Searcharxiv⌕ Search

arXiv subjects

B. G. Konopelchenko

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

At least 55 records · Page 3Linked to original sources
arXiv2006

Coisotropic deformations of associative algebras and dispersionless integrable hierarchies

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and the Hirota's tau function, and shows that the dispersionless Hirota's bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham's equations of genus zero due to Krichever.

nlin.SI↗
arXiv2006

On the heavenly equation hierarchy and its reductions

Second heavenly equation hierarchy is considered using the framework of hyper-Kähler hierarchy developed by Takasaki. Generating equations for the hierarchy are introduced, they are used to construct generating equations for reduced hierarchies. General $N$-reductions, logarithmic reduction and rational reduction for one of the Lax-Sato functions are discussed. It is demonstrated that rational reduction is equivalent to the symmetry constraint.

nlin.SI↗
arXiv2005

On the dbar-dressing method applicable to heavenly equation

The $\dbar$-dressing scheme based on local nonlinear vector $\dbar$-problem is developed. It is applicable to multidimensional nonlinear equations for vector fields, and, after Hamiltonian reduction, to heavenly equation. Hamiltonian reduction is described explicitely in terms of the $\dbar$-data. An analogue of Hirota bilinear identity for heavenly equation hierarchy is introduced, $τ$-function for the hierarchy is defined. Addition formulae (generating equations) for the $τ$-function are found. It is demonstrated that $τ$-function for heavenly equation hierarchy is given by the action for $\dbar$-problem evaluated on the solution of this problem.

nlin.SI↗
arXiv2004

On dispersionless BKP hierarchy and its reductions

Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the $τ$-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.

nlin.SI↗
arXiv2003

Conformal geometry of the (discrete) Schwarzian Davey-Stewartson II hierarchy

The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.

nlin.SI↗
arXiv2003

Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type

Symmetry constraints for (2+1)-dimensional dispersionless integrable equations are considered. It is demonstrated that they naturally lead to systems of hydrodynamic type which arise within the reduction method. One also easily obtaines an associated complex curve (Sato function) and corresponding generating equations. Dispersionless KP and 2DTL hierarchy are considered as illustrative examples.

nlin.SI↗
arXiv2003

Nonlinear Beltrami equation and tau-function for dispersionless hierarchies

It is proved that the action for nonlinear Beltrami equation (quasiclassical dbar-problem) evaluated on its solution gives a tau-function for dispersionless KP hierarchy. Infinitesimal transformations of tau-function corresponding to variations of dbar-data are found. Determinant equations for the function generating these transformations are derived. They represent a dispersionless analogue of singular manifold (Schwarzian) KP equations. Dispersionless 2DTL hierarchy is also considered.

nlin.SI↗
arXiv2002

Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves

We discuss the structure of shock singularities of the Burgers-Hopf hierarchy. It is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy. The stratification is associated with the Birkhoff decomposition of the Grassmannian given by the set of linear spaces spanned by the hierarchy. We then construct integrable hierarchy on each stratum and demonstrate that it describes a deformation of a hyperelliptic curve parametrizing the stratum. The hierarchy is called the hidden Burgers-Hopf hierarchy, and we found the Riemann invarint form and the hodograph solution.

nlin.SI↗
arXiv2001

Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy

A remarkable connection between soliton theory and an important and beautiful branch of the theory of graphical statics developed by Maxwell and his contemporaries is revealed. Thus, it is demonstrated that reciprocal triangles which constitute the simplest pair of reciprocal figures representing both a framework and a self-stress encapsulate the integrable discrete BKP equation and its Schwarzian version. The inherent Moebius invariant nature of the Schwarzian BKP equation is then exploited to define reciprocity in an inversive geometric setting. Integrable pairs of lattices of non-trivial combinatorics consisting of reciprocal triangles and their natural generalizations are discussed. Particular reductions of these BKP lattices are related to the integrable discrete versions of Darboux's 2+1-dimensional sine-Gordon equation and the classical Tzitzeica equation of affine geometry. Furthermore, it is shown that octahedral figures and their hexahedral reciprocals as considered by Maxwell likewise give rise to discrete integrable systems and associated integrable lattices.

nlin.SI↗
arXiv2001

Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.

nlin.SI↗
arXiv2001

On the unification of classical and novel integrable surfaces: I. Differential geometry

A novel class of integrable surfaces is recorded. This class of O surfaces is shown to include and generalize classical surfaces such as isothermic, constant mean curvature, minimal, `linear' Weingarten, Guichard and Petot surfaces and surfaces of constant Gaussian curvature. It is demonstrated that the construction of a Backlund transformation for O surfaces leads in a natural manner to an associated parameter-dependent linear representation. The classical pseudosphere and breather pseudospherical surfaces are generated.

nlin.SI↗
arXiv2000

On dbar-problem and integrable equations

Using the dbar-problem and dual dbar-problem, we derive bilinear relations which allows us to construct integrable hierarchies in different parametrizations, their Darboux-Bäcklund transformations and to analyze constraints for them ina very simple way. Scalar KP, BKP and CKP hierarchies are considered as examples.

nlin.SI↗
arXiv1999

Generalized KP hierarchy: Möbius Symmetry, Symmetry Constraints and Calogero-Moser System

Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that Möbius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary Bäcklund transformation. A more general class of multicomponent Möbius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent Möbius-type symmetries give rise to Calogero-Moser system.

solv-int↗
arXiv1999

On rigid string instantons in four dimensions

Generalized Weierstrass formulae for surfaces in four-dimensional space $\Bbb{R}^{4}$ are used to study (anti)self-dual rigid string configurations. It is shown that such configurations are given by superminimal immersions into $\Bbb{R}^{4}$. Explicit formulae for generic (anti)instantons are presented. Particular classes of surfaces are also analyzed.

hep-th↗