By Lisle I.G.
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Extra info for Equivalence Transformations for Classes of Differential Equations
Here the ﬁrst half of the decoupled system speciﬁes certain ‘arbitrary elements’ such as a = D(u) above, which are then inserted into the second half of the system to give the class C. g. g. ’s y¨ = ω(x, y, y)). ’s Let A be a system of σ of order κ in and ν µ diﬀerential equations independent variables w = (w1 , w2 , . . , wν ) dependent variables a = (a1 , a2 , . . , aµ ). described by the function g = (g 1 , g 2 , . . , g σ ) as g(w, a, a, . . , aκ ) = 0. 6) 1 Let E be the system of s equations f (x, u, u, .
5. A locally solvable class C of diﬀerential equations admits an augmented equivalence transformation τˆ if and only if τˆ leaves invariant (i) the surface A : g = 0 specifying the auxiliary system, and (ii) the surface E ∩ A : f = 0, g = 0. Proof. We have only to show the converse statement. Let τˆ be an augmented equivalence transformation. 19). 4) of A. 9 shows that τˆ leaves the surface A invariant. This establishes (i). 21) be a point on the surface E ∩ A : f = 0, g = 0. Local solvability (ii) of class C ensures there are functions θ(x), φ(w) such that Pˆ lies ˆ , φ) of a solution θ of equation E(φ) where φ solves A.
Equivalence transformations the arbitrary elements are independent of certain components of (x, u). However, both these papers rely on Ovsiannikov’s deﬁnition of equivalence transformation, so their formulation will fail for systems which are not locally solvable. 18) be a symmetry of the auxiliary system A. , the w component of τˆ is independent of a. This makes construction of such symmetries simpler than ﬁnding the full point symmetry group of A. 6. Point equivalence transformations for a locally solvable class C ˆ acting on augmented space (x, u, a), and of diﬀerential equations form a group Q a group Q acting on the base space (x, u).
Equivalence Transformations for Classes of Differential Equations by Lisle I.G.