D. Levi Dipartimento di Ingegneria Elettronica, Universit` degli Studi Roma Tre and Sezione INFN, Roma Tre, a Via della Vasca Navale 84, 00146 Roma, Italy E-mail: firstname.lastname@example.org P. Winternitz Centre de recherches math´matiques et e D´partement de math´matiques et statistique, Universit´ de Montr´al e e e e C.P. 6128, succ. Centre–ville,H3C 3J7, Montr´al (Qu´bec), Canada e e E-mail: email@example.com R.I. Yamilov Ufa Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russian Federation E-mail: RvlYamilov@matem.anrb.ru April 30, 2010
Abstract We present an algorithm for determining the Lie point symmetries of differential equations on ﬁxed non transforming lattices, i.e. equationsinvolving both continuous and discrete independent variables. The symmetries of a speciﬁc integrable discretization of the Krichever-Novikov equation, the Toda lattice and Toda ﬁeld theory are presented as examples of the general method.
arXiv:1004.5311v1 [math-ph] 29 Apr 2010
Two diﬀerent but equivalent inﬁnitesimal formalisms exist for calculating Lie point symmetriesof diﬀerential equations . One is that of ‘standard‘ vector ﬁelds
ξi (x, u)∂xi +
φα (x, u)∂uα
acting on the independent variables xi and dependent ones uα in the considered diﬀerential equation. The other is that of the evolutionary vector ﬁelds
ˆ XE =
Qα (x, u, ux )∂uα ,
acting only on the dependent variables. Theequivalence of the two formalisms is due to the fact that the total derivatives Dxi are themselves ‘generalized‘ symmetry operators, so for any diﬀerential equation E(xi , uα , uα,xi , · · · ) = 0 we have
ˆ ˆ prX E E|E=0 = (prX −
ξi Dxi )E|E=0 = 0.
ˆ ˆ ˆ ˆ Here prX E and prX are the appropriate diﬀerential prolongations of X E and X. Relation (4) implies that for pointtransformations we have
Qα = φα −
ξi uα,xi .
For all details we refer to e.g. P. Olver‘s textbook . An advantage of the standard formalism is its direct relation to the group transformations obtained by integrating the equations d˜i x d˜α u ˜ ˜ ˜ ˜ = ξi (x, u), = φα (x, u), dλ dλ xi |λ=0 = xi , ˜ uα |λ=0 = uα , i = 1, · · · , p, α = 1, · · · , q. ˜ (6)
One advantage of theevolutionary formalism is its direct relation to the existence of ﬂows commuting with the studied equation (3) d˜α u = Qα , dλ (7)
where Qα is the characteristic of the vector ﬁeld as in (5). Another advantage is that the evolutionary formalism can easily be adapted to the case of higher symmetries. Let us now consider a purely discrete equation, i.e. a diﬀerence equation. We restrict to thecase of one scalar function deﬁned on a two dimensional lattice umn . We shall view u as a continuous variable, introduce two further continuous variables x and t and consider (x, t, u) as being evaluated, or sampled at discrete points on 2
a lattice labelled by the indices m, n. We shall write (xmn , tmn , umn ) for values at these points. A diﬀerence system will consist of relations Ea(xm+k,n+l , tm+k,n+l , um+k,n+l ) = 0, a = 1, · · · , A km ≤ k ≤ kM , lm ≤ l ≤ lM , (8) between the variables x, t, and u evaluated at a ﬁnite number of points on a lattice. A Lie point symmetry of the system (8) will be generated by a vector ﬁeld of the form ˆ Xmn = ξ(xmn , tmn , umn )∂xmn + τ (xmn , tmn , umn )∂tmn + φ(xmn , tmn , umn )∂umn . (9)
We see that the vector ﬁeld (9) for diﬀerenceequations has the same form as (1) for diﬀerential ones. Its prolongation is however diﬀerent, namely ˆ prX =
ˆ Xm+k,n+l ,
where the sum is over all points ﬁguring in the system (8). In the continuous limit the system (8) goes into a partial diﬀerential equation, eq. (10) goes into the usual prolongation of a standard vector ﬁeld (i.e. it also acts on functions of derivatives). For...