintalo, M., Genty, G., Dudley, J.M.: Rogue-wave-like characteristics in femtosecond supercontinuum generation. Opt. Lett. 34, 2468 (2009)

    Article  Google Scholar 

  16. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Vector rogue waves in binary mixtures of Bose–Einstein condensates. Opt. Lett. 34, 3015 (2009)

    Article  Google Scholar 

  17. Newell, A.C.: Nonlinear tunnelling. J. Math. Phys. 19, 1126 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Serkin, V.N., Belyaeva, T.L.: High-energy optical Schrödinger solitons. JETP Lett. 74, 573 (2001)

    Article  Google Scholar 

  19. Serkin, V.N., Chapela, V.M., Persino, J., Belyaeva, T.L.: Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides. Opt. Commun. 192, 237 (2001)

    Article  Google Scholar 

  20. Wang, J.F., Li, L., Jia, S.T.: Nonlinear tunneling of optical similaritons in nonlinear waveguides. J. Opt. Soc. Am. B 25, 1254 (2008)

    Article  Google Scholar 

  21. Belyaeva, T.L., Serkin, V.N., Hernandez-Tenorio, C., Garcia-Santibanez, F.: Enigmas of optical and matter-wave soliton nonlinear tunneling. J. Mod. Opt. 57, 1087 (2010)

    Article  MATH  Google Scholar 

  22. Zhong, W.P., Belic, M.R.: Soliton tunneling in the nonlinear Schrödinger equation with variable coefficients and an external harmonic potential. Phys. Rev. E 81, 056604 (2010)

    Article  Google Scholar 

  23. Dai, C.Q., Chen, R.P., Zhang, J.F.: Analytical spatiotemporal similaritons for the generalized (3+1)-dimensional Gross–Pitaevskii equation with an external harmonic trap. Chaos Solitons Fractals 44, 862 (2011)

    Article  MathSciNet  Google Scholar 

  24. Wang, Y.Y., He, J.D., Dai, C.Q.: Spatiotemporal self-similar nonlinear tunneling effects in the (3+1)-dimensional inhomogeneous nonlinear medium with the linear and nonlinear gain. Opt. Commun. 284, 4738 (2011)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Akhmediev, N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)

    Google Scholar 

  27. Wu, L., Zhang, J.F., Li, L., Tian, Q., Porsezian, K.: Similaritons in nonlinear optical systems. Opt. Express 16, 6352 (2008)

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Belyaeva, T.L., Serkin, V.N.: Particle-wave duality of solitons: scaling symmetry breaking in soliton scattering, hidden role of the self-interaction energy and the de Broglie video-soliton. Nanociencia Moletronica 9, 1715 (2011)

    Google Scholar 

  30. Belyaeva, T.L., Serkin, V.N.: Wave-particle duality of solitons and solitonic analog of the Ramsauer–Townsend effect. Eur. Phys. J. D 66, 153 (2012)

    Article  Google Scholar 

  31. Schrödinger, E.: The Fundamental Idea of Wave Mechanics, Nobel Lecture, 1933, pp. 305–316, Biography, pp. 317–319. World Scientific, Singapore (1998)

    Google Scholar 

  32. Price, W.C., Chissick, S.S., Ravensdale, T., de Broglie, L.: Wave Mechanics: The First Fifty Years. Wiley, New York (1973)

    Google Scholar 

  33. Yan, Z.Y., Konotop, V.V., Akhmediev, N.: Three-dimensional rogue waves in nonstationary parabolic potentials. Phys. Rev. E 82, 036610 (2010)

    Article  MathSciNet  Google Scholar 

  34. Akhmediev, N., Eleonskii, V.M., Kulagin, N.E.: N-modulation signals in a single-mode optical waveguide under nonlinear conditions. Sov. Phys. JETP 62, 894 (1985)

    Google Scholar 

  35. Dai, C.Q., Wang, Y.Y., Zhang, J.F.: Nonlinear similariton tunneling effect in the birefringent fiber. Opt. Express 18, 17548 (2010)

    Article  Google Scholar 

  36. Dai, C.Q., Wang, Y.Y., Wang, X.G.: Ultrashort self-similar solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients in the inhomogeneous fiber. J. Phys. A, Math. Theor. 44, 155203 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. Dai, C.Q., Xu, Y.J., Chen, R.P., Zhang, J.F.: Self-similar optical beam in nonlinear waveguides. Eur. Phys. J. D 59, 457 (2010)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Applied nonlinear Science and Technology from the most important among all the top priority disciplines of Zhejiang Province.

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  1. School of Science, Zhejiang Lishui University, Lishui, Zhejiang, 323000, China

    Hai-Ping Zhu

  2. School of Science, Ningbo University, Ningbo, Zhejiang, 315211, China

    Hai-Ping Zhu

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Correspondence to Hai-Ping Zhu.

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Zhu, HP. Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn 72, 873–882 (2013). https://doi.org/10.1007/s11071-013-0759-1

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