Type II hidden symmetries for the homogeneous heat equation in some general classes of Riemannian spaces

Abstract

We study the reduction of the heat equation in Riemannian spaces which admit a gradient Killing vector, a gradient homothetic vector and in Petrov Type D,N,II and Type III space-times. In each reduction we identify the source of the Type II hidden symmetries. More specifically we find that a) If we reduce the heat equation by the symmetries generated by the gradient KV the reduced equation is a linear heat equation in the nondecomposable space. b) If we reduce the heat equation via the symmetries generated by the gradient HV the reduced equation is a Laplace equation for an appropriate metric. In this case the Type II hidden symmetries are generated from the proper CKVs. c) In the Petrov spacetimes the reduction of the heat equation by the symmetry generated from the nongradient HV gives PDEs which inherit the Lie symmetries hence no Type II hidden symmetries appear. We apply the general results to cases in which the initial metric is specified. We consider the case that the irreducible part of the decomposed space is a space of constant nonvanishing curvature and the case of the spatially flat Friedmann-Robertson-Walker space time used in Cosmology. In each case we give explicitly the Type II hidden symmetries provided they exist.

Lie symmetries for systems of evolution equations

“…There are various differences with the quasilinear system we studied in [24] the main one being that in [24] the CKVs of the metric g ab generate symmetries of the system. That means that in the case of static solutions of the system (1) the new "Type-II" hidden symmetries [37,38] are generated by the proper CKVs of the metric g ij , a result which generalizes the corresponding result of the one-dimensional case [39].…”