Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Valentyn Tychynin ; Olga Petrova ; Olesya Tertyshnyk

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal symmetries and formulae of nonlocal nonlinear superposition of solutions of these equations were used then for construction of chains of exact solutions. Linearization by means of the Legendre transformations of a second-order PDE with three independent variables allowed to obtain nonlocal superposition formulae for solutions of this equation, and to generate new solutions from group invariant solutions of a linear equation.

Détail

Keywords: Lie classical symmetry; nonlocal symmetries; formulae for generation of solutions; nonlinear superposition principle.

Bibliographie

[1] Anco S. C., Bluman G., “Derivation of conservation laws from nonlocal symmetries of differential equations”, J. Math. Phys., 37 (1996), 2361–2375 | DOI | MR | Zbl

[2] Moitsheki R. J., Broadbridge P., Edwards M. P., “Systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation”, J. Phys. A: Math. Gen., 37 (2004), 8279–8286 | DOI | MR | Zbl

[3] Galas F., “New nonlocal symmetries with pseudopotentials”, J. Phys. A: Math. Gen., 25 (1992), L981–L986 | DOI | MR | Zbl

[4] Papachristou C. J., Harrison K., “Nonlocal symmetries and Bäcklund transformations for the self-dual Yang–Mills system”, J. Math. Phys., 29 (1988), 238–243 | DOI | MR | Zbl

[5] Ibragimov N. H. (ed.), CRC Handbook of Lie group analysis of differential equations. Vol. 1. Symmetries, exact solutions and conservation laws, CRC Press, 1994 | MR | Zbl

[6] Olver P. J., Rosenau P., “The construction of special solutions to partial differential equations”, Phys. Lett. A, 114 (1986), 107–112 | DOI | MR | Zbl

[7] Olver P. J., Rosenau P., “Group-invariant solutions of differential equations”, SIAM J. Appl. Math., 47 (1987), 263–278 | DOI | MR | Zbl

[8] Fushchych W. I., Serov M. I., Tychynin V. A., Amerov T. K., “On nonlocal symmetry of nonlinear heat equation”, Proc. Acad. of Sci. Ukraine, 1992, no. 11, 27–33

[9] Krasil'shchik I. S., Vinogradov A. M., “Nonlocal trends in the geometry of differential equations: symmetries, conservation laws and Bäcklund transformations”, Acta Appl. Math., 15 (1989), 161–209 | DOI | MR

[10] Anderson R. L., Ibragimov N. H., Lie–Bäcklund transformations in applications, SIAM, Philadelphia, 1979 | MR

[11] Ibragimov N. H., Transformation groups applied to mathematical physics, Reidel, Dordrecht, 1985 | MR | Zbl

[12] Dzamay A. V., Vorob'ev E. M., “Infinitesimal weak symmetries of nonlinear differential equations in two independent variables”, J. Phys. A: Math. Gen., 27 (1994), 5541–5549 | DOI | MR

[13] Gandarias M. L., Bruzon M. S., “Symmetry analysis and solutions for a family of Cahn–Hillard equations”, Rep. Math. Phys., 46 (2000), 89–97 | DOI | MR | Zbl

[14] Clarkson P. A., Mansfield E. L., “Symmetry reduction and exact solutions of a class of nonlinear heat equations”, Phys. D, 70 (1993), 250–288 ; solv-int/9306002 | DOI | MR | MR

[15] Ibragimov N. H., Anderson R. L., “Lie–Bäcklund tangent transformations”, J. Math. Anal. Appl., 59 (1977), 145–162 | DOI | MR | Zbl

[16] Fok V. A., “Hydrogen atom and non-Euclidean geometry. Preliminary announcement”, Zs. Phys., 98 (1935), 145–154 | DOI

[17] Lowner C. A., A transformation theory of the partial differential equations of gas dynamics, Nat. Advis. Comm. Aeronat. Tech. Notes, no. 2065, Washington, 1950, 56 pp., ages

[18] Olver P. J., Applications of Lie groups to differential equations, Springer-Verlag, New York, 1993 | MR

[19] Guthrie G. A., “Recursion operators and nonlocal symmetries”, Proc. Roy. Soc. London A, 446 (1994), 107–114 | DOI | MR | Zbl

[20] Guthrie G. A., Hickman M. S., “Nonlocal symmetries of the KdV equation”, J. Math. Phys., 34 (1993), 193–205 | DOI | MR | Zbl

[21] Leo M., Leo R. A., Soliani G., Tempesta P., “On the relation between Lie symmetries and prolongation structures of nonlinear field equations”, Progr. Theoret. Phys., 105 (2001), 77–97 | DOI | MR | Zbl

[22] Olver P. J., Sanders J., Wang J.-P., “Ghost symmertries”, J. Nonlinear Math. Phys., 9, suppl. 1 (2002), 164–172 | DOI | MR

[23] Fushchich W. I., Serov N. I., “The conditional symmetry and reduction of the nonlinear heat conduction equation”, Proc. Acad. of Sci. Ukraine, Ser. A, 7 (1990), 24–27 | MR | Zbl

[24] Tychynin V. A., “Nonlocal symmetry and generating solutions for Harry–Dym type equations”, J. Phys. A: Math. Gen., 27 (1994), 2787–2797 | DOI | MR

[25] Pukhnachev V. V., “Nonlocal symmetries in nonlinear heat equations”, Energy Methods in Continuum Mechanics (Oviedo, 1994), Kluwer Acad. Publ., Dordrecht, 1996, 75–99 | MR

[26] Pukhnachev V. V., “Exact solutions of the hydrodynamic equations derived from partially invariant solutions”, J. Appl. Mech. Tech. Phys., 44 (2003), 317–323 | DOI | MR | Zbl

[27] Sophocleous C., “Transformation properties of variable-coefficient Burgers equation”, Chaos Solitons Fractals, 20 (2004), 1047–1057 | DOI | MR | Zbl

[28] Sophocleous C., “Further transformation properties of generalized inhomogeneous nonlinear diffusion equations with variable coefficients”, Phys. A, 345 (2005), 457–471

[29] Goard J. M., Broadbridge P., “Nonlinear superposition principles obtained by Lie symmetry methods”, J. Math. Anal. Appl., 214 (1997), 633–657 | DOI | MR | Zbl

[30] Jones S. E., Ames W. F., “Nonlinear superposition”, J. Math. Anal. Appl., 17 (1967), 484–487 | DOI | MR | Zbl

[31] Konopelchenko B. G., “On the general structure of nonlinear evolution equations and their Bäcklund transformations connected with the matrix non-stationary Schrödinger spectral problem”, J. Phys. A: Math. Gen., 15 (1982), 3425–3437 | DOI | MR | Zbl

[32] Rogers C., Shadwick W. F., Bäcklund transformations and their applications, Academic Press, New York, 1982 | MR

[33] Wahlquist H. D., Estabrook F. B., “Bäcklund transformation for solutions of the Korteweg–de Vries equation”, Phys. Rev. Lett., 31 (1973), 1386–1390 | DOI | MR

[34] Fushchych W. I., Tychynin V. A., “Generating of solutions for nonlinear equations by the Legendre transformation”, Proc. Acad. Sci. Ukraine, 7 (1992), 20–25 | MR

[35] Fairlie D. B., Mulvey J. A., “Integrable generalizations of the two-dimensional Born–Infeld equation”, J. Phys. A: Math. Gen., 27 (1994), 1317–1324 | DOI | MR | Zbl

[36] Kumei S., Bluman G. W., “When nonlinear differential equations are equivalent to linear differential equations”, SIAM J. Appl. Math., 42 (1982), 1157–1173 | DOI | MR | Zbl

[37] Ovsiannikov L. V., Group analysis of differential equations, Academic Press, 1982 | MR | Zbl

[38] Bluman G. W., Kumei S., Symmetries and differential equations, Springer, Berlin, 1989 | MR | Zbl

[39] Storm M. L., “Heat conduction in simple metals”, J. Appl. Phys., 22 (1951), 940–951 | DOI | MR | Zbl

[40] Bluman G., Kumei S., “On the remarkable nonlinear diffusion equation $(\partial/\partial x) [a(u+b)^{-2}(\partial u/\partial x)]-\partial u/\partial t=0$”, J. Math. Phys., 21 (1980), 1019–1023 | DOI | MR | Zbl