Group Classification of the General Evolution Equation: Local and Quasilocal Symmetries

Renat Zhdanov   ; Victor Lahno

Symmetry, integrability and geometry: methods and applications, Tome 1 (2005)

Résumé

We give a review of our recent results on group classification of the most general nonlinear evolution equation in one spatial variable. The method applied relies heavily on the results of our paper Acta Appl. Math., 69, 2001, in which we obtain the complete solution of group classification problem for general quasilinear evolution equation.

Détail

Keywords: group classification; symmetry; second order nonlinear evolution equation.

Bibliographie

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

[2] Bluman G., Cole J., Similarity methods for differential equations, Springer, Berlin, 1974 | MR | Zbl

[3] Olver P., Applications of Lie groups to differential equations, Springer, New York, 1986 | MR

[4] Fushchych W., Zhdanov R., Symmetries and exact solutions of nonlinear spinor equations, Mathematical Ukraina Publisher, Kyiv, 1997

[5] Ovsiannikov L. V., “Group properties of nonlinear heat equation”, Dokl. AN SSSR, 125 (1959), 492–495 (in Russian)

[6] Dorodnitsyn V. A., “On invariant solutions of non-linear heat equation with a source”, Zhurn. Vych. Matem. Matem. Fiziki, 22 (1982), 1393–1400 (in Russian) | MR | Zbl

[7] Oron A., Rosenau Ph., “Some symmetries of the nonlinear heat and wave equations”, Phys. Lett. A, 118 (1986), 172–176 | DOI | MR | Zbl

[8] Akhatov I. Sh., Gazizov R. K., Ibragimov N. H., “Group classification of equations of nonlinear filtration”, Dokl. AN SSSR, 293 (1987), 1033–1035 | MR | Zbl

[9] Edwards M. P., “Classical symmetry reductions of nonlinear diffusion-convection equations”, Phys. Lett. A, 190 (1994), 149–154 | DOI | MR | Zbl

[10] Yung C. M., Verburg K., Baveye P., “Group classification and symmetry reductions of the non-linear diffusion-convection equation $u_t=(D(u)u_x)_x-K'(u)u_x$”, Int. J. Non-Linear Mech., 29 (1994), 273–278 | DOI | MR | Zbl

[11] Pallikaros C., Sophocleous C., “On point transformations of generalized nonlinear diffusion equations”, J. Phys. A: Math. Gen., 28 (1995), 6459–6465 | DOI | MR | Zbl

[12] Gandarias M. L., “Classical point symmetries of a porous medium equation”, J. Phys. A: Math. Gen., 29 (1996), 607–633 | DOI | MR | Zbl

[13] Cherniha R., Serov M., “Symmetries, ansätze and exact solutions of nonlinear second-order evolutions equations with convection terms”, European J. Appl. Math., 9 (1998), 527–542 | DOI | MR | Zbl

[14] El-labany S. K., Elhanbaly A. M., Sabry R., “Group classification and symmetry reduction of variable coefficient nonlinear diffusion-convection equation”, J. Phys A: Math. Gen., 35 (2002), 8055–8063 | DOI | MR | Zbl

[15] Güngör F., “Group classification and exact solutions of a radially symmetric porous-medium equation”, Int. J. Non-Linear Mech., 37 (2002), 245–255 | DOI | MR | Zbl

[16] Reid G. J., “Finding abstract Lie symmetry algebras of differential equations without integrating determining equations”, European J. Appl. Math., 2 (1991), 319–340 | DOI | MR | Zbl

[17] Mubarakzjanov G. M., “On solvable Lie algebras”, Izv. Vyssh. Uchebn. Zaved. Matematika, 1963, no. 1, 114–123 (in Russian) | MR

[18] Mubarakzjanov G. M., “The classification of the real structure of five-dimensional Lie algebras”, Izv. Vyssh. Uchebn. Zaved. Matematika, 1963, no. 3(34), 99–106 (in Russian) | MR

[19] Fushchych W. I., “Symmetry in mathematical physics problems”, Algebraic-Theoretical Studies in Mathematical Physics, Inst. of Math. Acad. of Sci. Ukraine, Kyiv, 1981, 6–28

[20] Gagnon L., Winternitz P., “Symmetry classes of variable coefficient nonlinear Schrödinger equations”, J. Phys. A: Math. Gen., 26 (1993), 7061–7076 | DOI | MR | Zbl

[21] Zhdanov R. Z., Roman O. V., “On preliminary symmetry classification of nonlinear Schrödinger equation with some applications of Doebner–Goldin models”, Rep. Math. Phys., 45 (2000), 273–291 | DOI | MR | Zbl

[22] Zhdanov R. Z., Lahno V. I., “Group classification of heat conductivity equations with a nonlinear source”, J. Phys. A: Math. Gen., 32 (1999), 7405–7418 | DOI | MR | Zbl

[23] Basarab-Horwath P., Lahno V., Zhdanov R., “The structure of Lie algebras and the classification problem for partial differential equations”, Acta Appl. Math., 69 (2001), 43–94 | DOI | MR | Zbl

[24] Güngör F., Lagno V. I., Zhdanov R. Z., “Symmetry classification of KdV-type nonlinear evolution equations”, J. Math. Phys., 45 (2004), 2280–2313 | DOI | MR | Zbl

[25] Lahno V., Zhdanov R., “Group classification of nonlinear wave equations”, J. Math. Phys., 46 (2005), 053301, 37 pp., ages | DOI | MR | Zbl

[26] Lie S., “On integration of a class of linear differential equations by means of definite integrals”, Arch. Math., 6 (1981), 328–368 (in German)

[27] Akhatov I. S., Gazizov R. K., Ibragimov N. K., “Nonlocal symmetries: A heuristic approach”, J. Soviet Math., 55 (1991), 1401–1450 | DOI | MR

[28] Meirmanov A. M., Pukhnachov V. M., Shmarev S. I., Evolution equations and Lagrangian coordinates, Walter de Gruyter, Berlin, 1997 | MR