Abstract
Inspired by the mapping method and the direct method of symmetry reduction method, we present a new algorithm, nonlinear Schrödinger equation-based constructive method, to solve complex nonlinear evolution equations. This method can easily construct infinite solutions of the complex nonlinear evolution equations from abundant solutions of the nonlinear Schrödinger equation, including multi-soliton solutions with and without continuous wave background, rational solutions and periodic solutions, and so on. With the aid of symbolic computation, we choose (2+1)-dimensional and (3+1)-dimensional variable-coefficient nonlinear Schrödinger equations to illustrate the validity and advantages of the proposed method. According to exact solutions, we also graphically discuss some interesting soliton-like wave dynamic behaviors, which may be observable in the future experiments. These results are helpful to increase the bit-rate of optical communication.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Zhong, W.P., Belić, M.R., Huang, T.W.: Two-dimensional accessible solitons in PT-symmetric potentials. Nonlinear Dyn. 70, 2027–2034 (2012)
Geng, X.G., Lv, Y.Y.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dyn. 69, 1621–1630 (2012)
Biswas, A., Khalique, C.M.: Stationary solutions for non-linear dispersive Schrödinger’s equation. Nonlinear Dyn. 63, 623–626 (2011)
Zhang, Y., Yang, S., Li, C., Ge, J.Y., Wei, W.W.: Exact solutions and Painleve analysis of a new (2+1)-dimensional generalized KdV equation. Nonlinear Dyn. 68, 445–458 (2012)
Lü, X., Tian, B., Zhang, H.Q., Xu, T., Li, H.: Generalized (2+1)-dimensional Gardner model: bilinear equations, Backlund transformation, lax representation and interaction mechanisms. Nonlinear Dyn. 67, 2279–2290 (2012)
Wang, L., Gao, Y.T., Sun, Z.Y., Qi, F.H., Meng, D.X., Lin, G.D.: Solitonic interactions, Darboux transformation and double Wronskian solutions for a variable-coefficient derivative nonlinear Schrödinger equation in the inhomogeneous plasmas. Nonlinear Dyn. 67, 713–722 (2012)
Lai, X.J., Cai, X.O.: Chirped Wave Solutions of a Generalized (3+1)-Dimensional Nonlinear Schrödinger Equation. Z. Naturforschung A, J. Phys. Sci. 66, 392–400 (2011)
Yang, Z., Ma, S.H., Fang, J.P.: Exact solutions and soliton structures of (2+1)-dimensional Zakharov–Kuznetsov equation. Chin. Phys. B 60, 92–96 (2011)
Ma, S.H., Fang, J.P., Yang Z, R.Q.B.: Chaotic behaviors of the (2+1)-dimensional generalized Breor Kaup system. Chin. Phys. B 21, 050511 (2012)
Clarkson, P.A., Kruskal, M.D.: New similarity solutions of the Boussinesq equation. J. Math. Phys. 30, 2201–2213 (1989)
Lü, Z.S.: A Burgers equation-based constructive method for solving nonlinear evolution equations. Phys. Lett. A 353, 158–160 (2006)
Wang, D.S., Li, X.G.: Localized nonlinear matter waves in a Bose–Einstein condensate with spatially inhomogeneous two- and three-body interactions. J. Phys. B, At. Mol. Opt. Phys. 45, 105301 (2012)
Sun, K., Tian, B., Liu, W.J., Jiang, Y., Qu, Q.X., Wang, P.: Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction. Nonlinear Dyn. 67, 165–175 (2012)
Ma, S.H., Qiang, J.Y., Fang, J.P.: The interaction between solitons and chaotic behaviours of (2+1)-dimensional Boiti–Leon–Pempinelli system. Acta Phys. Sin. 56, 620–626 (2007) (in Chinese)
Chen, W.L., Zhang, W.T., Zhang, L.P., Dai, C.Q.: Non-completely elastic interactions in a (2+1)-dimensional dispersive long wave equation. Chin. Phys. B 21, 110507 (2012)
Li, H.Q., Wan, X.M., Fu, Z.T., Liu, S.K.: New special structures to the (2+1)-dimensional breaking soliton equations. Phys. Scr. 84, 035005 (2011)
Tian, Q., Wu, L., Zhang, J.F., Malomed, B.A., Mihalache, D., Liu, W.M.: Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity. Phys. Rev. E 83, 016602 (2011)
Zhang, J.F., Tian, Q., Wang, Y.Y., Dai, C.Q., Wu, L.: Self-similar optical pulses in competing cubic-quintic nonlinear media with distributed coefficients. Phys. Rev. A 81, 023832 (2010)
Sturdevant, B., Lott, D.A., Biswas, A.: Topological solitons in 1+2 dimensions with time-dependent coefficients. PIER Lett. 10, 69–75 (2009)
Malomed, B.A., Michalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B, Quantum Semiclass. Opt. 7, R53 (2005)
Petrović, N., Belić, M., Zhong, W.P., Xie, R.H., Chen, G.: Exact spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Schrödinger equation for both normal and anomalous dispersion. Opt. Lett. 34, 1609–1611 (2009)
Bélanger, N., Bélanger, P.A.: Bright solitons on a cw background. Opt. Commun. 124, 301–308 (1996)
Shin, H.J.: Soliton scattering from a finite conoidal wave train in a fiber. Phys. Rev. E 63, 026606 (2001)
Shin, H.J.: The dark soliton on a conoidal wave background. J. Phys. A, Math. Gen. 38, 3307–3315 (2005)
Akhmediev, N., Soto-Crespo, J.M., Ankiewicz, A.: How to excite a rogue wave. Phys. Rev. A 80, 043818 (2009)
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, 512–521 (2012)
Dai, C.Q., Zhou, G.Q., Zhang, J.F.: Controllable optical rogue waves in the femtosecond regime. Phys. Rev. E 85, 016603 (2012)
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No. 11005092 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. Y13F050037.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dai, CQ., Zhang, JF. Controllable dynamical behaviors for spatiotemporal bright solitons on continuous wave background. Nonlinear Dyn 73, 2049–2057 (2013). https://doi.org/10.1007/s11071-013-0921-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-013-0921-9