Skip to main content
SpringerLink
Log in

Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

Under investigation in this paper is a sextic nonlinear Schrödinger equation, which describes the pulses propagating along an optical fiber. Based on the symbolic computation, Lax pair and infinitely-many conservation laws are derived. Via the modiied Hirota method, bilinear forms and multi-soliton solutions are obtained. Propagation and interactions of the solitons are illustrated graphically: Initial position and velocity of the soliton are related to the coefficient of the sixth-order dispersion, while the amplitude of the soliton is not affected by it. Head-on, overtaking and oscillating interactions between the two solitons are displayed. Through the asymptotic analysis, interaction between the two solitons is proved to be elastic. Based on the linear stability analysis, the modulation instability condition for the soliton solutions is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  • Ankiewicz, A., Wang, Y., Wabnitz, S., N, Akhmediev: Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. Phys. Rev. E 89, 012907 (2014)

  • Ankiewicz, A., Kedziora, D.J., Chowdury, A., Bandelow, U., Akhmediev, N.: Infinite hierarchy of nonlinear Schrödinger equations and their solutions. Phys. Rev. E 93, 012206 (2016)

  • Bourkoff, E., Christodoulides, D.N., Zhao, W., Joseph, R.I., Christodoulides, D.N.: Evolution of femtosecond pulses in single-mode fibers having higher-order nonlinearity and dispersion. Opt. Lett. 12, 272–274 (1987)

    Article  ADS  Google Scholar 

  • Cai, L.Y., Wang, X., Wang, L., Li, M., Liu, Y., Shi, Y.Y.: Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Nonlinear Dyn. 90, 2221–2230 (2017)

    Article  Google Scholar 

  • Chai, J., Tian, B., Zhen, H.L., Sun, W.R.: Conservation laws, bilinear forms and solitons for a fifth-order nonlinear Schrödinger equation for the attosecond pulses in an optical fiber. Ann. Phys. 359, 371–384 (2015)

    Article  Google Scholar 

  • Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions. Phys. Rev. E 91, 022919 (2015)

  • Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Chen, L.: Vector spatiotemporal localized structures in (3+1)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)

    Article  MathSciNet  Google Scholar 

  • Dai, C.Q., Zhang, C.Q., Fan, Y., Chen, L.: Localized modes of the (n+1)dimensional Schrödinger equation with powerlaw nonlinearities in PT-symmetric potentials. Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  • Daniel, M., Kavitha, L., Amuda, R.: Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole-dipole interaction. Phys. Rev. B 59, 13774–13781 (1999)

    Article  ADS  Google Scholar 

  • de Oliveira, J.R., de Moura, M.A.: Analytical solution for the modified nonlinear Schrödinger equation describing optical shock formation. Phys. Rev. E 57, 4751–4756 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  • Guo, R., Zhao, X.J.: Discrete Hirota equation: discrete Darboux transformation and new discrete soliton solutions. Nonlinear Dyn. 84, 1901–1907 (2016)

    Article  MathSciNet  Google Scholar 

  • Guo, R., Liu, Y.F., Hao, H.Q., Qi, F.H.: Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium. Nonlinear Dyn. 80, 1221–1230 (2015)

    Article  MathSciNet  Google Scholar 

  • Guo, R., Zhao, H.H., Wang, Y.: A higher-order coupled nonlinear Schrödinger system: solitons, breathers, and rogue wave solutions. Nonlinear Dyn. 83, 2475–2484 (2016)

    Article  Google Scholar 

  • Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973a)

    Article  ADS  Google Scholar 

  • Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett. 23, 171–172 (1973b)

    Article  ADS  Google Scholar 

  • Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (1991)

    Google Scholar 

  • Lakoba, T.I., Kaup, D.J.: Hermite–Gaussian expansion for pulse propagation in strongly dispersion managed fibers. Phys. Rev. E 58, 6728–6741 (1998)

    Article  ADS  Google Scholar 

  • Lan, Z., Gao, B.: Solitons, breather and bound waves for a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide. Eur. Phys. J. Plus 132, 512 (2017)

  • Lan, Z.Z., Gao, Y.T., Zhao, C., Yang, J.W., Su, C.Q.: Dark soliton interactions for a fifth-order nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain. Superlattices Microstruct. 100, 191–197 (2016)

    Article  ADS  Google Scholar 

  • Li, P., Wang, L., Kong, L.Q., Wang, X., Xie, Z.Y.: Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation. Appl. Math. Lett. 85, 110–117 (2018)

    Article  MathSciNet  Google Scholar 

  • Liu, W.J., Tian, B., Jiang, Y., Sun, K., Wang, P., Li, M., Qu, Q.X.: Soliton solutions and Bäcklund transformation for the complex Ginzburg–Landau equation. Appl. Math. Comput. 217, 4369–4376 (2011)

    MathSciNet  MATH  Google Scholar 

  • Liu, W.J., Pan, N., Huang, L.G., Lei, M.: Soliton interactions for coupled nonlinear Schrödinger equations with symbolic computation. Nonlinear Dyn. 78, 755–770 (2014)

    Article  Google Scholar 

  • Liu, W.J., Pang, L.H., Yan, H., Ma, G.L., Lei, M., Wei, Z.Y.: High-order solitons transmission in hollow-core photonic crystal fibers. EPL 116, 64002 (2016)

    Article  ADS  Google Scholar 

  • Liu, W., Yu, W., Yang, C., Liu, M., Zhang, Y., Lei, M.: Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dyn. 89, 2933–2939 (2017)

    Article  MathSciNet  Google Scholar 

  • Lü, X., Lin, F.: Soliton excitations and shape-changing collisions in alphahelical proteins with interspine coupling at higher order. Commun. Nonlinear Sci. Numer. Simul. 32, 241–261 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  • Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016)

    Article  MathSciNet  Google Scholar 

  • Lü, X., Chen, S.T., Ma, W.X.: Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 86, 523–534 (2016)

    Article  MathSciNet  Google Scholar 

  • Nakatsuka, H., Grischkowsky, D., Balant, A.C.: Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion. Phys. Rev. Lett. 47, 910–913 (1981)

    Article  ADS  Google Scholar 

  • Nakazawa, M., Kubota, H., Suzuki, K., Yamada, E., Sahara, A.: Recent progress in soliton transmission technology. Chaos 10, 486–514 (2000)

    Article  ADS  Google Scholar 

  • Osman, M.S., Wazwaz, A.M.: An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients. Appl. Math. Comput. 321, 282–289 (2018)

    MathSciNet  Google Scholar 

  • Sun, W.R.: Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers. Ann. Phys. (Berlin) 529, 1–2 (2017)

    MATH  Google Scholar 

  • Sun, W.R., Tian, B., Zhen, H.L., Sun, Y.: Breathers and rogue waves of the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Nonlinear Dyn. 81, 725–732 (2015)

    Article  Google Scholar 

  • Szmytkowski, R.: Alternative approach to the solution of the momentum-space Schrödinger equation for bound states of the N-dimensional coulomb problem. Ann. Phys. (Berlin) 524, 345–352 (2012)

    Article  ADS  Google Scholar 

  • Wang, L., Li, X., Qi, F.H., Zhang, L.L.: Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell-Bloch equations. Ann. Phys. 359, 97–114 (2015a)

    Article  Google Scholar 

  • Wang, L., Zhu, Y.J., Qi, F.H., Li, M., Guo, R.: Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers. Chaos 25, 063111 (2015b)

    Article  MathSciNet  Google Scholar 

  • Wang, Q.M., Gao, Y.T., Su, C.Q., Mao, B.Q., Gao, Z., Yang, J.W.: Dark solitonic interaction and conservation laws for a higher-order (2+1)-dimensional nonlinear Schrödinger-type equation in a Heisenberg ferromagnetic spin chain with bilinear and biquadratic interaction. Ann. Phys. 363, 440–456 (2015c)

    Article  ADS  Google Scholar 

  • Wang, L., Zhang, J.H., Wang, Z.Q., Liu, C., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016a)

  • Wang, L., Zhu, Y.J., Wang, Z.Q., Xu, T., Qi, F.H., Xue, Y.S.: Asymmetric Rogue waves, breather-to-soliton conversion, and nonlinear wave interactions in the Hirota-Maxwell-Bloch system. J. Phys. Soc. Jpn. 85, 024001 (2016b)

    Article  ADS  Google Scholar 

  • Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)

    Article  MathSciNet  Google Scholar 

  • Wazwaz, A.M.: A two-mode modified KdV equation with multiple soliton solutions. Appl. Math. Lett. 70, 1–6 (2017a)

    Article  MathSciNet  Google Scholar 

  • Wazwaz, A.M.: Two-mode fifth-order KdV equations: necessary conditions for multiplesoliton solutions to exist. Nonlinear Dyn. 87, 1685–1691 (2017b)

    Article  Google Scholar 

  • Yang, J.W., Gao, Y.T., Wang, Q.M., Su, C.Q., Feng, Y.J., Yu, X.: Bilinear forms and soliton solutions for a fourth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous Heisenberg ferromagnetic spin chain or an alpha helical protein. Phys. B 481, 148–155 (2016)

    Article  ADS  Google Scholar 

  • Zhang, J.H., Wang, L., Liu, C.: Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc. R. Soc. A 473, 20160681 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  • Zhao, X.H., Tian, B., Liu, D.Y., Wu, X.Y., Chai, J., Guo, Y.J.: Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct. 100, 587–595 (2016)

    Article  ADS  Google Scholar 

  • Zhao, X.H., Tian, B., Guo, Y.J.: Solitons, Lax pair and infinitely-many conservation laws for a higher-order nonlinear Schrödinger equation in an optical fiber. Optik 132, 417–426 (2017)

    Article  ADS  Google Scholar 

  • Zhou, Q., Yao, D., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic law nonlinearities. J. Mod. Opt. 60, 1652–1657 (2013)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to all the members of our discussion group for their valuable comments. We thanks Prof. Y. T. Gao and Dr. J. J. Su for the timely and valuable comments. This work has been supported by the Science Research Project of Higher Education in Inner Mongolia Autonomous Region under Grant No. NJZZ18117, by the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No. 2018BS01004, and by the National Natural Science Foundation of China under Grant No. 11772017.

Author information

Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Zhong-Zhou Lan & Bo-Ling Guo

  2. School of Computer Information Management, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

    Zhong-Zhou Lan

Authors
  1. Zhong-Zhou Lan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bo-Ling Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhong-Zhou Lan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lan, ZZ., Guo, BL. Conservation laws, modulation instability and solitons interactions for a nonlinear Schrödinger equation with the sextic operators in an optical fiber. Opt Quant Electron 50, 340 (2018). https://doi.org/10.1007/s11082-018-1597-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-018-1597-7

Keywords

Search

Navigation