Skip to main content
SpringerLink
Log in

Control of dark and anti-dark solitons in the (2+1)-dimensional coupled nonlinear Schrödinger equations with perturbed dispersion and nonlinearity in a nonlinear optical system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we investigate the (2+1)-dimensional coupled nonlinear Schrödinger equations with perturbed dispersion and nonlinearity, which govern the transverse effects in a nonlinear optical system. Using symbolic calculation, the vector one- and two-soliton solutions are obtained via the Hirota method. By choosing the perturbation \(\alpha (t)\) of the dispersion rate of soliton transmission as different functions, we observe different dark and anti-dark soliton structures. Among other, the parabolic dark soliton, m-shaped and w-shaped anti-dark solitons, two kinds of s-shaped anti-dark solitons with different curvatures and an anti-dark soliton with a peak are displayed. Moreover, the effects of other free parameters on the phase shift and pulse width, and collision of solitons are discussed. These results are of potential significance for the study of ultrashort pulse lasers and optical logic switches.

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
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Wazwaz, A.M.: Abundant solutions of various physical features for the (2+1)-dimensional modified KdV-Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. 89, 1727–1732 (2017)

    Article  MathSciNet  Google Scholar 

  3. Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)

    Article  MathSciNet  Google Scholar 

  4. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40, 2277–2283 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3+1)-dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83(3), 1529–1534 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wazwaz, A.M.: A study on a two-wave mode Kadomtsev-Petviashvili equation: conditions for multiple soliton solutions to exist. Math. Methods Appl. Sci. 40, 4128–4133 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, N., Xia, T.C., Fan, E.G.: A Riemann-Hilbert approach to the Chen–Lee–Liu equation on the half line. Acta Math. Appl. Sin. 34(3), 493–515 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, N., Xia, T.C., Jin, Q.Y.: N-Fold Darboux transformation of the discrete Ragnisco–Tu system. Adv. Differ. Equ. 2018, 302 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Tao, M.S., Zhang, N., Gao, D.Z., Yang, H.W.: Symmetry analysis for three-dimensional dissipation Rossby waves. Adv. Differ. Equ. 2018, 300 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gu, J.Y., Zhang, Y., Dong, H.H.: Dynamic behaviors of interaction solutions of (3+1)-dimensional shallow mater wave equation. Comput. Math. Appl. 76(6), 1408–1419 (2018)

    Article  MathSciNet  Google Scholar 

  11. Liu, Y., Dong, H.H., Zhang, Y.: Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows. Anal. Math. Phys. 2018, 1–17 (2018)

    Google Scholar 

  12. Guo, M., Zhang, Y., Wang, M., Chen, Y.D., Yang, H.W.: A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism. Comput. Math. Appl. 143, 3589–3603 (2018)

    Article  MathSciNet  Google Scholar 

  13. Lu, C., Fu, C., Yang, H.W.: Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. Appl. Math. Comput. 327, 104–116 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Zhao, B.J., Wang, R.Y., Sun, W.J., Yang, H.W.: Combined ZK-mZK equation for Rossby solitary waves with complete Coriolis force and its conservation laws as well as exact solutions. Adv. Differ. Equ. 2018, 42 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yang, H.W., Chen, X., Guo, M., Chen, Y.D.: A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property. Nonlinear Dyn. 91, 2019–2032 (2018)

    Article  MATH  Google Scholar 

  16. Liu, X.Y., Triki, H., Zhou, Q., Liu, W.J., Biswas, A.: Analytic study on interactions between periodic solitons with controllable parameters. Nonlinear Dyn. 94, 1703–709 (2018)

    Article  Google Scholar 

  17. Zhang, Y.J., Yang, C.Y., Yu, W.T., Mirzazadeh, M., Zhou, Q., Liu, W.J.: Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94, 1351–1360 (2018)

    Article  Google Scholar 

  18. Liu, W.J., Zhang, Y.J., Triki, H., Mirzazadeh, M., Ekici, M., Zhou, Q., Biswas, A., Belic, M.: Interaction properties of solitonics in inhomogeneous optical fibers. Nonlinear Dyn. 95, 557–563 (2019)

    Article  Google Scholar 

  19. Liu, X.Y., Triki, H., Zhou, Q., Mirzazadeh, M., Liu, W.J., Biswas, A., Belic, M.: Generation and control of multiple solitons under the influence of parameters. Nonlinear Dyn. 95, 143–150 (2019)

    Article  Google Scholar 

  20. Yang, C.Y., Liu, W.J., Zhou, Q., Mihalache, D., Malomed, B.A.: One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 369–380 (2019)

    Article  Google Scholar 

  21. Zhang, C., Liu, J., Fan, X.W., Peng, Q.Q., Guo, X.S., Jiang, D.P., Qian, X.B., Su, L.B.: Compact passive Q-switching of a diode-pumped Tm, Y: CaF\(_{2}\) laser near 2 \(\iota \)m. Opt. Laser Technol. 103, 89–92 (2018)

    Article  Google Scholar 

  22. Liu, J., Wang, Y.G., Qu, Z.S., Fan, X.W.: 2 \(\mu \)m passive Q-switched mode-locked Tm\(^{3+}\): YAP laser with single-walled carbon nanotube absorber. Opt. Laser Technol. 44(4), 960–962 (2012)

    Article  Google Scholar 

  23. Lin, M.X., Peng, Q.Q., Hou, W., Fan, X.W., Liu, J.: 1.3 \(\iota \)m Q-switched solid-state laser based on few-layer ReS\(_{2}\) saturable absorber. Opt. Laser Technol. 109, 90–93 (2019)

    Article  Google Scholar 

  24. Zhang, F., Wu, Y.J., Liu, J., Pang, S.Y., Ma, F.K., Jiang, D.P., Wu, Q.H., Su, L.B.: Mode locked Nd\(^{3+}\) and Gd\(^{3+}\) co-doped calcium fluoride crystal laser at dual gain lines. Opt. Laser Technol. 100, 294–297 (2018)

    Article  Google Scholar 

  25. Wu, Y.J., Zhang, C., Liu, J.J., Zhang, H.N., Yang, J.M., Liu, J.: Silver nanorods absorbers for Q-switched Nd:YAG ceramic laser. Opt. Laser Technol. 97, 268–271 (2017)

    Article  Google Scholar 

  26. Zhang, F., Liu, J., Li, W.W., Mei, B.C., Jiang, D.P., Qian, X.B., Su, L.B.: Dual-wavelength continuous-wave and passively Q-switched Nd, Y: SrF\(_{2}\) ceramic laser. Opt. Eng. 55(10), 106114 (2016)

    Article  Google Scholar 

  27. Li, C., Fan, M.W., Liu, J., Su, L.B., Jiang, D.P., Qian, X.B., Xu, J.: Operation of continuous wave and Q-switching on diode-pumped Nd, Y: CaF\(_{2}\) disordered crystal. Opt. Laser Technol. 69, 140–143 (2015)

    Article  Google Scholar 

  28. Cai, W., Peng, Q.Q., Hou, W., Liu, J., Wang, Y.G.: Picosecond passively mode-locked laser of 532 nm by reflective carbon nanotube. Opt. Laser Technol. 58, 194–196 (2014)

    Article  Google Scholar 

  29. Wang, Y.G., Qu, Z.S., Liu, J., Tsang, Y.H.: Graphene oxide absorbers for watt-level high-power passive mode-locked Nd:GdVO\(_{4}\) laser operating at 1 \(\iota \)m. J. Lightwave Technol. 30(20), 3259–3262 (2012)

    Article  Google Scholar 

  30. Zhu, H.T., Zhao, L.N., Liu, J., Xu, S.C., Cai, W., Jiang, S.Z., Zheng, L.H., Su, L.B., Xu, J.: Monolayer graphene saturable absorber with sandwich structure for ultrafast solid-state laser. Opt. Eng. 55(8), 081304 (2016)

    Article  Google Scholar 

  31. Cai, W., Jiang, S.Z., Xu, S.C., Li, Y.Q., Liu, J., Li, C., Zheng, L.H., Su, L.B., Xu, J.: Graphene saturable absorber for diode pumped Yb:Sc\(_{2}\)SiO\(_{5}\) mode-locked laser. Opt. Laser Technol. 65, 1–4 (2015)

    Article  Google Scholar 

  32. Zhu, H.T., Liu, J., Jiang, S.Z., Xu, S.C., Su, L.B., Jiang, D.P., Qian, X.B., Xu, J.: Diode-pumped Yb, Y: CaF\(_{2}\) laser mode-locked by monolayer graphene. Opt. Laser Technol. 75, 83–86 (2015)

    Article  Google Scholar 

  33. Liu, W.J., Liu, M.L., Liu, B., Quhe, R.G., Lei, M., Fang, S.B., Teng, H., Wei, Z.Y.: Nonlinear optical properties of MoS\(_{2}\)-WS\(_{2}\) heterostructure in fiber lasers. Opt. Express 27, 6689–6699 (2019)

    Article  Google Scholar 

  34. Li, L., Lv, R.D., Liu, S.C., Chen, Z.D., Wang, J., Wang, Y.G., Ren, W., Liu, W.J.: Ferroferric-oxide nanoparticle based Q-switcher for a 1 \(\iota \)m region. Opt. Mater. Express 9, 731–738 (2019)

    Article  Google Scholar 

  35. Li, L., Lv, R.D., Wang, J., Chen, Z.D., Wang, H.Z., Liu, S.C., Ren, W., Liu, W.J., Wang, Y.G.: Optical nonlinearity of ZrS\(_{2}\) and applications in fiber laser. Nanomaterials 9, 315 (2019)

    Article  Google Scholar 

  36. Vasylchenkova, A., Prilepsky, J.E., Shepelsky, D., Chattopadhyay, A.: Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation. Commun. Nonlinear Sci. 68, 347–371 (2019)

    Article  MathSciNet  Google Scholar 

  37. Inui, T.: Asymptotic behavior of the nonlinear damped Schrödinger equation. Proc. Am. Math. Soc. 147(2), 763–773 (2019)

    Article  MATH  Google Scholar 

  38. Su, Y., Guo, Q.: Blow-up solutions to nonlinear Schrödinger system at multiple points. Z. Angew. Math. Phys. 70(1), 20 (2019)

    Article  MATH  Google Scholar 

  39. Wilson, J.P.: Generalized finite-difference time-domain method with absorbing boundary conditions for solving the nonlinear Schrödinger equation on a GPU. Comput. Phys. Commun. 235, 279–292 (2019)

    Article  MathSciNet  Google Scholar 

  40. Zayed, E.M.E., Shohib, R.M.A., Al-Nowehy, A.G.: Solitons and other solutions for higher-order NLS equation and quantum ZK equation using the extended simplest equation method. Comput. Math. Appl. 76(9), 2286–2303 (2018)

    Article  MathSciNet  Google Scholar 

  41. Zhang, J., Zhang, W., Xie, X.L.: Infinitely many solutions for a gauged nonlinear Schrödinger equation. Appl. Math. Lett. 88, 21–27 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  42. Chen, M.N., Guo, Q., Lu, D.Q., Hu, W.: Variational approach for breathers in a nonlinear fractional Schrödinger equation. Commun. Nonlinear Sci. 71, 73–81 (2019)

    Article  MathSciNet  Google Scholar 

  43. Ji, B.Q., Zhang, L.M.: Error estimates of exponential wave integrator Fourier pseudospectral methods for the nonlinear Schrödinger equation. Appl. Math. Comput. 343, 100–113 (2019)

    MathSciNet  Google Scholar 

  44. Zhang, Y.J., Yang, C.Y., Yu, W.T., Mirzazadeh, M., Zhou, Q., Liu, W.J.: Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dyn. 94(2), 1351–1360 (2018)

    Article  Google Scholar 

  45. Wu, H.L., Chen, J.Q., Li, Y.Q.: Existence of positive solutions to a linearly coupled Schrödinger system with critical exponent. Commun. Contemp. Math. 20, 7 (2018)

    Article  Google Scholar 

  46. Baleanu, D., Inc, M., Aliyu, A.I., Yusuf, A.: The investigation of soliton solutions and conservation laws to the coupled generalized Schrödinger-Boussinesq system. Wave. Random Complex 29(1), 77–92 (2019)

    Article  Google Scholar 

  47. Nath, D., Saha, N., Roy, B.: Stability of (1+1)-dimensional coupled nonlinear Schrödinger equation with elliptic potentials. Eur. Phys. J. Plus 133, 12 (2018)

    Article  Google Scholar 

  48. Du, Z., Tian, B., Chai, H.P., Yuan, Q.: Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein. Commun. Nonlinear Sci. 67, 49–59 (2019)

    Article  MathSciNet  Google Scholar 

  49. Jiang, Y., Qu, Q.X.: Some semirational solutions and their interactions on the zero-intensity background for the coupled nonlinear Schrödinger equations. Commun. Nonlinear Sci. 67, 403–413 (2019)

    Article  MathSciNet  Google Scholar 

  50. Cai, J.X., Bai, C.Z., Zhang, H.H.: Decoupled local/global energy-preserving schemes for the N-coupled nonlinear Schrödinger equations. J. Comput. Phys. 374, 281–299 (2018)

    Article  MathSciNet  Google Scholar 

  51. Kumar, S., Gupta, P.K., Uma, R., Sharma, R.P.: Enhancement in self-compression due to co-propagating laser pulse in plasma. Opt. Commun. 427, 37–43 (2018)

    Article  Google Scholar 

  52. Aliyu, A.I., Tchier, F., Inc, M., Yusuf, A., Baleanu, D.: Dynamics of optical solitons, multipliers and conservation laws to the nonlinear schrödinger equation in (2+1)-dimensions with non-Kerr law nonlinearity. J. Mod. Opt. 66(2), 136–142 (2019)

    Article  Google Scholar 

  53. Lan, Z.Z.: Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equation. Appl. Math. Lett. 86, 243–248 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  54. Cheng, W.G., Xu, T.Z.: Lump solutions and interaction behaviors to the (2+1)-dimensional extended shallow water wave equation. Mod. Phys. Lett. B 32, 31 (2018)

    MathSciNet  Google Scholar 

  55. Cai, Y.J., Bai, C.L., Luo, Q.L.: Exact soliton solutions for the (2+1)-dimensional coupled higher-order Nonlinear Schrödinger equations in birefringent optical-fiber communication. Commun. Theor. Phys. 67(3), 273 (2017)

    Article  MATH  Google Scholar 

  56. Wu, X.Y., Tian, B., Xie, X.Y., Chai, J.: Dark solitons and Bäcklund transformation for the (2+1)-dimensional coupled nonlinear Schrödinger equation with the variable coefficients in a graded-index waveguide. Superlattices Microstruct. 101, 117–126 (2017)

    Article  Google Scholar 

  57. Lan, Z.Z., Gao, B., Du, M.J.: Dark solitons behaviors for a (2+1)-dimensional coupled nonlinear Schrödinger system in an optical fiber. Chaos Solitons Fractals 111, 169–174 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  58. Mansfield, E.L., Reid, G.J., Clarkson, P.A.: Nonclassical reductions of a (3+1)-cubic nonlinear Schrödinger system. Comput. Phys. Commun. 115(2–3), 460–488 (1998)

    Article  MATH  Google Scholar 

  59. Abraham, N.B., Firth, W.J.: Overview of transverse effects in nonlinear-optical systems. J. Opt. Soc. Am. B 7(6), 951–962 (1990)

    Article  Google Scholar 

  60. Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: from Fibers to Photonic Crystals. Academic, San Diego (2003)

    Google Scholar 

  61. Su, J.J., Gao, Y.T.: Solitons for a (2+1)-dimensional coupled nonlinear Schrödinger system with time-dependent coefficients in an optical fiber. Wave. Random Complex Med. 28(4), 708–723 (2018)

    Article  Google Scholar 

  62. Su, J.J., Gao, Y.T.: Dark solitons for a (2+1)-dimensional coupled nonlinear Schrödinger system with time-dependent coefficients in an optical fiber. Superlattices Microstruct. 104, 498–508 (2017)

    Article  Google Scholar 

  63. Hirota, R.: Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

We acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 11674036 and 11875008), by the Beijing Youth Top-notch Talent Support Program (Grant No. 2017000026833ZK08), and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2017ZZ05). The work of MRB was supported by the grant NPRP8-028-1-001 from QNRF.

Author information

Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing, 100876, People’s Republic of China

    Weitian Yu & Wenjun Liu

  2. Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000, Annaba, Algeria

    Houria Triki

  3. School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan, 430212, People’s Republic of China

    Qin Zhou

  4. Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL, 35762, USA

    Anjan Biswas

  5. Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Anjan Biswas

  6. Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, 0008, South Africa

    Anjan Biswas

  7. Institute of Physics Belgrade, Pregrevica 118, 11080, Zemun, Serbia

    Milivoj R. Belić

Authors
  1. Weitian Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wenjun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Houria Triki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qin Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Anjan Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Milivoj R. Belić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qin Zhou.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

The authors declare that they have adhered to the ethical standards of research execution.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, W., Liu, W., Triki, H. et al. Control of dark and anti-dark solitons in the (2+1)-dimensional coupled nonlinear Schrödinger equations with perturbed dispersion and nonlinearity in a nonlinear optical system. Nonlinear Dyn 97, 471–483 (2019). https://doi.org/10.1007/s11071-019-04992-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-04992-w

Keywords

Search

Navigation