Recursion operators,conservation laws,and integrability conditions for difference equations

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.     

A partial Horn recursion in the cohomology of flag varieties

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.     

Explicitly recovering asymptotics of short range potentials

Inverse scattering for real-valued short range potentials on Rⁿ is studied. It is shown that the scattering matrix at fixed energy is the pull-back of a pseudo-differential operator and that the symbol of the operator determines the asymptotics of the potential. This is done by an explicit construction of the Poisson operator for the scattering problem as an oscillatory integral     

On Integrability of the Camassa–Holm Equation and Its Invariants

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar form, and in the form of 2×2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal). This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.     

Chern Forms on Mapping Spaces

We state a Chern–Weil type theorem which is a generalization of a Chern–Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1, 19, 20]. We specialize to the case to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group . In this last example, we precise Freed's construction [5] on the loop group: The cohomology of the first Chern form of any holomorphic connection in the class considered is given by the Kähler form.     

Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using Potentials

New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.     

Dirichlet to Neumann operator on differential forms

We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN map. The Hilbert transform connects boundary traces of conjugate co-closed forms.     

Bi-Hamiltonian systems of natural form

We propose a new method for constructing integrable systems of natural form. In this method, integrals of motion are solutions of an overdetermined system of algebraic and partial differential equations obtained from the compatibility condition for Poisson tensors polynomial in the momenta and from the condition that the bi-Lagrangian distribution corresponding to the integrals of motion is invariant under the action of the recursion operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 161–182, November, 2006.     

Traces on finite \mathcal{W} -algebras

We compute the space of Poisson traces on a classical W \mathcal{W} -algebra, i.e., linear functionals invariant under Hamiltonian derivations. Modulo any central character, this space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology of the corresponding quantum W \mathcal{W} -algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology, which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the centrally reduced classical W \mathcal{W} -algebra.     

Modified Differentials and Basic Cohomology for Riemannian Foliations

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincaré duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basic Euler characteristic and basic signature.     

Conservation laws for a class of third order evolutionary differential systems

Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights , , . The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.

     

Frobenius Poisson algebras

This paper is devoted to study Frobenius Poisson algebras. We introduce pseudo-unimodular Poisson algebras by generalizing unimodular Poisson algebras, and investigate Batalin-Vilkovisky structures on their cohomology algebras. For any Frobenius Poisson algebra, all Eatalin-Vilkovisky opera tors on its Poisson cochain complex are described explicitly. It is proved that there exists a Batalin-Vilkovisky operator on its cohomology algebra which is induced from a Batalin-Vilkovisky operator on the Poisson cochain complex, if and only if the Poisson st rue ture is pseudo-unimodular. The relation bet ween modular derivations of polynomial Poisson algebras and those of their truncated Poisson algebras is also described in some cases.     

An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models. The text was submitted by the authors in English.     

The variational Poisson cohomology

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.     

Inversion of integral of B-potential type with density from Φ<Subscript>γ</Subscript>

In [3], the inversion of an integral operator of potential type with constant characteristic generated by the many-dimensional generalized shift was obtained. In this paper, the author obtains a generalization of the results from [3] to the case of a shift of mixed type, i.e., on a part of the variable generalized shifts of integral nature adopted to deal with the Bessel singular differential operator act, whereas on the other part, the ordinary shift act. Also, it should be noted that in contrast to [3], the integral of B-potential type with homogeneous characteristic is considered in this paper. This generalization is attained by introducing general hypersingular integrals of the general form [8]. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

On the symmetry classification and conservation laws for quasilinear differential equations of second order

We obtain a sufficient condition for the absence of tangent transformations admitted by quasilinear differential equations of second order and a sufficient condition for the linear autonomy of the operators of the Lie group of transformations admitted by weakly nonlinear differential equations of second order. We prove a theorem concerning the structure of conservation laws of first order for weakly nonlinear differential equations of second order. We carry out the classification by first-order conservation laws for linear differential equations of second order with two independent variables.     

A basis of hierarchy of generalized symmetries and their conservation laws for the (3+1)-dimensional diffusion equation

We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of several orders. We construct a basis for conservation laws (with respect to the group admitted by the system of differential equations) and hence generate infinitely many conservation laws in each equivalence class.     

Poisson cohomology of broken Lefschetz fibrations

We compute the formal Poisson cohomology of a broken Lefschetz fibration by calculating it at fold and Lefschetz singularities. Near a fold singularity the computation reduces to that for a point singularity in 3 dimensions. For the Poisson cohomology around singular points we adapt techniques developed for the Sklyanin algebra. As a side result, we give compact formulas for the Poisson coboundary operator of an arbitrary Jacobian Poisson structure in 4 dimensions.     

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E→M. We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin-Kravchenko-Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M=S¹, we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S¹-pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111-127]. When dim(M)>1, we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.     

Cheeger–Chern–Simons Theory and Differential String Classes

We construct new concrete examples of relative differential characters, which we call Cheeger–Chern–Simons characters. They combine the well-known Cheeger–Simons characters with Chern–Simons forms. In the same way as Cheeger–Simons characters generalize Chern–Simons invariants of oriented closed manifolds, Cheeger–Chern–Simons characters generalize Chern–Simons invariants of oriented manifolds with boundary. We study the differential cohomology of compact Lie groups G and their classifying spaces BG. We show that the even degree differential cohomology of BG canonically splits into Cheeger–Simons characters and topologically trivial characters. We discuss the transgression in principal G-bundles and in the universal bundle. We introduce two methods to lift the universal transgression to a differential cohomology valued map. They generalize the Dijkgraaf–Witten correspondence between 3-dimensional Chern–Simons theories and Wess–Zumino–Witten terms to fully extended higher-order Chern–Simons theories. Using these lifts, we also prove two versions of a differential Hopf theorem. Using Cheeger–Chern–Simons characters and transgression, we introduce the notion of differential trivializations of universal characteristic classes. It generalizes well-established notions of differential String classes to arbitrary degree. Specializing to the class \({\frac{1}{2} p_1 \in H^4(B{\rm Spin}_n;\mathbb{Z})}\), we recover isomorphism classes of geometric string structures on Spin _n -bundles with connection and the corresponding spin structures on the free loop space. The Cheeger–Chern–Simons character associated with the class \({\frac{1}{2} p_1}\) together with its transgressions to loop space and higher mapping spaces defines a Chern–Simons theory, extended down to points. Differential String classes provide trivializations of this extended Chern–Simons theory. This setting immediately generalizes to arbitrary degree: for any universal characteristic class of principal G-bundles, we have an associated Cheeger–Chern–Simons character and extended Chern–Simons theory. Differential trivialization classes yield trivializations of this extended Chern–Simons theory.