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Embedded solitons in the \((2+1)\)-dimensional sine-Gordon equation

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Abstract

An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known equation to find solutions of the unknown. In this paper, we employ an improved similarity transformation to transform the \((2+1)\)-dimensional (D) sine-Gordon (SG) equation into the \((1+1)\)-D SG equation and obtain non-rational solutions of the \((2+1)\)-D SG equation by utilizing the known solutions of the \((1+1)\)-D SG equation. Based on the solutions obtained, and with the help of special choices of the involved solution parameters, several localized structures of the \((2+1)\)-D SG model are analyzed on a finite background, such as the embedded hourglass, split silo, dumbbell, and pie solitons. Their spatiotemporal profiles are displayed, and their properties are discussed.

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References

  1. Lamb, G.L.: Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev. Mod. Phys. 43, 99–124 (1971)

    Article  MathSciNet  Google Scholar 

  2. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Lett. 30, 1262–1264 (1973)

    Article  MathSciNet  Google Scholar 

  3. Aktosun, T., Demontis, F., Mee, C.: Exact solutions to the sine-Gordon equation. J. Math. Phys. 51(12), 123521 (2010)

    Article  MathSciNet  Google Scholar 

  4. Costabile, G., Parmentier, R.D., Savo, B., McLaughlin, D.W., Scott, A.C.: Exact solutions of the sine-Gordon equation describing oscillations in a long (but finite) Josephson junction. Appl. Phys. Lett. 32, 587–589 (1978)

    Article  Google Scholar 

  5. Bowcock, P., Corrigan, E., Zambon, C.: Some aspects of jump-defects in the quantum sine-Gordon model. J. High Energy Phys. 2005, 023 (2005)

    Article  MathSciNet  Google Scholar 

  6. Zhong, W.P., Belić, M., Petrović, M.: Solitary and extended waves in the generalized sinh-Gordon equation with a variable coefficient. Nonlinear Dyn. 76, 717–723 (2014)

    Article  MathSciNet  Google Scholar 

  7. Zhong, W.P., Belić, M., Zhang, Y.: Dark spatiotemporal optical solitary waves in self-defocusing nonlinear media. Nonlinear Dyn. 87(5), 2171–2177 (2016)

    Google Scholar 

  8. Zhong, W.P., Belić, M.: Resonance soliton clusters by azimuthally-modulation in self-focusing and self-defocusing materials. Nonlinear Dyn. 70, 2091–2102 (2013)

    Article  Google Scholar 

  9. Lamb, G.L.: Elements of Soliton Theory. Wiley, New York (1980)

    MATH  Google Scholar 

  10. Mukherjee, R., Balakrishnan, R.: Moving curves of the sine-Gordon equation: new links. Phys. Lett. A 373, 6347–6362 (2008)

    Article  MathSciNet  Google Scholar 

  11. Ferman, M.E., Kruglov, V.I., Thomsen, B.C., Dudley, J.M., Harvey, J.D.: Self-similar propagation and amplification of parabolic pulses in optical fibers. Phys. Rev. Lett. 84, 6010 (2000)

    Article  Google Scholar 

  12. Finot, C., Millot, G., Billet, C.: Experimental generation of parabolic pulses via Raman amplification in optical fiber. Opt. Express 11, 1547 (2003)

    Article  Google Scholar 

  13. Soljacic, M., Segev, M., Menyuk, C.R.: Self-Similarity and fractals in soliton-supporting systems. Phys. Rev. E 61, 1048 (2000)

    Article  Google Scholar 

  14. Zhong, W.P., Belić, M., Huang, T.: Rogue wave solutions to the generalized nonlinear Schrödinger equation with variable coefficients. Phys. Rev. E 87, 065201 (2013)

    Article  Google Scholar 

  15. Yang, Z.P., Zhong, W.P., Belić, M., Zhang, Y.: Controllable optical rogue waves via nonlinearity management. Opt. Express 26, 7587 (2018)

    Article  Google Scholar 

  16. Novikov, S.P., Veselov, A.P.: Two-dimensional Schrödinger operator: inverse scattering transform and evolutional equations. Physica D 18, 267–273 (1986)

    Article  MathSciNet  Google Scholar 

  17. Fokas, A.S., Santini, P.M.: Dromions and a boundary value problem for the Davey–tewartson equation. Physica D 44, 99–130 (1990)

    Article  MathSciNet  Google Scholar 

  18. Konopelchenko, B.G., Rogers, C.: On (\(2+1\))-dimensional NL system of Lowner type. Phys. Lett. A 158(8), 391–397 (1991)

    Article  MathSciNet  Google Scholar 

  19. Konopelchenko, B.G., Rogers, C.: On generalized Loewner systems: novel integrable equations in (\(2+1\))-dimensions. J. Math. Phys. 34, 214–242 (1993)

    Article  MathSciNet  Google Scholar 

  20. Leblond, H., Mihalache, D.: Ultra-short light bullets described by the two-dimensional sine-Gordon equation. Phys. Rev. A 81, 063815 (2010)

    Article  Google Scholar 

  21. Leblond, H., Triki, H., Mihalache, D.: Derivation of a generalized double-sine-Gordon equation describing ultrashort-soliton propagation in optical media composed of multilevel atom. Phys. Rev. A 86, 063825 (2012)

    Article  Google Scholar 

  22. Hirota, R.: Exact three-soliton solution of the two-dimensional sine-Gordon Equation. J. Phys. Soc. Jpn. 35, 1566 (1973)

    Article  Google Scholar 

  23. Kaliappan, P., Lakshmanan, M.: Kadomstev-Petviashvili and two-dimensional sine-Gordon equations: reduction to Painleve transcendents. J. Phys. A Math. Gen. 12(10), L249–L252 (1979)

    Article  Google Scholar 

  24. Leibbrandt, G.: New exact solutions of the classical sine-Gordon equation in \(2+1\) and \(3+1\) dimensions. Phys. Rev. Lett. 41, 435 (1978)

    Article  Google Scholar 

  25. Bogolyubskii, I.L., Makhankov, V.G.: Dynamics of spherically symmetrical pulson of large amplitude. JETP Lett. 25(2), 107–110 (1977)

    Google Scholar 

  26. Shi, R., Song, Z., Feng, T., Wang, G., Wang, X.: Analytical soliton solutions of the (\(2+1\))D sine-Gordon equation. Nonlinear Dyn. 88, 255–262 (2017)

    Article  MathSciNet  Google Scholar 

  27. Zhao, Z.L., Han, B.: Residual symmetry, Bäcklund transformation and CRE solvability of a (\(2+1\))D nonlinear system. Nonlinear Dyn. 94, 461–474 (2018)

    Article  Google Scholar 

  28. Wazwaz, A.M., Kaur, L.: Complex simplified Hirota’s forms and Lie symmetry analysis for multiple real and complex soliton solutions of the modified KdV-SG equation. Nonlinear Dyn. 95, 2209–2215 (2019)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 61775150 and 61701121 and by Guangdong Basic and Applied Basic Research Foundation (2017A030313346 and 2019A1515010749), China, and by Key projects of basic research and applied basic research in universities of Guangdong Province, China, under Grant No. 2018GKZDXM001. The work at the Texas A&M University at Qatar was supported by the NPRP 11S-1126-170033 project with the Qatar National Research Fund (a member of the Qatar Foundation). MRB acknowledges support from the Al Sraiya Holding Group.

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Authors and Affiliations

  1. School of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

    WenYe Zhong & Guofa Cai

  2. Department of Electronic Engineering, Shunde Polytechnic, Shunde, 528300, Guangdong Province, China

    Wei-Ping Zhong

  3. Texas A&M University at Qatar, 23874, Doha, Qatar

    Wei-Ping Zhong & Milivoj R. Belić

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  1. WenYe Zhong
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  2. Wei-Ping Zhong
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  3. Milivoj R. Belić
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  4. Guofa Cai
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Correspondence to Wei-Ping Zhong or Guofa Cai.

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Zhong, W., Zhong, WP., Belić, M.R. et al. Embedded solitons in the \((2+1)\)-dimensional sine-Gordon equation. Nonlinear Dyn 100, 1519–1526 (2020). https://doi.org/10.1007/s11071-020-05561-2

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