Skip to main content
SpringerLink
Log in

Nonlocal Transformations with Additional Variables. Forced Symmetries

  • Published:
Ukrainian Mathematical Journal Aims and scope

The concept of nonlocal transformation with additional variables is proposed, developed, and applied to the determination of additional symmetries of nonlinear partial differential equations. We consider possible schemes of relationships between the differential equations obtained by means of extended nonlocal transformations of this type and present several examples. The proposed method is used to construct algorithms and formulas for generating new solutions from the known solutions by using additional symmetries. These formulas are applied to find the exact solutions for some nonlinear equations.

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. L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).

    MATH  Google Scholar 

  2. P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1993).

    Book  MATH  Google Scholar 

  3. G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York (1989).

    Book  MATH  Google Scholar 

  4. G. W. Bluman and J. J. Cole, “The general similarity solution of the heat equation,” J. Math. Mech., No. 18, 1025–1042 (1968/69).

  5. W. I. Fushchich and N. I. Serov, “The conditional symmetry and reduction of the nonlinear heat equation,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 24–27 (1990).

  6. P. J. Olver and P. Rosenau, “The construction of special solutions to partial differential equations,” Phys. Lett. A, 114, No. 3, 107–112 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  7. W. I. Fushchich and A. G. Nikitin, Symmetry of Maxwell Equations [in Russian], Naukova Dumka, Kiev (1983).

    MATH  Google Scholar 

  8. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York (1982).

    MATH  Google Scholar 

  9. G. W. Bluman, G. J. Reid, and S. Kumei, “New classes of symmetries for partial differential equations,” J. Math. Phys., 29, No. 4, 806–811 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  10. I. M. Anderson, N. Kamran, and P. J. Olver, “Internal, external, and generalized symmetries,” Adv. Math., 100, No. 1, 53–100 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  11. W. I. Fushchych and V. A. Tychynin, Preprint No 82.33, Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1982).

  12. V. A. Tychynin, “Non-local symmetry and generating solutions for Harry–Dym-type equations,” J. Phys. A: Math. Gen., 27, No. 13, 4549–4556 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  13. V. A. Tychynin, O. V. Petrova, and O. M. Tertyshnyk, “Symmetries and generation of solutions for partial differential equations,” SIGMA, Symmetry Integrabil. Geom. Meth. Appl., 3, Paper 019, 0702033 (2007).

  14. V. A. Tychynin and O. V. Petrova, “Nonlocal symmetries and formulae for generation of solutions for a class of diffusion-convection equations,” J. Math. Anal. Appl., 382, No. 1, 20–33 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  15. E. G. Reyes, “Nonlocal symmetries and the Kaup–Kupershmidt equation,” J. Math. Phys., 46, No. 7, 073507 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Galas, “New nonlocal symmetries with pseudopotentials,” J. Phys. A: Math. Gen., 25, No. 15, L981–L986 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. R. Forsyth, Theory of Differential Equations, Vols. 5, 6, Dover Publication, New York (1959).

    Google Scholar 

  18. W. F. Ames, Nonlinear Partial Differential Equations in Engineering. Vol. 1, Academic Press, New York (1965).

    MATH  Google Scholar 

  19. N. H. Ibragimov and R. L. Anderson, “Lie–Bäcklund tangent transformations,” J. Math. Anal. Appl., 59, No. 1, 145–162 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  20. H. D.Wahlquist and F. B. Estabrook, “Bäcklund transformation for solution of the Korteweg–de Vries equation,” Phys. Rev. Lett., 31, No. 23, 1386–1389 (1973).

    Article  MathSciNet  Google Scholar 

  21. H. D. Wahlquist and F. B. Estabrook, “Prolongation structures of nonlinear evolution equations,” J. Math. Phys., 16, No. 1, 1–7 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  22. F. B. Estabrook, “Moving frames and prolongation algebras,” J. Math. Phys., 23, No. 11, 2071–2076 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Pirani, D. Robinson, and W. F. Shadwick, Jet Bundle Formulation of Backlund Transformations to Nonlinear Evolution Equations, D. Reidel Publishing Company, Dordrecht (1979).

    MATH  Google Scholar 

  24. R. Hermann, “The pseudopotentials of Estabrook and Wahlquist, the geometry of solutions, and the theory of connections,” Phys. Rev. Lett., No. 36, 835–836 (1976).

  25. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).

    Book  MATH  Google Scholar 

  26. I. Sh. Akhatov, R. K. Gazizov, and N. H. Ibragimov, “Nonlocal symmetries. A heuristic approach,” J. Sov. Math., 55, No. 1, 3–83 (1991); VINITI Series in Contemporary Problems of Mathematics. Newest Results [in Russian], No. 34 (1989), pp. 3–84.

  27. J. R. King, “Some non-local transformations between nonlinear diffusion equations,” J. Phys. A: Math. Gen., No. 23, 5441–5464 (1990).

  28. N. Euler, “Multipotentialisations and iterating-solution formulae: the Krichever–Novikov equation,” J. Nonlin. Math. Phys., 16, suppl. 1, 93–106 (2009).

    MathSciNet  MATH  Google Scholar 

  29. G.W. Bluman and P. Doran-Wu, “The use of factors to discover potential systems or linearizations,” Acta Appl. Math., No. 41, 21–43 (1995).

  30. G. W. Bluman, “Nonlocal extensions of similarity methods,” J. Nonlin. Math. Phys., 15, suppl. 1, 1–24 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  31. G. W. Bluman and A. F. Cheviakov, “Nonlocally related systems, linearization, and nonlocal symmetries for the nonlinear wave equation,” J. Math. Anal. Appl., No. 333, 93–111 (2007).

  32. G. W. Bluman, A. F. Cheviakov, and N. M. Ivanova, “Framework for nonlocally related partial differential equation systems and nonlocal symmetries: extension, simplification, and examples,” J. Math. Phys., No. 47, 113505, 1–23 (2006).

  33. R. O. Popovych and N. M. Ivanova, “Hierarchy of conservation laws of diffusion-convection equations,” J. Math. Phys., No. 46, 043502, 1–22 (2005); DOI: https://doi.org/10.1063/1.1865813.

  34. 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  MATH  Google Scholar 

  35. N. M. Ivanova, R. O. Popovych, C. Sophocleous, and O. O. Vaneeva, “Conservation laws and hierarchies of potential symmetries for certain diffusion equations,” Physica A, No. 388, 343–356 (2008).

  36. A. Clarkson, A. S. Fokas, and M. J. Ablowitz, “Hodograph transformations of linearizable partial differential equations,” SIAM J. Appl. Math., No. 49, 1188–1209 (1989).

  37. W. I. Fushchych and V. A. Tychynin, “Exact solutions and superposition principle for nonlinear wave equation,” Dokl. Akad. Nauk Ukr., Ser. A, No. 5, 32–36 (1990).

  38. V. A. Tychynin, “New nonlocal symmetries of diffusion-convection equations and their connection with generalized hodograph transformation,” Symmetry, 7, No. 4, 1751–1767 (2015); DOI: https://doi.org/10.3390/sym7041751.

    Article  MathSciNet  MATH  Google Scholar 

  39. W. Rzeszut, V. Vladimirov, O. M. Tertyshnyk, and V. A. Tychynin, “Linearizability and nonlocal superposition for nonlinear transport equation with memory,” Rep. Math. Phys., 72, No. 2, 235–252 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  40. V. A. Tychynin, “On construction of new exact solutions for nonlinear equations via known particular solutions,” in: Symmetry and Solutions of Equations of Mathematical Physics [in Russian], Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1989), pp. 86–89.

  41. V. A. Tychynin, Non-Local Symmetries and Solutions for Some Classes of Nonlinear Equations of Mathematical Physics, Candidate- Degree Thesis (Physics and Mathematics), Kiev (1994).

  42. V. A. Tychynin, “Adjoint solutions and superposition principle for linearizable Krichever–Novikov equation,” in: Collection of Works of the Institute of Mathematics “Symmetry and Integrability of the Equations of Mathematical Physics,” Kyiv, vol. 16, No. 1 (2019), pp. 181–192.

  43. A. Jeffrey, Applied Partial Differential Equations. An introduction, Academic Press, New York (2003).

  44. G. W. Bluman and P. Doran-Wu, “The use of factors to discover potential systems or linearisations,” Acta Appl. Math., No. 41, 21–43 (1995).

  45. V. A. Tychynin and O. N. Tertyshnik, “Nonlocal multiplication of solutions to one nonlinear telegraph equation,” in: Proc. of the Second All-Ukrainian Sci. Seminar “Ukrainian School of the Group Analysis of Differential Equations: Achievements and Prospects,” 19-20.10 (2012), pp. 129–140.

  46. V. I. Fushchich, V. A. Tychinin, and N. I. Serov, “A formula of the multiplication of solutions of the Korteweg-de Vries equations,” Ukr. Mat. Zh., 44, No. 5, 716–719 (1992); English translation: Ukr. Math. J., 44, No. 5, 649–651 (1992).

Download references

Author information

Authors and Affiliations

  1. Prydniprovska State Academy of Civil Engineering and Architecture, Dnipro, Ukraine

    V. A. Tychynin

Authors
  1. V. A. Tychynin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Tychynin.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 3, pp. 400–417, March, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i3.6995.

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

Tychynin, V.A. Nonlocal Transformations with Additional Variables. Forced Symmetries. Ukr Math J 74, 452–471 (2022). https://doi.org/10.1007/s11253-022-02075-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-022-02075-5

Search

Navigation