Abstract. For a system Y¢ of partial differential equations, the notion of a covering ~o ~ °2/~ is introduced where Y¢~ is infinite prolongation of ~'. Then nonlocal symmetries of ~ are defined as transformations of ~ which conserve the underlying contact structure. It turns out that generating functions of nonloeal symmetries are integro-differential-type operators.
AMS (MOS) subject classifications (1980
O. IntroductionIn [1] local symmetries and local conservation laws were discussed, i.e., such that are defined by differential operators. For example, any higher infinitesimal symmetry of an equation Y¢ c Jk(rr) is determined by its generating function ([ 1 ], Section 3.5), the latter being, in general, a nonlinear differential operator. As we saw, the local point of view effects in consistent and self-contained theory. Nevertheless, there are certain experimental facts, as well as purely theoretical considerations, which indicate its limitations. First of all it could be seen that the number of local symmetries and conservation laws in cases is just too small. Thus, an evolution equation u, --f(u, ux) + u~,,,, the form of which is quite similar to the Korteweg-de "Cries equation, as a rule has no local symmetries other than translations. The KdV equation U t ---UU x + Ux,~ itself is not completely integrable in the framework of the local theory.A natural geometric generalization of the local theory consists of constructing such extensions of objects like ~¢~ functions on which could be interpreted as some kind of generalized differential operators (e.g., as integro-differential operators). We call symmetries and conservation laws determined by such functions nonlocal.Nonlocal symmetries of special types were considered in a number of recent publications (see, for example, [2][3][4][5][6][7]
An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and ℓ * -covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples (the KdV equation and the Boussinesq system) and reconstruct for them the known Hamiltonian structures by our methods. For the coupled KdV-mKdV system, a new Hamiltonian structure is found and its uniqueness (in the class of polynomial (x, t)-independent structures) is proved. We also construct a nonlocal Hamiltonian structure for this system and prove its compatibility with the local one.
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