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B. G. Konopelchenko

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

At least 19 recordsLinked to original sources
arXiv2026

On solutions of the Euler equation for incoherent fluid on a rotating sphere

The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler equation parameterized by two arbitrary functions of two variables. Several particular explicit solutions are given. The blow-up curves, on which the derivatives of velocitiy blows up, are described. The limiting cases of slowly and rapidly rotating sphere are considered. The equation describing the deformations of elliptic functions modulus is presented.

math-ph↗
arXiv2025

On Euler equation for incoherent fluid in curved spaces

Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph equations. These hodograph equations provide us with various classes of solutions of the Euler equation, including stationary solutions. Particular cases of cone and sphere in the 3-dimensional Eulidean space are analysed in detail. Euler equation on the sphere in the 4-dimensional Euclidean space is considered too.

math-ph↗
arXiv2024

On pressureless Euler equation with external force

Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force.

math-ph↗
arXiv2024

Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation

It is shown that a canonical geometric setting of the integrable TED equation is a Kahlerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equation arises naturally in this context. In a particular 4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein spaces of neutral signature governed by Plebanski's first heavenly equation. In another reduction, the affine manifolds are Hessian, governed by compatible general heavenly equations. The Legendre invariance of the latter gives rise to a (dual) Hessian structure. Foliations of affine manifolds in terms of self-dual Einstein spaces are also shown to arise in connection with a natural 5-dimensional reduction.

nlin.SI↗
arXiv2023

On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two- dimensional homogeneous Euler equations are studied. Two regimes of approaching a blowup points, respectively, with variable or fixed time are analysed. It is shown that in the $n$-dimensional ($n=2,3$) generic case the blowups of degrees $1,..,n$ at the variable time regime and of degrees $1/2,..,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realisable.

math-ph↗
arXiv2022

On the fine structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation

Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible ranks of certain matrix generated by the initial data. Blow-ups form a hierarchy composed by $n+1$ levels with the strongest singularity of derivatives given by $\partial u_i/\partial x_k \sim |δ\mathbf{x}|^{-(n+1)/(n+2)}$ along certain critical directions. It is demonstrated that in the multi-dimensional case there are certain bounded linear superposition of blow-up derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.

nlin.SI↗
arXiv2021

Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$ associated with the $n$-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case ($n>1$) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson $n$. Several concrete examples in two- and three-dimensional cases are analysed. Properties of $\mathbb{R}^n_{\underline{u}} \to \mathbb{R}^n_{\underline{x}}$ mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

math-ph↗
arXiv2021

On universality of homogeneous Euler equation

Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type equations. Integrable one dimensional systems in terms of Riemann invariants and its extensions, multidimensional equations describing isoenthalpic and polytropic motions and shallow water type equations are among them.

nlin.SI↗
arXiv2020

Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates

Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebanski's first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4+4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries.

nlin.SI↗
arXiv2019

Integrability properties of symmetric 4+4-dimensional heavenly type equation

We demonstrate that the dispersionless $\bar\partial$-dressing method developed before for general heavenly equation is applicable to the $4+4$ and $2N+2N$ - dimensional symmetric heavenly type equations. We introduce generating relation and derive the two-form defining the potential and equation for it. We develop the dressing scheme, calculate a class of special solutions and demonstrate that reduction from $4+4$-dimensional equation to four-dimensional general heavenly equation can be effectively performed on the level of the dressing data. We consider also the extension of the proposed scheme to $2N+2N$-dimensional case.

nlin.SI↗
arXiv2019

On the plane into plane mappings of hydrodynamic type. Parabolic case

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-liner partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach to such case are discussed. Hierarchy of singularities is analysed by double-scaling expansion method for the simplest $2$-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by $(k+1,k+2)$ curves which are of cusp type for $k=2n+1$, $n=1,2,3,\dots$. Regularization of these singularities by deformation of plane into plane mappings into surface $S^{2+k} (\subset \mathbb{R}^{2+k} ) $ to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted.

math-ph↗
arXiv2019

On an integrable multi-dimensionally consistent 2n+2n-dimensional heavenly-type equation

Based on the commutativity of scalar vector fields, an algebraic scheme is developed which leads to a privileged multi-dimensionally consistent 2n+2n-dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The "universal" character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalising to higher dimensions a great variety of well-known integrable equations such as the dispersionless KP and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.

nlin.SI↗
arXiv2018

Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain

Non-standard parabolic regularization of gradient catastrophes for the Burgers-Hopf equation is proposed. It is based on the analysis of all (generic and higher order) gradient catastrophes and their step by step regularization by embedding the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan block. Probabilistic realization of such procedure is presented. The complete regularization of the Burgers-Hopf equation is achieved by embedding it into the infinite parabolic Jordan chain. It is shown that the Burgers equation is a particular reduction of the Jordan chain. Gradient catastrophes for the parabolic Jordan systems are also studied.

math-ph↗
arXiv2015

Jordan form, parabolicity and other features of change of type transition for hydrodynamic type systems

Changes of type transitions for the two-component hydrodynamic type systems are discussed. It is shown that these systems generically assume the Jordan form (with 2 X 2 Jordan block) on the transition line with hodograph equations becoming parabolic. Conditions which allow or forbid the transition from hyperbolic domain to elliptic one are discussed. Hamiltonian systems and their special subclasses and equations, like dispersionless nonlinear Schroedinger, dispersionless Boussinesq, one-dimensional isentropic gas dynamics equations and nonlinear wave equations are studied. Numerical results concerning the crossing of transition line for the dispersionless Boussinesq equation are presented too.

math-ph↗
arXiv2014

Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations

Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalised) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.

nlin.SI↗
arXiv2013

Projective differential geometry of multidimensional dispersionless integrable hierarchies

We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $Ω_m$ with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.

nlin.SI↗
arXiv2013

Birkhoff strata of Sato Grassmannian and algebraic curves

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum $Σ_S$ contains a subset $W_{\hat{S}}$ of points for which each fiber of the corresponding tautological subbundle $TB_{W_S}$ is closed with respect to multiplication. Algebraically $TB_{W_S}$ is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle $TB_{W_\varnothing}$ represents the tower of families of normal rational (Veronese) curves of all degrees. For $W_1$ such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles $TB_{W_{1,2,\dots,n}}$ represent families of plane $(n+1,n+2)$ curves (trigonal curves at $n=2$) and space curves of genus $n$. Two methods of regularization of singular curves contained in $TB_{W_{\hat{S}}}$, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.

math-ph↗