Residual Symmetry

Let Δ = 0 be a differential system of maximal rank and of order n over the space M ⫅ (space of independent variables x = {x ⁱ, i = 1, ..., I}) × (space of dependent variables u = {u _j , j = 1,...,J}), with Δ = {Δ_ν (x, u ⁽ⁿ⁾), ν = 1,...,p}. If G is a local group of transformations acting on M, it follows that G is a symmetry group of Δ if for any element X of the Lie algebra of G [1]:

with functions ϕⁱ and ψ^j, the nth order prolongation of X on the n-jet space associated to M annihilates Δ, on Δ = 0 [1]:

Let H be a subgroup of the symmetry group G acting regularly and transversally on M, then there is a reduced (or residual, or factor) system of equations, denoted Δ /H, over M/H, for which solutions correspond to H-invariant solutions : u = f(x), of Δ = 0, which means that the action of any element of H preserves (locally) the graph {(x, f(x))} [1,2]. The H-invariant solutions are linked to H-invariants on M.

The reduced differential system Δ /H, will in turn be left invariant under...