Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 065, 15 pages      nlin.SI/0608038      http://dx.doi.org/10.3842/SIGMA.2006.065

On the Linearization of Second-Order Differential and Difference Equations

Vladimir Dorodnitsyn
Keldysh Institute of Applied Mathematics of Russian Academy of Science, 4 Miusskaya Sq., Moscow, 125047 Russia

Received November 28, 2005, in final form July 13, 2006; Published online August 15, 2006

Abstract
This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and Noether-type integration technique. It turned out that there exist nonlinear superposition principles for solutions of special second-order ordinary difference equations which possess Lie group symmetries. This superposition springs from the linearization of second-order ordinary difference equations by means of non-point transformations which act simultaneously on equations and meshes. These transformations become some sort of contact transformations in the continuous limit.

Key words: non-point transformations; second-order ordinary differential and difference equations; linearization; superposition principle.

pdf (216 kb)   ps (143 kb)   tex (15 kb)

References

  1. Dorodnitsyn V., Kozlov R., Winternitz P., Lie group classification of second order difference equations, J. Math. Phys., 2000, V.41, 480-504.
  2. Dorodnitsyn V., Kozlov R., Winternitz P., Continuous symmetries of Lagrangians and exact solutions of discrete equations, J. Math. Phys., 2004, V.45, 336-359, nlin.SI/0307042.
  3. Lie S., Klassifikation und Integration von gewöhnlichen Differentialgleichunden zwischen x, y die eine Gruppe von Transformationen gestatten, Math. Ann., 1888, V.32, 213-281.
  4. Noether E., Invariante Variationsprobleme, Nachr. Konig. Gesell. Wissen., Gottingen, Math.-Phys. Kl., 1918, Heft 2, 235-257.
  5. Dorodnitsyn V., Noether-type theorems for difference equations, Appl. Numer. Math., 2001, V.39, 307-321.
  6. Dorodnitsyn V., The group properties of difference equations, Moscow, Fizmatlit, 2001 (in Russian).
  7. Kumei S., Bluman G.W., When nonlinear differential equations are equivalent to linear differential equations, SIAM J. Appl. Math., 1982, V.42, 1157-1173.
  8. Lie S., Klassifikation und Integration von gewöhnlichen Differentialgleichunden zwischen x, y die eine Gruppe von Transformationen gestatten. I, II, III, IV, Reprinted in Lie's Gessamelte Abhandlungen, Vol. 5, Leipzig, Teubner, 1924.
  9. Ames W.F., Anderson R.L., Dorodnitsyn V.A., Ferapontov E.V., Gazizov R.K., Ibragimov N.H., Svirshchevskii S.R., CRC hand-book of Lie group analysis of differential equations, Vol. I: Symmetries, exact solutions and conservation laws, Editor N.H. Ibragimov, CRC Press, 1994.
  10. Maeda S., The similarity method for difference equations, IMA J. Appl. Math., 1987, V.38, 129-134.
  11. Byrnes G.B., Sahadevan R., Quispel G.R.W., Factorizable Lie symmetries and the linearization of difference equations, Nonlinearity, 1995, V.8, 44-59.

Previous article   Next article   Contents of Volume 2 (2006)