Infinite Jet Space Research Articles

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

A Foundational Approach to the Lie Theory for Fractional Order Partial Differential Equations

AbstractWe provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable.

Classification of rational functions in symplectic and metric spaces

We consider symplectic and orthogonal classifications of rational functions of many variables. The main idea of these classifications consists in the use of methods of the differential geometry and the geometric theory of jet spaces. Namely, we consider group actions on an infinite jet space (rather than on functions), which allows us to find fields of differential invariants of these groups. Finally, we prove that dependencies between basic differential invariants and their derivatives completely determine the orbit of the corresponding function.

Homological evolutionary vector fields in Korteweg–de Vries, Liouville, Maxwell, and several other models

We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg–de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler–Lagrange gauge theories (such as the Yang–Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.

Variational Lie algebroids and homological evolutionary vector fields

We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields.

A convenient criterion under which ℤ2-graded operators are Hamiltonian

We formulate a simple and convenient criterion under which skew-adjoint ℤ2-graded total differential operators are Hamiltonian, provided that their images are closed under commutation in the Lie algebras of evolutionary vector fields on the infinite jet spaces for vector bundles over smooth manifolds.

Rationality of the Poincaré series in Arnold's local problems of analysis

For any smooth action of a Lie pseudo-group we construct a domain (in the corresponding infinite jet space) consisting of finitely many open sets (atoms) such that all points in each atom have the same rational Poincaré series. We also prove that these series can be calculated algorithmically.

Uniqueness results for operators in the variational sequence

We prove that the most interesting operators in the Euler–Lagrange complex from the variational bicomplex in infinite order jet spaces are determined up to multiplicative constant by the naturality requirement, provided the fibres of fibred manifold

Iterated differential forms: The Λ k − 1 -spectral sequence on infinite jets

In the preceding note math.DG/0610917 the $\Lambda_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence for infinitely prolonged PDE's will be given in the subsequent note.

Iterated differential forms IV: The ℰ-spectral sequence

For the multiple differential algebra of iterated differential forms (see math.DG/0605113 and math.DG/0609287) on a diffiety (O,C) an analogue of C-spectral sequence is constructed. The first term of it is naturally interpreted as the algebra of secondary iterated differential forms on (O,C). This allows to develop secondary tensor analysis on generic diffieties, some simplest elements of which are sketched here. The presented here general theory will be specified to infinite jet spaces and infinitely prolonged PDEs in subsequent notes.

Domains in infinite jet spaces: -spectral sequences

Domains in infinite jets present the simplest class of diffieties with boundary. In this note some basic elements of geometry of these domains are introduced and an analogue of the C-spectral sequence in this context is studied. This, in particular, allows cohomological interpretation and analysis of initial data, boundary conditions, etc, for general partial differential equations and of transversality conditions in calculus of variations. This kind applications and extensions to arbitrary diffieties will be considered in subsequent publications.

On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

AbstractWe study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one‐dimensional systems of hyperbolic PDEsvt+ [ϕ(v)]x= 0. Under certain genericity assumptions it is proved that anybi‐Hamiltonianperturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space dependingrationallyon the derivatives. The main tool is in constructing the so‐called quasi‐Miura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi‐Hamiltonian structure of hydrodynamic type (the quasi‐triviality theorem). We also describe, following [35], the invariants of such bi‐Hamiltonian structures with respect to the group of Miura‐type transformations dependingpolynomiallyon the derivatives. © 2005 Wiley Periodicals, Inc.

On symmetries and cohomological invariants of equations possessing flat representations

We study the equation E fc of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1–1 correspondence with morphisms ϕ : E→ E fc , E and E fc being treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang–Mills equations and their reductions are of this type. With each flat representation ϕ we associate a complex C ϕ of vector-valued differential forms such that H 1( C ϕ ) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Bäcklund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c S of C ϕ . Symmetries with exact c S form a subalgebra reflecting some geometric properties of E and ϕ. We show that the complex corresponding to E fc itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of E fc are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.

On complexes related with calculus of variations

We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.

Iterated BRST Cohomology

The iterated BRST cohomology is studied by computing cohomology of the variational complex on the infinite order jet space of a smooth fibre bundle. This computation also provides a solution of the global inverse problem of the calculus of variations in Lagrangian field theory.

Symmetry reduction of variational bicomplexes and the principle of symmetric criticality

Consider a system of differential equations Δ = 0 which is invariant under a Lie group G of point transformations acting on the space E of independent and dependent variables. By a method due to Lie, the G invariant solutions of these differential equations are found by solving a reduced system of differential equations [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] = 0 on the space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] of invariants of G . In this paper we explore the relationship between the G invariant conservation laws and variational principles for the system of equations Δ = 0 and the conservation laws and variational principles for the reduced equations [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] = 0. This problem translates into one of constructing a certain cochain map [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /] between the G invariant variational bicomplex for the infinite jet space on E and the free variational bicomplex for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /]. We prove that such a cochain map exists locally if and only if the relative Lie algebra cohomology condition H q (Γ, Γ 0 ( e 0 )) ≠ 0 is satisfied, where q is the orbit dimension of G , Γ the Lie algebra of vector fields on E which generate the infinitesimal action of G , and Γ 0 ( e 0 ) the linear isotropy subalgebra of Γ at e 0 . As a simple consequence we prove that the vanishing of H q (Γ, Γ 0 ( e 0 )) is the only local obstruction to Palais' principle of symmetric criticality.

Classification of local generalized symmetries for the vacuum Einstein equations

A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's “exact set of fields” for the vacuum equations.

An algebraic construction of the generic singularities of Boardman-Thom

In this paper we study some of the functorial properties of the infinite jet space in order to give a coordinate free algebraic definition of the generic singularities of Boardman-Thom. More precisely, suppose thatk is a commutative ring with an identity and suppose that A is a commutative ring with an identity which is ak-algebra. An A-k-Lie algebra L is ak-Lie algebra with ak-Lie algebra map ϕ from L to the algebra ofk-derivations of A to itself such that ford, d′εL anda, a′εA, then $$[ad'],a 'd'] = a(\varphi (d')ad' - a'(\varphi (d')a)d' + aa'[d',\;d'].$$ . There is a universal enveloping algebra for such Lie algebras which we denote by E(L). Denote byL-alg the category of A-algebras B which have L and hence E(L) acting as left operators such that foraεA,dεL, (da)i B=d(a.i B). If F is the forgetful functor fromL-alg to the category of A-algebras, we show that F has a left adjoint J(L, ·) which is the natural algebraic translation of the infinite jet space. In the third section of this paper we construct a theory of singularities for a derivation from a ring to a module and then we apply this construction to J(L, C) where C is an A-algebra. These singularities are subschemas with defining sheaf of ideals given by Fitting invariants of appropriately chosen modules when A and B are polynomial rings over a fieldk and C=A⊗ k B; these are the generic singularities of Boardman-Thom. Finally we show that, under some rather general conditions on the structure of C as an A-algebra, the generic singularities are regular immersions in the sense of Berthelot.