Conditional symmetry and new representations of the Galilean algebra for nonlinear parabolic equations (Q1344975)

The method of conditional symmetry for partial differential equations obtains group invariant solutions of sets of differential equations which do not admit the group as a Lie symmetry group. This is often expressed by demanding that a system extended by additional conditions admits the symmetry. Evenso the operators of conditional symmetry do not generally form an algebra until supplemented by the Lie symmetries of the system. In the current paper, the authors give a constructive method for finding an algebra of conditional invariance for equations of the form \((\partial_t u_a \partial_a) h(x_1,\dots, x_n; u)- \Delta u= F(u)\) under the requirement that the conditional and Lie symmetries together form a basis for the Galilean algebra. They do this by identifying those equations whose Lie symmetries can be extende to the Galilean algebra, constructing a representation for which the equation is an invariant and then finding the supplementary conditions under which the Galilean algebra becomes the algebra of both conditional and Lie symmetries.