Skip to main content
SpringerLink
Log in

Several categories of exact solutions of the third-order flow equation of the Kaup–Newell system

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

Abstract

In this paper, we introduce the third-order flow equation of the Kaup–Newell (KN) system. We study this equation, and we obtain different types of solutions by using the Darboux transformation (DT) and the extended DT of the KN system, such as solitons, positons, breathers, and rogue waves. The extended DT is obtained by taking the degenerate eigenvalues \( \lambda _{i} \rightarrow \lambda _{1} (i=3,5,7,\ldots ,2k-1)\) and by performing the Taylor expansion near \(\lambda _{1}\) of the determinants of DT. Some analytic expressions are explicitly given for the first-order solutions. We study the unique waveforms of both the first-order and higher-order rogue-wave solutions for special choices of parameters, and we find different types of such wave structures: fundamental pattern, triangular, modified-triangular, pentagram, ring, ring-triangular, and multi-ring wave patterns. We conclude that the third-order dispersion and quintic nonlinear term of the KN system modify both the trajectories and speeds of the solutions as compared with those corresponding to the second-order flow equation of the KN system.

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
Fig. 14

Similar content being viewed by others

References

  1. Mollenauer, L.F., Stolen, R.H., Gordon, J.P.: Experimental observations of picosecond plise narrowing and solitons in optical fibers. IEEE J. Quantum Electron. 17, 2378–2378 (1980)

    Google Scholar 

  2. Strecker, K.E., Partridge, G.B., Truscott, A.G., et al.: Formation and propagation of matter-wave soliton trains. Nature 417, 150–153 (2002)

    Google Scholar 

  3. Dudley, J.M., Genty, G., Coen, S.: Supercontinuum generation is photonic crystal fiber. Rev. Mod. Phys. 78, 1135–1148 (2006)

    Google Scholar 

  4. Lin, Q., Painter, O.J., Agrawal, G.P.: Nonlinear optical phenomena in silicon waveguides: modeling and applications. Opt. Express 15, 16604–16644 (2007)

    Google Scholar 

  5. Solli, D.R., Ropers, C., Jalali, B.: Active control of rogue waves for stimulated supercontinuum generation. Phys. Rev. Lett. 101, 233902 (2008)

    Google Scholar 

  6. Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)

    Google Scholar 

  7. Agrawal, G.P.: Nonlinear Fiber Optics, 5th edn. Academic Press, Oxford (2013)

    MATH  Google Scholar 

  8. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  9. 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)

    MathSciNet  Google Scholar 

  10. Liu, J.G., He, Y.: Abundant lump and lump-kink solutions for the new (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Nonlinear Dyn. 92, 1103–1108 (2018)

    Google Scholar 

  11. Sergyeyev, A.: Integrable (3+1)-dimensional systems with rational Lax pairs. Nonlinear Dyn. 91, 1677–1680 (2018)

    MATH  Google Scholar 

  12. Xu, G.Q., Wazwaz, A.M.: Characteristics of integrability, bidirectional solitons and localized solutions for a (3+1)-dimensional generalized breaking soliton equation. Nonlinear Dyn. 96, 1989–2000 (2019)

    Google Scholar 

  13. Ding, C.C., Gao, Y.T., Deng, G.F.: Breather and hybrid solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation for the water waves. Nonlinear Dyn. 97, 2023–2040 (2019)

    MATH  Google Scholar 

  14. Chen, S., Zhou, Y., Baronio, F., Mihalache, D.: Special types of elastic resonant soliton solutions of the Kadomtsev–Petviashvili II equation. Rom. Rep. Phys. 70, 102 (2018)

    Google Scholar 

  15. Kaur, L., Wazwaz, A.M.: Bright-dark lump wave solutions for a new form of the (3+1)-dimensional BKP-Boussinesq equation. Rom. Rep. Phys. 71, 102 (2019)

    Google Scholar 

  16. Malomed, B.A., Mihalache, D.: Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)

    Google Scholar 

  17. Hasegawa, A., Kodama, Y.: Solitons in Optical Communication. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  18. Hasegawa, A.: Optical solitons in communications: from integrability to controllability. Acta. Appl. Math. 39, 85–90 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Hasegawa, A.: An historical review of application of optical solitons for high speed communications. Chaos 10, 475–485 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Hasegawa, A.: Soliton-based optical communications: an overview. IEEE J. Sel. Top. Quantum Electron. 6, 1161–1172 (2000)

    Google Scholar 

  21. Mollenauer, L.F., Gordon, J.P.: Solitons in Optical Fibers: Fundamentals and Applications. Academic Press, London (2006)

    Google Scholar 

  22. Hasegawa, A., Matsumoto, M.: Optical Solitons in Fibers. Springer, Berlin (2010)

    Google Scholar 

  23. Chiao, R.Y., Garmire, E., Townes, C.H.: Self-trapping of optical beams. IEEE J. Quantum Electron. 13, 479–482 (1964)

    Google Scholar 

  24. Zakharov, V.E.: Stability of perodic waves of finite amplitude on the surface of a deepfluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)

    Google Scholar 

  25. 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, 4791–4803 (1998)

    Google Scholar 

  26. Kundu, A.: Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger-type equations. J. Math. Phys. 25, 3433–3438 (1984)

    MathSciNet  Google Scholar 

  27. Wang, X., Yang, B., Chen, Y., Yang, Y.Q.: Higher-order rogue wave solutions of the Kundu–Eckhaus equation. Phys. Scr. 89, 095210 (2014)

    Google Scholar 

  28. Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809 (1973)

    MathSciNet  MATH  Google Scholar 

  29. Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81, 046602 (2010)

    MathSciNet  Google Scholar 

  30. Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. Lett. 90, 113902 (2003)

    Google Scholar 

  31. Wang, L.H., Porsezian, K., He, J.S.: Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys. Rev. E 87, 053202 (2013)

    Google Scholar 

  32. Xu, S.W., He, J.S., Wang, L.H.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A Math. Theor. 44, 305203 (2011)

    MATH  Google Scholar 

  33. Zhang, Y.S., Guo, L.J., Xu, S.W., Wu, Z.W., He, J.S.: The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 19, 1706–1722 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Xiang, Y.J., Dai, X.Y., Wen, S.C., Guo, J., Fan, D.Y.: Controllable Raman soliton self-frequency shift in nonlinear metamaterials. Phys. Rev. A 84(3), 2484–2494 (2011)

    Google Scholar 

  35. Saha, M., Sarma, A.K.: Modulation instability in nonlinear metamaterials induced by cubic-quintic nonlinearities and higher-order dispersive effects. Opt. Commun. 291, 321–324 (2013)

    Google Scholar 

  36. Mohamadou, A., Latchio-Tiofack, C.G., Kofane, T.C.: Wave train generation of solitons in systems with higher-order nonlinearities. Phys. Rev. E 82, 016601 (2010)

    Google Scholar 

  37. Choudhuri, A., Porsezian, K.: Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation. Phys. Rev. A 85(3), 1431–1435 (2012)

    Google Scholar 

  38. Renninger, W.H., Chong, A., Wise, F.W.: Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A 77, 023814 (2008)

    Google Scholar 

  39. Peng, J.S., Zhan, L., Gu, Z.C., Qian, K., Luo, S.Y., Shen, Q.S.: Experimental observation of transitions of different pulse solutions of the Ginzburg–Landau equation in a mode-locked fiber laser. Phys. Rev. A 86, 033808 (2012)

    Google Scholar 

  40. Akhmediev, N., Afanasjev, V.V.: Novel arbitrary-amplitude soliton solutions of the cubic–quintic complex Ginzburg–Landau equation. Phys. Rev. Lett. 75, 2320–2323 (1995)

    Google Scholar 

  41. Akhmediev, N., Afanasjev, V.V., Soto-Crespo, J.M.: Singularities and special soliton solutions of the cubic–quintic complex Ginzburg–Landau equation. Phys. Rev. E 53, 1190–1200 (1996)

    Google Scholar 

  42. Soto-Crespo, J.M., Akhmediev, N., Afanasjev, V.V.: Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation. J. Opt. Soc. Am. B 13, 1439–1449 (1996)

    Google Scholar 

  43. Kharif, C., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B Fluids 22, 603–634 (2003)

    MathSciNet  MATH  Google Scholar 

  44. Yu, W., Liu, W., Triki, H., Qin, Z., Biswas, A., Belić, R.M.: 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)

    MATH  Google Scholar 

  45. Yu, W., Liu, W., Triki, H., Qin, Z., Biswas, A.: Phase shift, oscillation and collision of the anti-dark solitons for the (3+1)-dimensional coupled nonlinear Schrödinger equation in an optical fiber communication system. Nonlinear Dyn. 97, 1253–1262 (2019)

    MATH  Google Scholar 

  46. Xie, X.Y., Meng, G.Q.: Dark solitons for a variable-coefficient AB system in the geophysical fluids or nonlinear optics. Eur. Phys. J. Plus 134, 359 (2019)

    Google Scholar 

  47. Xie, X.Y., Yang, S.K., Ai, C.H., Kong, L.C.: Integrable turbulence for a coupled nonlinear Schrödinger system. Phys. Lett. A 384(5), 126119 (2020)

    Google Scholar 

  48. Kenji, I.: Generalization of the Kaup–Newell inverse scattering formulation and Darboux transformation. J. Phys. Soc. Jpn. 68, 355–359 (1999)

    MathSciNet  MATH  Google Scholar 

  49. Hopkin, M.: Sea snapshots will map frequency of freak waves. Nature 430, 492–492 (2004)

    Google Scholar 

  50. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)

    Google Scholar 

  51. Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of Peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)

    Google Scholar 

  52. Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80, 033610 (2009)

    Google Scholar 

  53. Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675–678 (2009)

    MATH  Google Scholar 

  54. Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean. Springer, Berlin (2009)

    MATH  Google Scholar 

  55. Akhmediev, N., Dudley, J.M., Solli, D.R., Turitsyn, S.K.: Recent progress in investigating optical rogue waves. J. Opt. 15, 060201 (2013)

    Google Scholar 

  56. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47–89 (2013)

    MathSciNet  Google Scholar 

  57. Chen, S., Baronio, F., Soto-Crespo, J.M., Grelu, P., Mihalache, D.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017)

    MathSciNet  MATH  Google Scholar 

  58. Liu, W., Zhang, Y., He, J.: Rogue wave on a periodic background for Kaup–Newell equation. Rom. Rep. Phys. 70, 106 (2018)

    Google Scholar 

  59. Charalampidis, E.G., Cuevas-Maraver, J., Frantzeskakis, D.J., Kevrekidis, P.G.: Rogue waves in ultracold bosonic seas. Rom. Rep. Phys. 70, 504 (2018)

    Google Scholar 

  60. Li, Z.D., Wei, H.C., He, P.B.: Rogue wave structure and formation mechanism in the coupled nonlinear Schrödinger equations. Rom. Rep. Phys. 71, 110 (2019)

    Google Scholar 

  61. Ward, C.B., Kevrekidis, P.G.: Rogue waves as self-similar solutions on a background: a direct calculation. Rom. J. Phys. 64, 112 (2019)

    Google Scholar 

  62. Liu, W., Wazwaz, A.M.: Dynamics of fusion and fission collisions between lumps and line solitons in the Maccari’s System. Rom. J. Phys. 64, 111 (2019)

    Google Scholar 

  63. Wang, Z.H., He, L.Y., Qin, Z.Y., Grimshaw, R., Mu, G.: High-order rogue waves and their dynamics of the Fokas–Lenells equation revisited: a variable separation technique. Nonlinear Dyn. 98, 2067–2077 (2019)

    MATH  Google Scholar 

  64. Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank. Phys. Rev. Lett. 106, 204502 (2011)

    Google Scholar 

  65. Yeom, D.I., Eggleton, B.J.: Photonics: rogue waves surface in light. Nature 450, 953–954 (2007)

    Google Scholar 

  66. He, J.S., Wang, L.H., Li, L.J., Porsezian, K., Erdelyi, R.: Few-cycle optical rogue waves: complex modified Korteweg–de Vries equation. Phys. Rev. E 89, 062917 (2014)

    Google Scholar 

  67. He, J.S., Zhang, H.R., Wang, L.H., Porsezian, K., Fokas, A.S.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013)

    Google Scholar 

Download references

Funding

This work is supported by the NSF of China under Grant No. 11671219.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, Zhejiang, People’s Republic of China

    Huian Lin & Lihong Wang

  2. Institute for Advanced Study, Shenzhen University, Shenzhen, 518060, Guangdong, People’s Republic of China

    Jingsong He

  3. Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, 077125, Bucharest, Magurele, Romania

    Dumitru Mihalache

Authors
  1. Huian Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jingsong He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lihong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Mihalache
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingsong He.

Ethics declarations

Conflict statement

We declare we have no conflict of interests.

Ethical statement

Authors declare that they comply with ethical standards.

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

Lin, H., He, J., Wang, L. et al. Several categories of exact solutions of the third-order flow equation of the Kaup–Newell system. Nonlinear Dyn 100, 2839–2858 (2020). https://doi.org/10.1007/s11071-020-05650-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05650-2

Keywords

Search

Navigation