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.
Localized structures Modified tanh-function method Different ansätz Variable-coefficient breaking soliton model
This is a preview of subscription content, log in to check access.
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.
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)CrossRefMATHGoogle Scholar
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)MathSciNetGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
Dai, C.Q., Xu, Y.J.: Exact solutions for a Wick-type stochastic reaction Duffing equation. Appl. Math. Model. 39, 7420–7426 (2015)MathSciNetCrossRefGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
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)Google Scholar
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)MathSciNetCrossRefMATHGoogle Scholar
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)MathSciNetMATHGoogle Scholar
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)MathSciNetCrossRefMATHGoogle Scholar
Kudryashov, N.A.: Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3507–3529 (2009)MathSciNetCrossRefMATHGoogle Scholar
Parkes, E.J.: Observations on the tanh–coth expansion method for finding solutions to nonlinear evolution equations. Appl. Math. Comput. 217, 1749–1754 (2010)MathSciNetMATHGoogle Scholar
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)MathSciNetCrossRefMATHGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
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)MathSciNetCrossRefMATHGoogle Scholar
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)MathSciNetCrossRefMATHGoogle Scholar
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)MathSciNetCrossRefMATHGoogle Scholar