On (2)-relative cohomology of the Lie algebra of vector fields and differential operators

Abstract

Let Vect(?) be the Lie algebra of smooth vector fields on ?. The space of symbols Pol(T*?) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(?)-module that becomes trivial once the action is restricted to (2) ? Vect(?). The deformations of Pol(T*?), which become trivial once the action is restricted to (2) and such that the Vect(?)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D _λ,μ = Homdiff(F _λ, F _μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).     

Differential Operators,Symmetries and the Inverse Problem for Second-Order Differential Equations

Abstract

With each second-order differential equation Z in the evolution space J ¹(M _n+1) we associate, using the natural f(3, ?1)-structure and the f(3, 1)-structure K, a group of automorphisms of the tangent bundle T (J ¹(M _n+1)), with isomorphic to a dihedral group of order 8. Using the elements of and the Lie derivative, we introduce new differential operators on J ¹(M _n+1) and new types of symmetries of Z. We analyze the relations between the operators and the “dynamical” connection induced by Z. Moreover, we analyze the relations between the various symmetries, also in connection with the inverse problem for Z. Both the approach based on the Poincaré–Cartan two forms and the one relying on the introduction of the so-called metrics compatible with Z are explicitly worked out.     

A note on Bernoulli polynomials and solitons

Abstract

The dependence on time of the moments of the one-soliton KdV solutions is given by Bernoulli polynomials. Namely, we prove the formula expressing the moments of the one-soliton function sech²(x-t) in terms of the Bernoulli polynomials Bn(x). We also provide an alternative short proof to the Grosset-Veselov formula connecting the one-soliton to the Bernoulli numbers (D?=?d/dx) published recently in this journal.     

Interrelations of discrete Painlevé equations through limiting procedures

We study the discrete Painlevé equations associated to the affine Weyl group which can be obtained by the implementation of a special limits of -associated equations. This study is motivated by the existence of two -associated discrete both having a double ternary dependence in their coefficients and which have not been related before. We show here that two equations correspond to two different limits of a -associated discrete Painlevé equation. Applying the same limiting procedures to other -associated equations we obtained several -related equations most of which have not been previously derived.