Lie points symmetries of differential

Lie point symmetries of differential–difference equations
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: levi@roma3.infn.it 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: wintern@crm.umontreal.ca 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

1

Introduction

Two different but equivalent infinitesimal formalisms exist for calculating Lie point symmetriesof differential equations [18]. One is that of ‘standard‘ vector fields
p q

ˆ X=
i=1

ξi (x, u)∂xi +
α=1

φα (x, u)∂uα

(1)

1

acting on the independent variables xi and dependent ones uα in the considered differential equation. The other is that of the evolutionary vector fields
q

ˆ XE =
α=1

Qα (x, u, ux )∂uα ,

(2)

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
p

(3)

ˆ ˆ prX E E|E=0 = (prX −
i=1

ξi Dxi )E|E=0 = 0.

(4)

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
p

Qα = φα −
i=1

ξi uα,xi .

(5)

For all details we refer to e.g. P. Olver‘s textbook [18]. 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 =
k,l

ˆ Xm+k,n+l ,

(10)

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...