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Symmetry Reduction of Nonlinear Differential Equations

In recent years the symmetry method is often used for reduction of partial differential equations to the equations with fewer number of independent variables and thus for construction of exact solutions for different mathematical physics problems. To construct a corresponding ansatz generators of classical Lie point transformations are used as well as operators of conditional symmetry. In this connection the application of combination of conditional and generalized symmetry is fruitful as was shown in [1,2] on the examples of evolution equation in two-dimensional case (see also [3]). In [4] Svirshchevskii proposed the symmetry reduction method based on the invariance of linear ordinary differential equations (see also [5]). It is the symmetry explanation of “nonlinear” separation of variables [6] for the evolution-type equations. Here we propose an approach applicable for symmetry reduction of partial differential equations which are not restricted to evolution type ones. It can be used in multi-dimensional case. This approach is the generalization of the method introduced in [4].