Wahlquist-Estabrook type coverings over the heat equation (Q913109)

The Wahlquist-Estabrook coverings and their symmetries are studied for the nonlinear heat equation \(H_ t=(B(H)H_ x)_ x\) both in the nonlinear \((B(H)\neq 1)\) and the linear \((B(H)=1)\) case. It is proved that in the nonlinear case the group of local symmetries for such an equation may be extended with the use of some nonlocal symmetries for a sufficiently wide class of functions \(B(H)\). All the pairwise nonequivalent coverings of the type are calculated and their group behavior is analyzed. In the linear case any p-dimensional irreducible Wahlquist-Estabrook covering is shown to be isomorphic to the infinite extension of the heat equation \(H_ t=H_{xx}\). As a consequence, the group of local symmetries for this equation is found to be isomorphic to the group of its nonlocal symmetries corresponding to each such a covering.