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Discussions on localized structures based on equivalent solution with different forms of breaking soliton model

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Abstract

Although the modified tanh-function method was proposed to construct variable separation solutions many years ago, two important issues have not been emphasized: (1) the equivalence of variable separation solutions with different expressions by means of the modified tanh-function method; and (2) the universality of lack of physical meanings for some localized structures constructed by variable separation solutions. In this paper, we construct eleven types of variable separation solutions for the variable-coefficient breaking soliton system by means of the modified tanh-function method with three different ansätz, that is, positive-power ansatz, radical sign combined ansatz, and positive and negative power-symmetric ansatz. However, all other solutions with different forms can be re-derived from one of these solutions by means of some re-definitions of p and q. Therefore, solutions with different forms obtained by different ansätz of the modified tanh-function method are essentially equivalent. From this perspective, different ansätz are not really effective to construct so-called new solutions. Moreover, we study some localized coherent structures such as dromion, peakon and compacton for all components of the same model. We find if there is no divergent phenomenon for the other component, these localized coherent structures such as dromion, peakon and compacton are physical, otherwise, these localized coherent structures lose their values in the real application. We hope that these results have potential values to deeply investigate exact solutions and the related localized structures of nonlinear models in physics, engineering and biophysics.

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References

  1. Mirzazadeh, M., Ekici, M., Sonmezoglu, A.: Optical solitons with complex Ginzburg–Landau equation. Nonlinear Dyn. 85, 1979–2016 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zhou, Q., Mirzazadeh, M., Ekici, M., Sonmezoglu, A.: Optical solitons in media with time-modulated nonlinearities and spatiotemporal dispersion. Nonlinear Dyn. 86, 623–638 (2016)

    Article  MATH  Google Scholar 

  3. Xu, S.L., Petrovic, N., Belic, M.R.: Analytical study of solitons in non-Kerr nonlinear negative-index materials. Nonlinear Dyn. 81, 574–579 (2016)

    MathSciNet  Google Scholar 

  4. 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 

  5. 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 

  6. 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 

  7. 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 

  8. Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A.H., Zerrad, E., Biswas, A.: Thirring combo-solitons with cubic nonlinearity and spatio-temporal dispersion. Waves Random Complex Media 26, 204–210 (2016)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Song, L.N., Wang, Q., Zheng, Y., Zhang, H.Q.: A new extended Riccati equation rational expansion method and its application. Chaos Solitons Fractals 31, 548–556 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Tang, X.Y., Lou, S.Y., Zhang, Y.: Localized exicitations in \((2+1)\)-dimensional systems. Phys. Rev. E 66, 046601 (2002)

    Article  MathSciNet  Google Scholar 

  13. Zheng, C.L., Fang, J.P., Chen, L.Q.: New variable separation excitations of a \((2+1)\)-dimensional Broer–Kaup–Kupershmidt system obtained by an extended mapping approach. Z. Natur. A 59, 912–918 (2004)

  14. Dai, C.Q., Wang, Y.Y.: Notes on the equivalence of different variable separation approaches for nonlinear evolution equations. Commun. Nonlinear Sci. Numer. Simul. 19, 19–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dai, C.Q., Wang, Y.Y.: The novel solitary wave structures and interactions in the \((2+1)\)-dimensional Kortweg–de Vries system. Appl. Math. Comput. 208, 453–461 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Dai, C.Q., Wang, Y.Y.: Combined wave solutions of the \((2+1)\)-dimensional generalized Nizhnik–Novikov–Veselov system. Phys. Lett. A 372, 1810–1815 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kudryashov, N.A.: Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3507–3529 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Parkes, E.J.: Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput. 217, 1749–1754 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Parkes, E.J.: A note on solitary travelling-wave solutions to the transformed reduced Ostrovsky equation. Commun. Nonlinear Sci. Numer. Simul. 15, 2769–2771 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, Y.Y., Dai, C.Q.: Caution with respect to “new” variable separation solutions and their corresponding localized structures. Appl. Math. Model. 40, 3475–3482 (2016)

    Article  MathSciNet  Google Scholar 

  21. 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 

  22. Dai, C.Q., Zhang, J.F.: Novel variable separation solutions and localized excitations via the ETM in nonlinear soliton systems. J. Math. Phys. 47, 043501 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dai, C.Q., Zhang, J.F.: New types of interactions based on variable separation solutions via the general projective Riccati equation method. Rev. Math. Phys. 19, 195–226 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sun, F.W., Cai, J.X., Gao, Y.T.: Analytic localized solitonic excitations for the \((2+1)\)-dimensional variable-coefficient breaking soliton model in fluids and plasmas. Nonlinear Dyn. 70, 1889–1901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Arai, A.: Exactly solvable supersymmetric quantum mechanics. J. Math. Anal. Appl. 158, 63–79 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ruan, H.Y.: Some discussions about the variable separating method for solving nonlinear models. Chin. Phys. B 19, 050204 (2010)

    Article  Google Scholar 

  27. Wang, Y.Y., Zhang, Y.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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050011), the National Natural Science Foundation of China (Grant No. 11375007) and the Zhejiang Province Undergraduate Scientific and Technological Innovation Project (Grant No. ). Dr. Chao-Qing Dai is also sponsored by the Foundation of New Century “151 Talent Engineering” of Zhejiang Province of China and Youth Top-notch Talent Development and Training Program of Zhejiang A&F University.

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

    Bin Zhang, Xue-Long Zhang & Chao-Qing Dai

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  1. Bin Zhang
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  2. Xue-Long Zhang
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Correspondence to Chao-Qing Dai.

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Zhang, B., Zhang, XL. & Dai, CQ. Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonlinear Dyn 87, 2385–2393 (2017). https://doi.org/10.1007/s11071-016-3197-z

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  • DOI: https://doi.org/10.1007/s11071-016-3197-z

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