Skip to main content
SpringerLink
Log in

N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

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

Abstract

In this paper, the Sasa-Satsuma equation in fluid dynamics and nonlinear optics is investigated. Starting from the first-order Darboux transformation, we construct an N-fold generalized Darboux transformation (GDT), where N is a positive integer. Through the obtained N-fold GDT, we derive three kinds of the semirational solutions, which describe the second-order degenerate solitons, third-order degenerate solitons and interaction between the second-order degenerate solitons and one soliton, respectively. We graphically illustrate the above three kinds of semirational solutions and investigate them through the asymptotic analysis, from which we find that the characteristic lines of the semirational solutions are composed of the straight lines and curves. Expressions of the characteristic lines, positions, amplitudes, slopes, positions and phase shifts of the asymptotic solitons are presented through the asymptotic analysis. The above discussions might be extended to the higher-order solitons, and to the relevant analysis on the degenerate breathers.

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

Similar content being viewed by others

References

  1. Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  2. Chabchoub, A., Kibler, B., Finot, C., Millot, G., Onorato, M., Dudley, J.M., Babanin, A.V.: The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface. Ann. Phys. 361, 490 (2015)

    Article  MATH  Google Scholar 

  3. Congy, T., El, G.A., Hoefer, M.A., Shearer, M.: Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure. Stud. Appl. Math. 142, 241 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chabchoub, A., Grimshaw, R.H.J.: The hydrodynamic nonlinear Schrödinger equation: Space and time. Fluids 23, 1 (2016)

    Google Scholar 

  5. El, G., Tovbis, A.: Spectral theory of soliton and breather gases for the focusing nonlinear Schrödinger equation. Phys. Rev. E 101, 052207 (2020)

    Article  MathSciNet  Google Scholar 

  6. Sasa, N., Satsuma, J.: New-type of soliton solutions for a higher-order nonlinear Schrödinger equation. J. Phys. Soc. Japan 60, 409 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Xu, T., Wang, D., Li, M., Liang, H.: Soliton and breather solutions of the Sasa–Satsuma equation via the Darboux transformation. Phys. Scr. 89, 075207 (2014)

    Article  Google Scholar 

  8. Nimmo, J.J.C., Yilmaz, H.: Binary Darboux transformation for the Sasa–Satsuma equation. J. Phys. A-Math. Theor. 48, 425202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yang, B., Chen, Y.: High-order soliton matrices for Sasa–Satsuma equation via local Riemann-Hilbert problem. Nonlinear Anal.-Real World Appl. 45, 918 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ghosh, S., Kundu, A., Nandy, S.: Soliton solutions, Liouville integrability and gauge equivalence of Sasa–Satsuma equation. J. Math. Phys. 40, 1993 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Xu, T., Li, M., Li, L.: Anti-dark and Mexican-hat solitons in the Sasa–Satsuma equation on the continuous wave background. Europhys. Lett. 109, 30006 (2015)

    Article  Google Scholar 

  12. Zhao, L.C., Li, S.C., Ling, L.: Rational W-shaped solitons on a continuous-wave background in the Sasa–Satsuma equation. Phys. Rev. E 89, 023210 (2014)

    Article  Google Scholar 

  13. Zhao, L.C., Li, S.C., Ling, L.: W-shaped solitons generated from a weak modulation in the Sasa–Satsuma equation. Phys. Rev. E 93, 032215 (2016)

    Article  MathSciNet  Google Scholar 

  14. Chen, S.: Twisted rogue-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 023202 (2013)

    Article  Google Scholar 

  15. Soto-Crespo, J.M., Devine, N., Hoffmann, N.P., Akhmediev, N.: Rogue waves of the Sasa–Satsuma equation in a chaotic wave field. Phys. Rev. E 90, 032902 (2014)

    Article  Google Scholar 

  16. Bandelow, U., Akhmediev, N.: Sasa–Satsuma equation: Soliton on a background and its limiting cases. Phys. Rev. E 86, 026606 (2012)

    Article  Google Scholar 

  17. Liu, L., Tian, B., Chai, H.P., Yuan, Y.Q.: Certain bright soliton interactions of the Sasa–Satsuma equation in a monomode optical fiber. Phys. Rev. E 95, 032202 (2017)

    Article  Google Scholar 

  18. Yang, D.Y., Tian, B., Tian, H.Y., Wei, C.C., Shan, W.R., Jiang, Y.: Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber. Chaos Solitons Fract. 156, 111719 (2022)

  19. Wu, X.H., Gao, Y.T., Yu, X., Liu, L.Q., Ding, C.C.: Vector breathers, rogue and breather-rogue waves for a coupled mixed derivative nonlinear Schrödinger system in an optical fiber. Nonlinear Dyn. 111, 5641 (2023)

  20. Zhou, T.Y., Tian, B.: Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Appl. Math. Lett. 133, 108280 (2022)

  21. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C., Hu, L., Li, L.Q.: Binary Darboux transformation, solitons, periodic waves and modulation instability for a nonlocal Lakshmanan-Porsezian-Daniel equation. Wave Motion 114, 103036 (2022)

  22. Yang, D.Y., Tian, B., Hu, C.C., Liu, S.H., Shan, W.R.. Jiang, Y.: Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber. Wave. Random Complex (2023) in press. https://doi.org/10.1080/17455030.2021.1983237

  23. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C.: N-fold generalized Darboux transformation and soliton interactions for a three-wave resonant interaction system in a weakly nonlinear dispersive medium. Chaos Solitons Fract. 165, 112786 (2022)

  24. Gao, X.Y., Guo, Y.J., Shan, W.R.: Reflecting upon some electromagnetic waves in a ferromagnetic film via a variable-coefficient modified Kadomtsev-Petviashvili system. Appl. Math. Lett. 132, 108189 (2022)

  25. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Considering the shallow water of a wide channel or an open sea through a generalized (2+1)-dimensional dispersive long-wave system. Qual. Theory Dyn. Syst. 21, 104 (2022)

  26. Shen, Y., Tian, B., Liu, S.H., Zhou, T.Y. : Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Nonlinear Dyn. 108, 2447–2460 (2022)

  27. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C.: Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Nonlinear Dyn. 108, 1599–1616 (2022)

  28. Cheng, C.D., Tian, B., Zhou, T.Y., Shen, Y.: Wronskian solutions and Pfaffianization for a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma. Phys. Fluids 35, 037101 (2023)

  29. Tao, Y., He, J.: Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E 85, 026601 (2012)

    Article  Google Scholar 

  30. Chen, S.S., Tian, B., Tian, H.Y., Yang, D.Y.: N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfv\(\rm \acute{e}\)n waves in a plasma. Nonlinear Dyn. 108, 1561 (2022)

    Article  Google Scholar 

  31. Guo, B., Ling, L., Liu, Q.P.: Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  32. Wang, L., Zhu, Y.J., Wang, Z.Z., Qi, F.H., Guo, R.: Higher-order semirational solutions and nonlinear wave interactions for a derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 33, 218 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lin, M., Yue, X., Xu, T.: Multi-pole solutions and their asymptotic analysis of the focusing Ablowitz–Ladik equation. Phys. Scr. 95, 055222 (2020)

    Article  Google Scholar 

  34. Xu, T., Lan, S., Li, M., Li, L.L., Zhang, G.W.: Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation. Physica D 47, 390 (2019)

    MATH  Google Scholar 

  35. Kodama, Y.: Optical solitons in a monomode fiber. J. Stat. Phys. 39(5), 597–614 (1985)

    Article  MathSciNet  Google Scholar 

  36. Kodama, Y., Hasegawa, A.: Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Q. Electron. 23(5), 510–524 (1987)

    Article  Google Scholar 

  37. Sedletsky, Y.V.: The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid. J. Exp. Theor. Phys. 97, 180 (2003)

    Article  Google Scholar 

  38. Slunyaev, A.V.: A high-order nonlinear envelope equation for gravity waves in finite-depth water. J. Exp. Theor. Phys. 101, 926 (2005)

    Article  Google Scholar 

  39. Potasek, M.J., Tabor, M.: Exact solutions for an extended nonlinear Schrödinger equation. Phys. Lett. A 154, 449 (1991)

    Article  MathSciNet  Google Scholar 

  40. Cavalcanti, S.B., Cressoni, J.C., da Cruz, H.R., Gouveia-Neto, A.S.: Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation. Phys. Rev. A 43, 6162 (1991)

    Article  Google Scholar 

  41. Trippenbach, M., Band, Y.B.: Effects of self-steepening and self-frequency shifting on short-pulse splitting in dispersive nonlinear media. Phys. Rev. A 57, 4791 (1998)

    Article  Google Scholar 

  42. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: N-fold Darboux transformation and solitonic interactions for the Kraenkel-Manna-Merle system in a saturated ferromagnetic material. Nonlinear Dyn. 111, 2641–2649 (2023)

    Article  Google Scholar 

  43. Gao, X.Y., Guo, Y.J., Shan, W.R.: Symbolically computing the shallow water via a (2+1)-dimensional generalized modified dispersive water-wave system: similarity reductions, scaling and hetero-Bäcklund transformations. Qual. Theory Dyn. Syst. 22, 17 (2023)

  44. Zhou, T.Y., Tian, B., Chen, Y.Q., Shen, Y.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions of a (2+1)-dimensional generalized Burgers system with the variable coefficients in a fluid. Nonlinear Dyn. 108, 2417–2428 (2022)

  45. Cheng C.D., Tian, B., Shen, Y., Zhou, T.Y.: Bilinear form and Pfaffian solutions for a (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics and plasma physics. Nonlinear Dyn. 111, 6659 (2023)

  46. Liu, F.Y., Gao, Y.T.: Lie group analysis for a higher-order Boussinesq-Burgers system. Appl. Math. Lett. 132, 108094 (2022)

  47. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Lump and hybrid solutions for a (3+1)-dimensional Boussinesq-type equation for the gravity waves over a water surface. Chin. J. Phys. 83, 515 (2023)

  48. Gao, X.T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

  49. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: Pfaffian solutions and nonlinear waves of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics. Phys. Fluids 35, 025103 (2023)

    Article  Google Scholar 

  50. Gao, X.Y., Guo, Y.J., Shan, W. R.: Shallow-water investigations: Bilinear auto-Bäcklund transformations for a (3+1)-dimensional generalized nonlinear evolution system. Appl. Comput. Math. 22, 133–142 (2023)

    Google Scholar 

  51. Zhou, T.Y., Tian, B., Zhang, C.R., Liu, S. H.: Auto-Bäcklund transformations, bilinear forms, multiple-soliton, quasi-soliton and hybrid solutions of a (3+1)-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in an electron-positron plasma. Eur. Phys. J. Plus 137, 912 (2022)

    Article  Google Scholar 

  52. Gao, X.T., Tian, B., Feng, C.H.: In oceanography, acoustics and hydrodynamics: investigations on an extended coupled (2+1)-dimensional Burgers system. Chin. J. Phys. 77, 2818–2824 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11772017 and by the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Xi-Hu Wu, Yi-Tian Gao, Xin Yu & Cui-Cui Ding

  2. Shen Yuan Honors College, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Xi-Hu Wu

Authors
  1. Xi-Hu Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi-Tian Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xin Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cui-Cui Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yi-Tian Gao or Xin Yu.

Additional information

Publisher's Note

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

Appendix

Appendix

When considering the linear pulses in an optical transmission system, dispersion character of the fiber material has been considered to limit the maximum value of the bit rate of transmission (i.e., channel capacity of an optical fiber) [35]. To overcome this limitation, the nonlinear change of dielectric (the so-called Kerr effect) of the fiber has been used to compensate for the dispersion effect [35]. Afterward, the researchers have analyzed an optical pulse propagating through a cylindrical fiber with the Kerr effect [35, 36]. Starting from the three-dimensional vector Maxwell equation for the electric field in a fiber with an inhomogeneous dielectric constant, the researchers have derived the one-dimensional scalar perturbation nonlinear Schrödinger in an appropriate asymptotic sense, i.e.,

$$\begin{aligned} \begin{aligned}&i\frac{\partial u}{\partial Z}+\frac{1}{2}\frac{\partial ^2 u}{\partial T^2}+|u|^2u\\&\quad =i\epsilon \left[ \alpha _1\frac{\partial ^3 u}{\partial T^3}+\alpha _2 |u|^2\frac{\partial u}{\partial T}+\alpha _3u^2\frac{\partial u^*}{\partial T} \right] , \end{aligned} \end{aligned}$$
(28)

where the meanings of the variables are offered in Refs. [35, 36]. In fact, the derivation process details in Refs. [35, 36] have been shown to be really complicated. On the basis of Eq. (28), the feasibility of a long-distance-high-bit-rate optical transmission system by the use of solitons has been discussed [35].

Actually, when \(\alpha _1=\alpha _2=\alpha _3=0\), Eq. (28) has been reduced to the NLS equation [1]. That is to say, Eq. (28) has been considered as the result of adding higher-order terms to the NLS equation [17]. As the real parameters \(\alpha _1,~\alpha _2\) and \(\alpha _3\) satisfy certain conditions, e.g., \(\alpha _1:\alpha _2:\alpha _3=0:1:1\), \(\alpha _1:\alpha _2:\alpha _3=0:1:0\) and \(\alpha _1:\alpha _2:\alpha _3=1:6:0\), Eq. (28) has been shown to be integrable and solvable through the inverse scattering transform scheme [6].

Later, under the condition \(\alpha _1:\alpha _2:\alpha _3=1:6:3\), the Sasa–Satsuma equation has been derived [6]. With the higher-order nonlinear terms concluded, the Sasa–Satsuma equation has been considered to have the applications as follows:

  • Amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium- and large-depth (compared to the wavelength) fluid layer [37].

  • Gravity surface waves in the finite-depth water by assuming small wave steepness, narrow-band spectrum, and small depth as compared to the modulation length [38].

  • A train of the femtosecond pulses with repetition rates of a few terahertz in the nonlinear optical fibers [39, 40].

  • Propagation of the short optical pulses through a dispersive media with a cubic self-focusing nonlinear polarization, which incorporates the self-steepening and self-frequency shifting [41].

  • Other relevant symbolic-computation studies have been reported, e.g., in Refs. [42,43,44,45,46,47,48,49,50,51,52].

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, XH., Gao, YT., Yu, X. et al. N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics. Nonlinear Dyn 111, 16339–16352 (2023). https://doi.org/10.1007/s11071-023-08533-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08533-4

Keywords

Search

Navigation