Abstract
The coupled mixed derivative nonlinear Schrödinger equations, correlated with Lax pairsinvolving \(3\times 3\) matrices, arise as a significant integrable system in many physical contexts. By constructing the Darboux transformation, breathing bright–dark solitons, mixed kink solutions, mixed periodic solutions, semi-rational rogue wave solutions and various types of mixed soliton solutions are attained. Furthermore, breather fusion, breather fission and higher-order solutions are derived, whose dynamic behaviors are illustrated graphically to distinguish different parameter values.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Data availability
The data that support the findings of this article are available from the corresponding author upon reasonable request.
References
Li, M., Xiao, J.H., Liu, W.J., Wang, P., Qin, B., Tian, B.: Mixed-type vector solitons of the \(N\)-coupled mixed derivative nonlinear Schrödinger equations from optical fibers. Phys. Rev. E 87(3), 032914 (2013)
Li, M., Xiao, J.H., Qin, B., Wang, M., Tian, B.: Vector-soliton bound states for the coupled mixed derivative nonlinear Schrödinger equations in optical fibers. Wave Motion 50(1), 1–10 (2013)
Janutka, A.: Collisions of optical ultra-short vector pulses. J. Phys. A 41(28), 285204 (2008)
Li, M., Tian, B., Liu, W.J., Jiang, Y., Sun, K.: Dark and anti-dark vector solitons of the coupled modified nonlinear Schrödinger equations from the birefringent optical fibers. Eur. Phys. J. D 59(2), 279–289 (2010)
Li, M., Xiao, J.H., Jiang, Y., Wang, M., Tian, B.: Bound-state dark/antidark solitons for the coupled mixed derivative nonlinear Schrödinger equations in optical fibers. Eur. Phys. J. D 66(11), 1–14 (2012)
Zhang, H.Q., Tian, B., Lü, X., Li, H., Meng, X.H.: Soliton interaction in the coupled mixed derivative nonlinear Schrödinger equations. Phys. Lett. A 373(47), 4315–4321 (2009)
Porsezian, K.: Soliton models in resonant and nonresonant optical fibers. Pramana J. Phys. 57(5), 1003–1039 (2001)
Geng, X.G., Li, R.M., Xue, B.: A vector general nonlinear Schrödinger equation with \((m+n)\) components. J. Nonlinear Sci. 30, 991 (2020)
Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger-type equation. J. Math. Phys. 19, 798 (1978)
Zhang, Y., Yang, J.W.: Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation. Nonlinear Anal-Real. 33, 237–252 (2017)
Wang, L., Liu, C., Wu, X., Wang, X., Sun, W.R.: Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation. Nonlinear Dyn. 94, 977 (2018)
Guo, R., Hao, H.Q., Zhang, L.L.: Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74(3), 701–709 (2013)
Ji, T., Zhai, Y.: Soliton, breather and rogue wave solutions of the coupled Gerdjikov-Ivanov equation via Darboux transformation. Nonlinear Dyn. 101(1), 619–631 (2020)
Tajiri, M., Watanabe, Y.: Breather solutions to the focusing nonlinear Schrödinger equation. Phys. Rev. E 57(3), 3510 (1998)
Kivshar, Y.S., Flach, S.: Focus issue: nonlinear localized modes: physics and applications. Chaos 13, 586 (2003)
Xie, X.Y., Yang, S.K., Ai, C.H., Kong, L.C.: Integrable turbulence for a coupled nonlinear Schrödinger system. Phys. Lett. A 384, 126119 (2020)
Chen, J.B., Pelinovsky, D.E.: Rogue periodic waves of the focusing nonlinear Schrödinger equation. Proc. R. Soc. A 474, 20170814 (2018)
Frisquet, B., Kibler, B., Morin, P., Baronio, F., Conforti, M., Millot, G., Wabnitz, S.: Optical dark rogue wave. Sci. Rep. 6, 20785 (2016)
Ankiewicz, A., Wang, Y., Wabnitz, S., Akhmediev, N.: Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. Phys. Rev. E 89(1), 012907 (2014)
Hisakado, M., Iizuka, T., Wadati, M.: Coupled hybrid nonlinear Schrödinger equation and optical solitons. J. Phys. Soc. Jpn. 63(8), 2887–2894 (1994)
Hisakado, M., Wadati, M.: Integrable multi-component hybrid nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 64(2), 408–413 (1995)
Wu, R., Jiang, W., Li, L.: Homoclinic orbits for coupled modified nonlinear Schrödinger equations. Chaos, Solitons & Fractals 38(4), 1093–1103 (2008)
Zhang, H.Q.: Darboux transformation and \(N\)-soliton solution for the coupled modified nonlinear Schrödinger equations. Z. Naturforsch. A 67(12), 711–722 (2012)
Zhang, H.Q.: Energy-exchange collisions of vector solitons in the \(N\)-coupled mixed derivative nonlinear Schrödinger equations from the birefringent optical fibers. Opt. Commun. 290, 141–145 (2013)
Hu, B., Xia, T.: A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line. Math. Methods Appl. Sci. 41(13), 5112–5123 (2018)
Yan, X.W.: Lax pair, Darboux-dressing transformation and localized waves of the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber. Appl. Math. Lett. 107, 106414 (2020)
Hang, C., Wu, Q.L., Zhang, H.Q.: Breathers and double-pole solutions of coupled mixed derivative nonlinear Schrödinger equations from optical fibers. Mod. Phys. Lett. B 35(22), 2150373 (2021)
Song, N., Lei, Y., Cao, D.: Dynamics analysis of higher-order soliton solutions for the coupled mixed derivative nonlinear Schrödinger equation. Acta Mech. Sinica 38(5), 1–7 (2022)
Song, N., Lei, Y.X., Zhang, Y.F., Zhang, W.: Localized waves for the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber. J. Nonlinear Math. Phys. 29, 318–330 (2022)
Kanna, T., Lakshmanan, M.: Exact soliton solutions, shape changing collisions, and partially coherent solitons in coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 86, 5043 (2001)
Degasperis, A., Lombardo, S.: Multicomponent integrable wave equations II. Soliton solutions. J. Phys. A 42, 385206 (2009)
Kaup, D.J., Malomed, B.A., Tasgal, R.S.: Internal dynamics of a vector soliton in a nonlinear optical fiber. Phys. Rev. E 48, 3049 (1993)
Yin, H.M., Tian, B., Zhao, X.C.: Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system. Comput. Math. Appl. 368, 124768 (2020)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant No. 11371326 and Grant No. 11975145).
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 11371326 and Grant No. 11975145).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jin, J., Zhang, Y., Ye, R. et al. The breather and semi-rational rogue wave solutions for the coupled mixed derivative nonlinear Schrödinger equations. Nonlinear Dyn 111, 633–643 (2023). https://doi.org/10.1007/s11071-022-07834-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-022-07834-4