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

SIGMA 2 (2006), 065, 15 pages      nlin.SI/0608038

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

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.

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