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


SIGMA 3 (2007), 019, 14 pages      math-ph/0702033      http://dx.doi.org/10.3842/SIGMA.2007.019

Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Valentyn Tychynin a, Olga Petrova b and Olesya Tertyshnyk b
a) Prydniprovs'ka State Academy of Civil Engineering and Architecture, 24a Chernyshevsky Str., Dnipropetrovsk, 49005 Ukraine
b) Dnipropetrovsk National University, 13 Naukovyi Per., Dnipropetrovsk, 49050 Ukraine

Received January 06, 2006, in final form January 17, 2007; Published online February 06, 2007

Abstract
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.

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

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