Skip to main content
SpringerLink
Log in

Nonlocal symmetries of some nonlinear partial differential equations with third-order Lax pairs

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

By including spectral functions, we obtain nonlocal symmetries equivalent to Lie point symmetries of the corresponding extended systems for the Boussinesq equation, the modified generalized Vakhnenko equation, the Hirota–Satsuma equation, and the Sawada–Kotera equation. All considered equations have third-order Lax pairs, which allows studying their nonlocal symmetries in a unified way.

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

Similar content being viewed by others

References

  1. A. M. Vinogradov and I. S. Krasil’shchik, “A method of computing higher symmetries of nonlinear evolution equations, and nonlocal symmetries,” Sov. Math. Dokl., 22, 235–239 (1980).

    MATH  Google Scholar 

  2. I. S. Krasil’shchik and A. M. Vinogradov, “Nonlocal symmetries and the theory of coverings: An addendum to A. M. Vinogradov’s ‘local symmetries and conservation laws’,” Acta Appl. Math., 2, 79–96 (1984).

    Article  MathSciNet  Google Scholar 

  3. I. S. Krasil’shchik and A. M. Vinogradov, “Nonlocal trends in the geometry of differential equations: Symmetries, conservation laws, and Bäcklund transformations,” Acta Appl. Math., 15, 161–209 (1989).

    Article  MathSciNet  Google Scholar 

  4. Y.-Q. Li, J.-C. Chen, Y. Chen, and S.-Y. Lou, “Darboux transformations via Lie point symmetries: KdV equation,” Chin. Phys. Lett., 31, 010201 (2014).

    Article  ADS  Google Scholar 

  5. F. Galas, “New nonlocal symmetries with pseudopotentials,” J. Phys. A: Math. Gen., 25, L981–L986 (1992).

    Article  MathSciNet  ADS  Google Scholar 

  6. K. Kiso, “Pseudopotentials and symmetries of evolution equations,” Hokkaido Math. J., 18, 125–136 (1989).

    Article  MathSciNet  Google Scholar 

  7. I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Nonlocal symmetries: Heuristic approach,” J. Soviet Math., 55, 1401–1450 (1991).

    Article  Google Scholar 

  8. M. C. Nucci, “Pseudopotentials, Lax equations, and Bäcklund transformations for nonlinear evolution equations,” J. Phys. A: Math. Gen., 21, 73–79 (1988).

    Article  ADS  Google Scholar 

  9. S.-Y. Lou and X.-B. Hu, “Non-local symmetries via Darboux transformations,” J. Phys. A: Math. Gen., 30, L95–L100 (1997).

    Article  MathSciNet  Google Scholar 

  10. S. C. Anco, E. D. Avdonina, A. Gainetdinova, L. R. Galiakberova, N. H. Ibragimov, and T. Wolf, “Symmetries and conservation laws of the generalized Krichever–Novikov equation,” J. Phys. A: Math. Theor., 49, 105201 (2016).

    Article  MathSciNet  ADS  Google Scholar 

  11. E. G. Reyes, “Nonlocal symmetries and the Kaup–Kupershmidt equation,” J. Math. Phys., 46, 073507 (2005).

    Article  MathSciNet  ADS  Google Scholar 

  12. R. Hernández-Heredero and E. G. Reyes, “Nonlocal symmetries and a Darboux transformation for the Camassa–Holm equation,” J. Phys. A: Math. Theor., 42, 182002 (2009).

    Article  MathSciNet  ADS  Google Scholar 

  13. R. Hernández Heredero and E. G. Reyes, “Nonlocal symmetries, compacton equations, and integrability,” Internat. J. Geom. Meth. Modern Phys., 10, 1350046 (2013).

    Article  MathSciNet  ADS  Google Scholar 

  14. S. B. Leble and N. V. Ustinov, “Third order spectral problems: Reductions and Darboux transformations,” Inverse Problems, 10, 617–633 (1994).

    Article  MathSciNet  ADS  Google Scholar 

  15. A. S. Fokas and M. J. Ablowitz, “On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane,” J. Math. Phys., 25, 2494–2505 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  16. B. G. Konopelchenko, “The two-dimensional second-order differential spectral problem: Compatibility conditions, general BTs, and integrable equations,” Inverse Problems, 4, 151–163 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  17. M. Leo, R. A. Leo, G. Soliani, and P. Tempesta, “On the relation between Lie symmetries and prolongation structures of nonlinear field equations: Non-local symmetries,” Progr. Theor. Phys., 105, 77–97 (2001).

    Article  MathSciNet  ADS  Google Scholar 

  18. A.-M. Wazwaz, “Multiple-soliton solutions for the Boussinesq equation,” App. Math. Comput., 192, 479–486 (2007).

    Article  MathSciNet  Google Scholar 

  19. C. Gu, H. Hu, and Z. Zhou, Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry (Math. Phys. Stud., Vol. 26), Springer, Dordrecht (2005).

    Book  Google Scholar 

  20. X.-B. Hu and S.-Y. Lou, “Nonlocal symmetries of nonlinear integrable models,” in: Symmetry in Nonlinear Mathematical Physics (Proc. Inst. Math. NAS Ukraine, Vol. 30, No. 1), Inst. Math. NAS Ukraine, Kiev (2000), pp. 120–126.

    MathSciNet  MATH  Google Scholar 

  21. S.-Y. Lou, “A note on the new similarity reductions of the Boussinesq equation,” Phys. Lett. A, 151, 133–135 (1990).

    Article  MathSciNet  ADS  Google Scholar 

  22. V. O. Vakhnenko and E. J. Parkes, “The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method,” Chaos Solitons Fractals, 13, 1819–1826 (2002).

    Article  MathSciNet  ADS  Google Scholar 

  23. Y. Wang and Y. Chen, “Integrability of the modified generalised Vakhnenko equation,” J. Math. Phys., 53, 123504 (2012).

    Article  MathSciNet  ADS  Google Scholar 

  24. V. O. Vakhnenko, E. J. Parkes, and A. J. Morrison, “A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation,” Chaos Solitons Fractals, 17, 683–692 (2003).

    Article  MathSciNet  ADS  Google Scholar 

  25. P. G. Estevez, “Reciprocal transformations for a spectral problem in 2+1 dimensions,” Theor. Math. Phys., 159, 763–769 (2009).

    Article  MathSciNet  Google Scholar 

  26. R. Hirota and J. Satsuma, “\(N\)-Soliton solutions of model equations for shallow water equation,” J. Phys. Soc. Japan, 40, 611-612 (1976).

    MATH  ADS  Google Scholar 

  27. P. A. Clarkson and E. L. Mansfield, “Symmetry reductions and exact solutions of shallow water wave equations,” Acta Appl. Math., 39, 245–276 (1995).

    Article  MathSciNet  Google Scholar 

  28. P. Deift, C. Tomei, and E. Trubowitz, “Inverse scattering and the Boussinesq equation,” Commun. Pure Appl. Math., 35, 567–628 (1982).

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research was supported by the Natural Science Foundation of Zhejiang Province (No. LQ20A010009), General Scientific Research of Zhejiang Province (No. 201909003329), National Natural Science Foundation of China (No. 11675055), and Natural Science Foundation of Shanghai (No. 19ZR1414000).

Author information

Authors and Affiliations

  1. College of Science, Zhejiang University of Technology, Hangzhou, China

    Xiazhi Hao

Authors
  1. Xiazhi Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiazhi Hao.

Ethics declarations

The author declares no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hao, X. Nonlocal symmetries of some nonlinear partial differential equations with third-order Lax pairs. Theor Math Phys 206, 119–127 (2021). https://doi.org/10.1134/S004057792102001X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S004057792102001X

Keywords

Search

Navigation