Equivalence classes of related evolution equations and Lie symmetries

Abstract

This is an extension of earlier work (SIAM J. Math. Anal. vol.16, p.221, 1985) by the authors which gives a correspondence between Lie symmetry operators (with non-trivial time dependence) for a given evolution equation and those evolution equations related to the given one by a change of independent and dependent coordinates. They work out the correspondence between symmetries of a system of evolution equations nu t=K(y, v) and those systems us=J(x, u) related to it by a change of coordinates t=T(s, x, u), y=Y(s, x, u), v=V(s, x, u) and show (extending ideas of Humi (1986) and Rosencrans (1976)) how to determine the related equations directly from the symmetry operators without solving a system of differential equations. In general there are multiple evolution equations associated with a given symmetry; for the case of scalar evolution equations they compute explicitly the structure of each equivalence class.