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B. G. Konopelchenko

Publications and source records attributed to B. G. Konopelchenko.

At least 37 records · Page 2Linked to original sources
arXiv2013

Cohomological, Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of Sato Grassmannian

Cohomological and Poisson structures associated with the special tautological subbundles $TB_{W_{1,2,\dots,n}}$ for the Birkhoff strata of Sato Grassmannian are considered. It is shown that the tangent bundles of $TB_{W_{1,2,\dots,n}}$ are isomorphic to the linear spaces of $2-$coboundaries with vanishing Harrison's cohomology modules. Special class of 2-coboundaries is provided by the systems of integrable quasilinear PDEs. For the big cell it is the dKP hierarchy. It is demonstrated also that the families of ideals for algebraic varieties in $TB_{W_{1,2,\dots,n}}$ can be viewed as the Poisson ideals. This observation establishes a connection between families of algebraic curves in $TB_{W_{\hat{S}}}$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of hydrodynamical type systems like dKP hierarchy. Interrelation between cohomological and Poisson structures is noted.

math-ph↗
arXiv2013

Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the "inverse" hierarchy and equations associated with the real-analytic Eisenstein series $E(β,\bar{β};1/2)$among them. Specific bi-Hamiltonian structure of these equations is also discussed.

math-ph↗
arXiv2013

Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $Ω_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deformations of (N-1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as Liouville equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3), dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4) and others. Gauge invariant part of the forms $Ω_{N-1}$ provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2,N+1) are defined via the requirement of the closedness of the dual forms $Ω_{N-1}^{\star}$. It is shown that at N=3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form $Ω_{2}$, coincides with the Maxwell equations for orthogonal electric and magnetic fields.

nlin.SI↗
arXiv2012

Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter

Quasiclassical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation the governing equations are given by elliptic system of quasi-linear PDEs of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painleve' I equation.

nlin.SI↗
arXiv2011

Gradient catastrophe and flutter in vortex filament dynamics

Gradient catastrophe and flutter instability in the motion of vortex filament within the localized induction approximation are analyzed. It is shown that the origin if this phenomenon is in the gradient catastrophe for the dispersionless Da Rios system which describes motion of filament with slow varying curvature and torsion. Geometrically this catastrophe manifests as a rapid oscillation of a filament curve in a point that resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painlevé-I equation.

nlin.SI↗
arXiv2011

Algebraic varieties in Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Harrison cohomology and integrable systems

Local properties of families of algebraic subsets $W_g$ in Birkhoff strata $Σ_{2g}$ of Gr$^{(2)}$ containing hyperelliptic curves of genus $g$ are studied. It is shown that the tangent spaces $T_g$ for $W_g$ are isomorphic to linear spaces of 2-coboundaries. Particular subsets in $W_g$ are described by the intergrable dispersionless coupled KdV systems of hydrodynamical type defining a special class of 2-cocycles and 2-coboundaries in $T_g$. It is demonstrated that the blows-ups of such 2-cocycles and 2-coboundaries and gradient catastrophes for associated integrable systems are interrelated.

math-ph↗
arXiv2011

Birkhoff strata of the Grassmannian Gr$\mathrm{^{(2)}}$: Algebraic curves

Algebraic varieties and curves arising in Birkhoff strata of the Sato Grassmannian Gr${^{(2)}}$ are studied. It is shown that the big cell $Σ_0$ contains the tower of families of the normal rational curves of all odd orders. Strata $Σ_{2n}$, $n=1,2,3,...$ contain hyperelliptic curves of genus $n$ and their coordinate rings. Strata $Σ_{2n+1}$, $n=0,1,2,3,...$ contain $(2m+1,2m+3)-$plane curves for $n=2m,2m-1$ $(m \geq 2)$ and $(3,4)$ and $(3,5)$ curves in $Σ_3$, $Σ_5$ respectively. Curves in the strata $Σ_{2n+1}$ have zero genus.

math-ph↗
arXiv2010

Hamiltonian motions of plane curves and formation of singularities and bubbles

A class of Hamiltonian deformations of plane curves is defined and studied. Hamiltonian deformations of conics and cubics are considered as illustrative examples. These deformations are described by systems of hydrodynamical type equations. It is shown that solutions of these systems describe processes of formation of singularities (cusps, nodes), bubbles, and change of genus of a curve.

math-ph↗
arXiv2009

Discrete integrable systems and deformations of associative algebras

Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformation Driving Algebra and governed by the central system of equations. It is demonstrated that many discrete equations like discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BKP equations, discrete Hirota-Miwa equations for KP and BKP hierarchies are particular realizations of the central system. An interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful.An interpretation of the Menelaus relation (discrete Schwarzian KP equation), discrete Hirota-Miwa equation for KP hierarchy, consistency around the cube as the associativity conditions and the concept of gauge equivalence, for instance, between the Menelaus and KP configurations are particular examples.

nlin.SI↗
arXiv2009

Coisotropic deformations of algebraic varieties and integrable systems

Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space algebraic curves are governed, in particular, by the dKP, WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems. Particular attention is paid to the study of two- and three-dimensional deformations of elliptic curves. Problem of an appropriate choice of Poisson structure is discussed.

nlin.SI↗
arXiv2009

On the deformation theory of structure constants for associative algebras

Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.

math.RA↗
arXiv2008

Continuous-discrete integrable equations and Darboux transformations as deformations of associative algebras

Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed scheme of deformations.

nlin.SI↗
arXiv2008

Discrete, q-difference deformations of associative algebras and integrable systems

Discrete and q-difference deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by a central system of discrete or q-difference equations which in particular cases represent discrete and q-difference versions of the oriented associativity equation. It is demonstrated also that the celebrated Hirota-Miwa bilinear equation for the AKP and BKP hierarchies describes discrete deformations of certain finite-dimensional algebras.

nlin.SI↗
arXiv2008

Quantum deformations of associative algebras and integrable systems

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.

nlin.SI↗
arXiv2007

Quasiclassical generalized Weierstrass representation and dispersionless DS equation

Quasiclassical generalized Weierstrass representation (GWR) for highly corrugated surfaces with slow modulation in the four-dimensional Euclidean space is proposed. Integrable deformations of such surfaces are described by the dispersionless Davey-Stewartson hierarchy. Quasiclassical GWRs for other four-dimensional spaces and dispersionless DS system are discussed too.

nlin.SI↗
arXiv2006

Dispersionless integrable equations as coisotropic deformations. Extensions and reductions

Interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systInterpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level. ems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level.

nlin.SI↗