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Vladimir Novikov

Publications and source records attributed to Vladimir Novikov.

13 recordsLinked to original sources
arXiv2026

Global Results on the Classification of Two-Component Integrable Evolutionary Systems

We derive necessary and sufficient integrability conditions for two-component polynomial evolutionary systems of odd order in $(1+1)$ dimensions. Integrable systems are members of infinite hierarchies of commuting symmetries, which are characterised by their spectral invariants. We prove that there are precisely $24$ possible spectral classes of integrable hierarchies. As an application, we obtain a complete classification of integrable homogeneous hierarchies whose lowest-order equations are of order 3 and 5. The resulting classification naturally splits into two classes. The C--integrable systems are reduced, by means of differential substitutions, to linear--triangular form, while the S--integrable systems are related, through linear changes of variables and differential substitutions, to canonical Drinfeld--Sokolov KdV-type systems associated with affine Lie algebras of rank two.

nlin.SI↗
arXiv2025

Unveiling Mechanical Motions in Non-linear Optical Organic Micro Ring Resonators

This work demonstrates the multifaceted dynamic motion and three-dimensional (3D) spatial manipulation of organic micro ring resonators (MRRs). The MRRs are fabricated via surface-tension-assisted self-assembly of 2,2'-((1E,1'E)-hydrazine-1,2-diylidenebis(methaneylylidene))bis(3,5-dibromophenol) (HDBP), exhibiting nonlinear optical (NLO) emission and frequency comb-type whispering gallery modes. Interestingly, the MRRs are micromechanically reconfigurable into various strained architectures, including lifting, transferring, vertical standing, axial spinning, and wheel-like rolling, using an atomic force microscopy cantilever tip. The MRRs retained their photonic traits throughout these dynamic motions, underscoring their mechanical robustness. Notably, the demonstration of axial spinning and rolling locomotion extends the manipulation capabilities beyond two-dimensional control, enabling complete 3D spatial control. These results establish a foundational platform for next-generation NLO mechanophotonic systems, where reconfigurable smart and soft photonic elements can be dynamically controlled with high spatial precision.

physics.optics↗
arXiv2024

Integrability of Nonabelian Differential-Difference Equations: the Symmetry Approach

We propose a novel approach to tackle integrability problem for evolutionary differential-difference equations (D$Δ$Es) on free associative algebras, also referred to as nonabelian D$Δ$Es. This approach enables us to derive necessary integrability conditions, determine the integrability of a given equation, and make progress in the classification of integrable nonabelian D$Δ$Es. This work involves establishing symbolic representations for the nonabelian difference algebra, difference operators, and formal series, as well as introducing a novel quasi-local extension for the algebra of formal series within the context of symbolic representations. Applying this formalism, we solve the classification problem of integrable skew-symmetric quasi-linear nonabelian equations of orders $(-1,1)$, $(-2,2)$, and $(-3,3)$, consequently revealing some new equations in the process.

nlin.SI↗
arXiv2021

Second-order PDEs in 3D with Einstein-Weyl conformal structure

Einstein-Weyl geometry is a triple (D,g,w), where D is a symmetric connection, [g] is a conformal structure and w is a covector such that: (i) connection D preserves the conformal class [g], that is, Dg=wg; (ii) trace-free part of the symmetrised Ricci tensor of D vanishes. Three-dimensional Einstein-Weyl structures arise naturally on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, the covector w is a somewhat more mysterious object, recovered from the Einstein-Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge-Ampere type), the covector w is also expressible in terms of the equation, thus providing an efficient dispersionless integrability test. The knowledge of g and w provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein-Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein-Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

nlin.SI↗
arXiv2020

Second-order PDEs in 4D with half-flat conformal structure

We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere property. Since half-flatness of the conformal structure is equivalent to the existence of a nontrivial dispersionless Lax pair, our result explains the observation that all known scalar second-order integrable dispersionless PDEs in dimensions four and higher are of Monge-Ampere type. Some partial classification results of Monge-Ampere equations in 4D with half-flat conformal structure are also obtained.

math.DG↗
arXiv2017

Integrable systems in 4D associated with sixfolds in Gr(4,6)

Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an $n$-dimensional vector space $V$. A submanifold $X\subset Gr(d, n)$ gives rise to a differential system $Σ(X)$ that governs $d$-dimensional submanifolds of $V$ whose Gaussian image is contained in $X$. We investigate a special case of this construction where $X$ is a sixfold in $Gr(4, 6)$. The corresponding system $Σ(X)$ reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems $Σ(X)$. These naturally fall into two subclasses. (1) Systems of Monge-Ampère type. The corresponding sixfolds $X$ are codimension 2 linear sections of the Plücker embedding $Gr(4,6)\subset\mathbb{P}^{14}$. (2) General linearly degenerate systems. The corresponding sixfolds $X$ are the images of quadratic maps $\mathbb{P}^6- Gr(4, 6)$ given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system $Σ(X)$ gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

nlin.SI↗
arXiv2017

On a class of integrable systems of Monge-Ampère type

We investigate a class of multi-dimensional two-component systems of Monge-Ampère type that can be viewed as generalisations of heavenly-type equations appearing in self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Ampère type turn out to be integrable, and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Ampère type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Ampère property.

nlin.SI↗
arXiv2016

On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, etc. We prove the equivalence of the four different approaches to integrability, revealing a remarkable correspondence with Einstein-Weyl geometry and the theory of GL(2,R) structures.

math.DG↗
arXiv2016

Two-component generalizations of the Camassa-Holm equation

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators is carried out, which leads to bi-Hamiltonian structures for the same systems of equations. Some exact solutions and Lax pairs are also constructed for the systems considered.

nlin.SI↗
arXiv2013

On Classification of Integrable Davey-Stewartson Type Equations

This paper is devoted to the classification of integrable Davey-Stewartson type equations. A list of potentially deformable dispersionless systems is obtained through the requirement that such systems must be generated by a polynomial dispersionless Lax pair. A perturbative approach based on the method of hydrodynamic reductions is employed to recover the integrable systems along with their Lax pairs. Some of the found systems seem to be new.

nlin.SI↗
arXiv2009

Generalisations of the Camassa-Holm equation

We classify generalised Camassa-Holm type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of orders 2, 3 and 5. We present the corresponding Lax representations or linearisation transformations for these equations. Some of the obtained equations seem to be new.

nlin.SI↗
arXiv2006

Does frequency-temperature superposition hold in deeply super-cooled liquids?

The temperature evolution of the broadband $10^{-6}$-$10^{10}$ Hz dielectric susceptibility of the paradigmatic glass formers glycerol, propylene carbonate, and fluoro-aniline is analyzed assuming a three-step relaxation due to the $α$-process, its excess wing, and a $β$-process. We find that the $α$-peak and the wing can be described by susceptibility functions with temperature-independent high-frequency exponents, while the relative weight of these contributions does depend on the temperature. The excess wing and the $β$-process are distinct phenomena; in particular, the relaxation strength of the excess wing grows with decreasing the temperature, contrary to that of the $β$-process. In our interpretation, the frequency-temperature superposition of the $α$-process is valid for all temperatures; in the case of glycerol, a typical $β$-process is unambiguously identified for the first time.

cond-mat.dis-nn↗