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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).
P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York (1993).
G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York (1989).
G. W. Bluman and J. J. Cole, “The general similarity solution of the heat equation,” J. Math. Mech., No. 18, 1025–1042 (1968/69).
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).
P. J. Olver and P. Rosenau, “The construction of special solutions to partial differential equations,” Phys. Lett. A, 114, No. 3, 107–112 (1986).
W. I. Fushchich and A. G. Nikitin, Symmetry of Maxwell Equations [in Russian], Naukova Dumka, Kiev (1983).
C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York (1982).
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).
I. M. Anderson, N. Kamran, and P. J. Olver, “Internal, external, and generalized symmetries,” Adv. Math., 100, No. 1, 53–100 (1993).
W. I. Fushchych and V. A. Tychynin, Preprint No 82.33, Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1982).
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).
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).
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).
E. G. Reyes, “Nonlocal symmetries and the Kaup–Kupershmidt equation,” J. Math. Phys., 46, No. 7, 073507 (2005).
F. Galas, “New nonlocal symmetries with pseudopotentials,” J. Phys. A: Math. Gen., 25, No. 15, L981–L986 (1992).
A. R. Forsyth, Theory of Differential Equations, Vols. 5, 6, Dover Publication, New York (1959).
W. F. Ames, Nonlinear Partial Differential Equations in Engineering. Vol. 1, Academic Press, New York (1965).
N. H. Ibragimov and R. L. Anderson, “Lie–Bäcklund tangent transformations,” J. Math. Anal. Appl., 59, No. 1, 145–162 (1977).
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).
H. D. Wahlquist and F. B. Estabrook, “Prolongation structures of nonlinear evolution equations,” J. Math. Phys., 16, No. 1, 1–7 (1975).
F. B. Estabrook, “Moving frames and prolongation algebras,” J. Math. Phys., 23, No. 11, 2071–2076 (1982).
F. Pirani, D. Robinson, and W. F. Shadwick, Jet Bundle Formulation of Backlund Transformations to Nonlinear Evolution Equations, D. Reidel Publishing Company, Dordrecht (1979).
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).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).
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.
J. R. King, “Some non-local transformations between nonlinear diffusion equations,” J. Phys. A: Math. Gen., No. 23, 5441–5464 (1990).
N. Euler, “Multipotentialisations and iterating-solution formulae: the Krichever–Novikov equation,” J. Nonlin. Math. Phys., 16, suppl. 1, 93–106 (2009).
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).
G. W. Bluman, “Nonlocal extensions of similarity methods,” J. Nonlin. Math. Phys., 15, suppl. 1, 1–24 (2008).
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).
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).
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.
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).
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).
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).
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).
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.
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).
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.
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).
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.
A. Jeffrey, Applied Partial Differential Equations. An introduction, Academic Press, New York (2003).
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).
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.
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).
Author information
Authors and Affiliations
Corresponding author
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.
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02075-5