arXiv subjects
Publications and source records attributed to Arthemy V. Kiselev.
We define a jet-space analog of the BV-Laplacian, avoiding delta-functions and infinite constants; instead we show that the main properties of the BV-Laplacian and its relation to the Schouten bracket originate from the underlying jet-space geometry.
The Schouten bracket (or antibracket) plays a central role in the Poisson formalism and the Batalin-Vilkovisky quantization of gauge systems. There are several (in)equivalent ways to realize this concept on jet spaces. In this paper, we compare the definitions, examining in what ways they agree or disagree and how they relate to the case of usual manifolds.
We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.
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.
We formulate a simple and convenient criterion under which skew-adjoint Z_2-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.
We define Lie algebroids over infinite jet spaces and establish their equivalent representation through homological evolutionary vector fields.
We generalize the notion of a Lie algebroid over infinite jet bundle by replacing the variational anchor with an N-tuple of differential operators whose images in the Lie algebra of evolutionary vector fields of the jet space are subject to collective commutation closure. The linear space of such operators becomes an algebra with bi-differential structural constants, of which we study the canonical structure. In particular, we show that these constants incorporate bi-differential analogues of Christoffel symbols.
The generators and commutation relations are calculated explicitly for higher symmetry algebras of a class of hyperbolic Euler-Lagrange systems of Liouville type (in particular, for 2D Toda chains associated with semi-simple complex Lie algebras). Mathematics Subject Classification (2000): 35Q53, 37K05, 37K30.
For the 3-component dispersionless Boussinesq-type system, we construct two compatible nontrivial finite deformations for the Lie algebra structure in the symmetry algebra.
For the root system of each complex semi-simple Lie algebra of rank two, and for the associated 2D Toda chain $E=\{u_{xy}=\exp(K u)\}$, we calculate the two first integrals of the characteristic equation $D_y(w)=0$ on $E$. Using the integrals, we reconstruct and make coordinate-independent the $(2\times 2)$-matrix operators $\square$ in total derivatives that factor symmetries of the chains. Writing other factorizations that involve the operators $\square$, we obtain pairs of compatible Hamiltonian operators that produce KdV-type hierarchies of symmetries for $\cE$. Having thus reduced the problem to the Hamiltonian case, we calculate the Lie-type brackets, transferred from the commutators of the symmetries in the images of the operators $\square$ onto their domains. With all this, we describe the generators and derive all the commutation relations in the symmetry algebras of the 2D Toda chains, which serve here as an illustration for a much more general algebraic and geometric set-up.
Let F be a smooth real manifold with a linear connection in the tangent bundle. How can we extend the coefficients of the connection to bi-differential operators that incorporate the original structure at zero order? Take a constant mapping of F to a point. Suppose that the point belongs to another manifold M^n. Consider a fibre bundle over M^n with fibres F. Denote the projection of the total space to M^n by π. Construct the space of infinite jets $J^\infty(π)$ of sections for this bundle. The $C^\infty(J^\infty(π))$-module of π-vertical evolutionary fields on the jet space is the analogue of the $C^\infty(F)$-module of vector fields on F, which now `feels' the presence of the base M^n. We define and classify the matrix linear differential operators whose images in the module of evolutionary fields are closed w.r.t. commutation. We define the compatibility for such operators with collective commutation closure. We prove that, under reparameterizations in their domain, the bi-differential structural constants, which describe the commutation of the fields in the sum of their images, obey an analogue of the transformation rule for Christoffel symbols. We show that the new structure is the connection in the pair of Lie algebras (the domain and the sum of images of the operators) related by homomorphisms (points in the linear space spanned by the operators). The application of this theory to geometry of PDE solves a long-standing problem. We give a complete description of symmetry algebras for 2D Toda chains associated with semi-simple Lie algebras: we obtain all symmetry generators and calculate all commutation relations. As a by-product, we expose a geometric scheme for derivation of bi-Hamiltonian KdV-type hierarchies related to root systems.
We prove that Mathieu's N=2 supersymmetric Korteweg-de Vries equations with a=1 or a=4 admit Hirota's n-supersoliton solutions, whose nonlinear interaction does not produce any phase shifts. For initial profiles that can not be distinguished from a one-soliton solution at times t<<0, we reveal the possibility of a spontaneous decay and, within a finite time, transformation into a solitonic solution with a different wave number. This paradoxal effect is realized by the completely integrable N=2 super-KdV systems, whenever the initial soliton is loaded with other solitons that are virtual and become manifest through the tau-function as the time grows. Key words and phrases: Hirota's solitons, N=2 supersymmetric KdV, Krasil'shchik-Kersten system, phase shift, spontaneous decay.
An algebraic definition of Gardner's deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. An exactly solvable two-component extension of the Liouville equation is found.
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
Using the algebraic method of Gardner's deformations for completely integrable systems, we construct the recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these systems, we obtain new integrable equations adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.
We construct new integrable coupled systems of N=1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equation, and unusual properties of a recursion operator for its Hamiltonian symmetries are described. Also, we construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.
In this paper, we investigate the algebraic and geometric properties of the hyperbolic Toda equations $u_{xy}=\exp(Ku)$ associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogs to the potential modified Korteweg-de Vries equation $u_t=u_{xxx}+u_x^3$ is constructed, and its relation with the hierarchy for the Korteweg-de Vries equation $T_t=T_{xxx}+TT_x$ is established. Group-theoretic structures for the dispersionless (2+1)-dimensional Toda equation $u_{xy}=\exp(-u_{zz})$ are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems $Ψ_t = iΨ_{xx} + i f(|Ψ|) Ψ$ (multi-soliton complexes) are described.