Skip to main content
SpringerLink
Log in

Partial-limit solutions and rational solutions with parameter for the Fokas-Lenells equation

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

Abstract

A partial-limit method is developed to understand solutions related to real eigenvalues for the Fokas–Lenells equation. By applying a partial-limit procedure to soliton solutions of the Fokas–Lenells equation, new multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop \(|u|^2\), the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly changing amplitudes. The partial-limit procedure may be extended to other integrable systems and generate new solutions.

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

Similar content being viewed by others

References

  1. Kaup, D.J., Newell, A.C.: On the Coleman correspondence and the solution of the massive Thirring model. Lett. AL Nuovo Cimento 20, 325–331 (1977)

    Article  MathSciNet  Google Scholar 

  2. Gerdjikov, V.S., Ivanov, M.I., Kulish, P.P.: Quadratic bundle and nonlinear equations. Theor. Math. Phys. 44, 784–795 (1980)

    Article  Google Scholar 

  3. Nijhoff, F.W., Capel, H.W., Quispel, G.R.W., van der Linden, J.: The derivative nonlinear Schrödinger equation and the massive Thirring model. Phys. Lett. A 93, 455–458 (1983)

    Article  MathSciNet  Google Scholar 

  4. Thirring, W.E.: A soluble relativistic field theory. Ann. Phys. 3, 91–112 (1958)

    Article  MathSciNet  Google Scholar 

  5. Wightman, A.S.: Introduction to some aspects of the relativistic dynamics of quantized fields. In: Levy, M. (ed) 1964 Carg\(\grave{\rm e}\)se Summer School Lectures, pp. 171–291. Gordon and Breach, New York (1967)

  6. Mikhailov, A.V.: Integrability of the two-dimensional Thirring model. JETP Lett. 23, 320–323 (1976)

    Google Scholar 

  7. Gerdjikov, V.S., Ivanov, M.I.: The quadratic pencil of general type and the nonlinear evolution. Hierarchies of Hamiltonian structures. JINR preprint E2-82-595, Dubna, USSR (1982)

  8. Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)

    Article  Google Scholar 

  9. Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215–232 (2009)

    Article  MathSciNet  Google Scholar 

  10. Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145–150 (1995)

    Article  MathSciNet  Google Scholar 

  11. Lenells, J., Fokas, A.S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity 22, 11–27 (2009)

    Article  MathSciNet  Google Scholar 

  12. Ai, L.P., Xu, J.: On a Riemann–Hilbert problem for the Fokas–Lenells equation. Appl. Math. Lett. 87, 57–63 (2019)

    Article  MathSciNet  Google Scholar 

  13. Zhao, Y., Fan, E.G.: Inverse scattering transformation for the Fokas–Lenells equation with nonzero boundary conditions. J. Nonlinear Math. Phys. 28, 38–52 (2021)

    Article  MathSciNet  Google Scholar 

  14. Lenells, J.: Dressing for a novel integrable generalization of the nonlinear Schrödinger equation. J. Nonlinear Sci. 20, 709–722 (2010)

    Article  MathSciNet  Google Scholar 

  15. Zhao, P., Fan, E.G., Hou, Y.: Algebro-geometric solutions and their reductions for the Fokas–Lenells hierarchy. J. Nonlinear Math. Phys. 20, 355–393 (2013)

    Article  MathSciNet  Google Scholar 

  16. He, J.S., Xu, S.W., Porsezian, K.: Rogue waves of the Fokas–Lenells equation. J. Phys. Soc. Jpn. 81, 124007 (2012)

  17. Wang, Y., Xiong, Z.J., Ling, L.: Fokas–Lenells equation: three types of Darboux transformation and multi-soliton solutions. Appl. Math. Lett. 107, 106441 (2020)

    Article  MathSciNet  Google Scholar 

  18. Matsuno, Y.: A direct method of solution for the Fokas–Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions. J. Phys. A: Math. Theor. 45, 235202 (2012)

    Article  Google Scholar 

  19. Matsuno, Y.: A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. J. Phys. A: Math. Theor. 45, 475202 (2012)

    Article  Google Scholar 

  20. Liu, S.Z., Wang, J., Zhang, D.J.: The Fokas-Lenells equations: bilinear approach. arXiv:2104.04938v2, preprint (2021), to appear in Stud. Appl. Math. https://doi.org/10.1111/sapm.12454

  21. Lamb, G.L., JR.: Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev. Mod. Phys. 43, 99–124 (1971)

  22. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. JETP 34, 62–69 (1972)

    MathSciNet  Google Scholar 

  23. Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer-Verlag, Berlin (1991)

    Book  Google Scholar 

  24. Zhang, D.J.: Notes on solutions in Wronskian form to soliton equations: KdV-type. arxiv: nlin.SI/0603008, preprint (2006)

  25. Shchesnovich, V.S., Yang, J.K.: Higher-order solitons in the \(N\)-wave system. Stud. Appl. Math. 110, 297–332 (2003)

    Article  MathSciNet  Google Scholar 

  26. Zhang, D.J., Zhao, S.L.: Solutions to the ABS lattice equations via generalized Cauchy matrix approach. Stud. Appl. Math. 131, 72–103 (2013)

    Article  MathSciNet  Google Scholar 

  27. Zhang, D.J., Zhao, S.L., Sun, Y.Y., Zhou, J.: Solutions to the modified Korteweg-de Vries equation. Rev. Math. Phys. 26, 14300064 (2014)

    Article  MathSciNet  Google Scholar 

  28. Hirota, R.: A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys. 52, 1498–1512 (1974)

    Article  Google Scholar 

  29. Nimmo, J.J.C.: A bilinear Bäcklund transformation for the nonlinear Schrödinger equation. Phys. Lett. A 99, 279–280 (1983)

    Article  MathSciNet  Google Scholar 

  30. Ankiewicz, A., Akhmediev, N.: Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions. Nonlinear Dyn. 91, 1931–1938 (2018)

    Article  Google Scholar 

  31. Xing, Q.X., Wu, Z.W., Mihalache, D., He, J.S.: Smooth positon solutions of the focusing modified Korteweg-de Vries equation. Nonlinear Dyn. 89, 2299–2310 (2017)

    Article  MathSciNet  Google Scholar 

  32. Zhang, Z., Yang, X.Y., Li, B.: Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation. Nonlinear Dyn. 100, 1551–1557 (2020)

    Article  Google Scholar 

  33. Nimmo, J.J.C., Freeman, N.C.: Rational solution of the Korteweg-de Vries equation in Wronskian form. Phys. Lett. A 99, 443–446 (1983)

    Article  MathSciNet  Google Scholar 

  34. Wu, H., Zhang, D.J.: Mixed rational-soliton solutions of two differential-difference equations in Casorati determinant form. J. Phys. A: Gen. Math. 36, 4867–4873 (2003)

    Article  MathSciNet  Google Scholar 

  35. Chen, P., Wang, G.S., Zhang, D.J.: The limit solutions of the difference-difference KdV equation. Chaos Solitons Fractals 40, 376–381 (2009)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Z., Li, B., Chen, J.C., Guo, Q.: Construction of higher-order smooth positons and breather positons via Hirota’s bilinear method. Nonlinear Dyn. 105, 2611–2618 (2021)

  37. Zhang, R.F., Bilige, S.D.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to \(p\)-gBKP equatuon. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  Google Scholar 

  38. Zhang, R.F., Bilige, S.D., Temuer, C.L.: Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method. J. Syst. Sci. Complex. 34, 122–139 (2021)

    Article  MathSciNet  Google Scholar 

  39. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the \((2+1)\)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the \((3+1)\)-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

Download references

Acknowledgements

The author is grateful to Prof. Da-jun Zhang for discussion. The author’s sincere thanks are also extended to the referees for their invaluable comments. This project is supported by the NSF of China (Nos. 12171308 and 11875040).

Funding

This project is supported by the NSF of China (Nos. 12171308 and 11875040).

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Hua Wu

Authors
  1. Hua Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hua Wu.

Ethics declarations

Conflict of interest

The author declares that there is no conflict of interests regarding the publication of this paper.

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

Wu, H. Partial-limit solutions and rational solutions with parameter for the Fokas-Lenells equation. Nonlinear Dyn 106, 2497–2508 (2021). https://doi.org/10.1007/s11071-021-06911-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06911-4

Keywords

Search

Navigation