arXiv subjects

S. P. Novikov

Publications and source records attributed to S. P. Novikov.

At least 19 recordsLinked to original sources

arXiv2023

Geometry of quasiperiodic functions on the plane

The present article proposes a review of the most recent results obtained in the study of Novikov's problem on the description of the geometry of the level lines of quasi-periodic functions in the plane. Most of the paper is devoted to the results obtained for functions with three quasi-periods, which play a very important role in the theory of transport phenomena in metals. In this part, along with previously known results, a number of new results are presented that significantly refine the general description of the picture that arises in this case. New statements are also presented for the case of functions with more than three quasi-periods, which open up approaches to the further study of Novikov's problem in the most general formulation. The role of Novikov's problem in various fields of mathematical and theoretical physics is also discussed.

math-ph↗

arXiv2022

Open level lines of a superposition of periodic potentials on a plane

We consider here open level lines of potentials resulting from the superposition of two different periodic potentials on the plane. This problem can be considered as a particular case of the Novikov problem on the behavior of open level lines of quasi-periodic potentials on the plane with four quasi-periods. At the same time, the formulation of this problem may have many additional features that arise in important physical systems related to it. Here we will try to give a general description of the emerging picture both in the most general case and in the presence of additional restrictions. The main approach to describing the possible behavior of the open level lines will be based on their division into topologically regular and chaotic level lines.

math-ph↗

arXiv2021

Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter

The paper is devoted to the questions connected with the investigation of the S.P. Novikov problem of the description of the geometry of level lines of quasiperiodic functions on a plane with different numbers of quasiperiods. We consider here the history of the question, the current state of research in this field, and a number of applications of this problem to various physical problems. The main attention is paid to the applications of the results obtained in the field under consideration to the theory of transport phenomena in electron systems.

math-ph↗

arXiv2021

The theory of closed 1-forms, levels of quasiperiodic functions and transport phenomena in electron systems

The paper is devoted to the applications of the theory of dynamical systems to the theory of transport phenomena in metals in the presence of strong magnetic fields. More precisely, we consider the connection between the geometry of the trajectories of dynamical systems, arising at the Fermi surface in the presence of an external magnetic field, and the behavior of the conductivity tensor in a metal in the limit $\, ω_{B} τ\rightarrow \infty $. The paper contains a description of the history of the question and the investigation of special features of such behavior in the case of the appearance of trajectories of the most complex type on the Fermi surface of a metal.

math-ph↗

arXiv2020

Poisson brackets of hydrodynamic type and their generalizations

In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms of the corresponding Poisson brackets and the connection of such structures with the theory of integration of systems of hydrodynamic type.

math-ph↗

arXiv2020

Misha Shubin. 1944 -- 2020

The article describes the biography and manifold contributions to research in mathematics of Mikhail Aleksandrovich Shubin.

math.HO↗

arXiv2015

On the s-meromorphic OD operators

We consider linear spectral-meromorphic (s-meromorphic) OD operators at the real axis such that all local solutions to the eigenvalue problems are meromorphic for all $λ$. By definition, rank one algebro-geometrical operator $L$ admit an OD operator $A$ such that $[L,A]=0$ and rank of this commuting pair is equal to one. All of them are s-meromorphic. In particular, second order ``singular soliton'' operators satisfy to this condition. Operator $L^+$ formally adjoint to s-meromorphic operator $L$ is also s-meromorphic. For singular eigenfunctions of operators $L,L^+$ following scalar product $ =\int_R f\bar{g}dx$ is well-defined such that $ = $ avoiding isolated singular points. For the case $L=L^+$ this formula defines indefinite inner product on the spaces of singular functions $f,g\in F_L$ associated with operator $L$. They are $C^{\infty}$ outside of singularities and have isolated singularities of the same type as eigenfunctions $Lf=λf$. Every s-meromorphic operator can be approximated by algebro-geometric rank one operators in any finite interval

math-ph↗

arXiv2015

Singular Soliton Operators and Indefinite Metrics

The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All algebro-geometrical or "singular finite-gap" potentials satisfy to this condition. A Spectral Theory is constructed here for the periodic and rapidly decreasing cases in the special classes of functions with singularities and indefinite inner product. It has a finite number of negative squares if the unimodular Bloch multipliers are fixed in the periodic case, and in the rapidly decreasing case. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform for the Riemann Surfaces preserving remarkable multiplicative properties of the ordinary (i.e. genus zero) Fourier Transform based on the standard exponential basis, leads to such operators as it was shown in our previous works.

math-ph↗

arXiv2013

Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold

Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds and equilateral triangle lattices were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, new discretization of Complex Analysis and new discretization of $GL_n$ connections on the triangulated $n$-manifolds. The general theory of the new type discrete $GL_n$ connections was developed. However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$ connections such that $\det A=\pm 1$) was not selected properly. As we prove in this work, it plays fundamental role (similar to magnetic field in the continuous case) in the theory of self-adjoint discrete Schrodinger operators for the equilateral triangle lattice in $\RR^2$. In Appendix~1 we present a complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections. Therefore we correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection). Using communications of Korepanov we completely clarify connection of classical theory of electric chains and star-triangle with discrete Laplace transformation on the triangle lattices

math-ph↗

arXiv2009

New Discretization of Complex Analysis: The Euclidean and Hyperbolic Planes

Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for the analogs of equilateral triangle lattice in Hyperbolic plane. Some specific very interesting "dynamical phenomena" appear in this case solving most fundamental boundary problems. Mike Boyle from the University of Maryland helped to use here the methods of symbolic dynamics

math.GT↗

arXiv2007

Dynamical Systems and Differential Forms. Low Dimensional Hamiltonian Systems

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flows. Some invariant forms can be naturally considered as "forms on the quotient." As a space, this quotient may be very bad in the conventional topological sense. These considerations lead to an analog of the de Rham cohomology theory for manifolds carrying smooth dynamical system. The cohomology theory for quotients, called "basic cohomology" in the literature, appears naturally in our approach. We define also new exotic cohomology groups associated with the so-called cohomological equation in dynamical systems and find exact sequences connecting them with the cohomology of quotients. Explicit computations are performed for geodesic and horocycle flows of compact surfaces of constant negative curvature. Are these famous systems Hamiltonian for a 3D manifold with a Poisson structure? Below, we discuss exotic Poisson structures on 3-manifolds having complicated Anosov-type Casimir foliations. We prove that horocycle flows are Hamiltonian for such exotic structures. The geodesic flow is non-Hamiltonian in the 3D sense.

math.GT↗

arXiv2006

Reality problems in the soliton theory

This is a survey article dedicated mostly to the theory of real regular "finite-gap" (algebro-geometrical) periodic and quasiperiodic Sine-Gordon solutions. Long period this theory remained unfinished and ineffective, and by that reason practically had no applications. Even for such simple physical quantity as "Topological Charge" no formulas existed expressing it through the "Inverse Spectral Data". Few years ago the present authors solved this problem and made this theory effective. This article contains description of the history and recent achievements. It describes also the reality problems for several other fundamental soliton systems.

nlin.SI↗

arXiv2005

Topology of the Generic Hamiltonian Foliations on the Riemann Surface

Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). This approach allows us to present a convenient combinatorial model of the whole topology of the flow, especially effective for g=2. A maximal abelian covering over the Riemann Surface is needed here. The complete combinatorial model of the flow is constructed. It consists of the Plane Diagram and g straight line flows in the 2-tori ''with obstacles''. The Fundamental Semigroup of positive closed paths transversal to foliation, and the topological characteristics of trajectories, are studied. This work contains an improved exposition of the results presented in the authors recent preprint (arXiv math.GT/0105338) and new results calculating all TCB in the 2-torus with obstacle, in terms of Continued Fractions.

math.GT↗

arXiv2005

Topology of Foliations given by the real part of holomorphic 1-forms

Topology of Foliations of the Riemann Surfaces given by the real part of generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB) instead of using just one closed transversal curve as in the classical approach of the ergodic theory. In some cases the TCB approach allows us to present a convenient combinatorial model of the whole topology of the flow, especially effective for g=2. A maximal abelian covering over the Riemann Surface provided by the Abel Map, plays a key role in this work. The behavior of our system in the Fundamental Domain of that covering can be easily described in the sphere with $g$ holes. It leads to the Plane Diagram of our system. The complete combinatorial model of the flow is constructed. It is based on the Plane Diagram and g straight line flows in the planes corresponding to the $g$ canonically adjoint pairs of cycles in the Transversal Canonical Basis. These pairs do not cross each other. Making cuts along them, we come to the maximal abelian fundamental domain (associated with Abel Map and Theta-functions) instead of the standard $4g$-gon in the Hyperbolic Plane and its beautiful "flat" analogs which people used for the study of geodesics of the flat metrics with singularities. Topological splitting of the flow into torical pieces is constructed. Several mistakes are corrected. Algebraic description of transversal canonical bases on the torus with obstacles is given. The appendix is extended: a reference to a work of G. Levitt is added which allowed to prove that every foliation of this class admits a TCB.

math.GT↗

arXiv2004

Discrete connections on the triangulated manifolds and difference linear equations

Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference equation acting on the scalar functions of vertices in any simplicial manifold. This theory appeared as a by-product of the new type of discretization of the special Completely Integrable Systems, such as the famous 2D Toda Lattice and corresponding 2D stationary Schrodinger operators. A nonstandard discretization of the 2D Complex Analysis based on these ideas was developed in our recent work closely connected with this one. A complete classification theory is constructed here for the Discrete DG Connections based on the mixture of the abelian and nonabelian features.

math-ph↗

arXiv2004

On the Metric Independent Exotic Homology

Different types of nonstandard homology groups based on the various subcomplexes of differential forms are considered as a continuation of the recent authors works. Some of them reflect interesting properties of dynamical systems on the compact manifolds. In order to study them a Special Perturbation Theory in the form of Spectral Sequences is developed. In some cases a convenient fermionic formalism of dealing with differential forms is used originated from the work of Witten in the Morse Theory (1982)and the authors work where some nonstandard analog of Morse Inequalities for vector fields was found (1986).

math.DG↗

arXiv2003

Topology, Quasiperiodic functions and the Transport phenomena

In this article we give the basic concept of the "Topological Numbers" in theory of quasiperiodic functions. The main attention is paid to apperance of such values in transport phenomena including Galvanomagnetic phenomena in normal metals (Chapter 1) and the modulations of 2D electron gas (Chapter 2). We give just the main introduction to both of these areas and explain in a simple way the appearance of the "integral characteristics" in both of these problems. The paper can not be considered as the detailed survey article in the area but explains the main basic features of the corresponding phenomena.

cond-mat↗

arXiv2003

2D Toda Chain, Commuting Difference Operators and Holomorphic Bundles

High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics of Tyurin Parameters characterizing the stable holomorphic vector bundles over the algebraic curves (Riemann Surfaces).

math-ph↗