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The paper considers the semiclassical dynamics of electrons on complex Fermi surfaces in the presence of strong magnetic fields. The reconstructions of the general topological structure of such dynamics are accompanied by the appearance of closed extremal trajectories of a special form, closely related to geometry and topology of the Fermi surface. The study of oscillation phenomena on such trajectories allows, in particular, to propose a relatively simple method for refining the parameters of the dispersion relation in metals with complex Fermi surfaces., 34 pages, 56 figures, revtex

The results of recent studies in the theory of dynamical systems related to the motion of electrons on complex Fermi surfaces in normal metals are presented. The problem considered is closely related to the description of electron transport phenomena in strong magnetic fields and is therefore of great interest from the viewpoint of topology and dynamical systems theory. We will try to give a brief overview of the state of the art in this research area, as well as point out a number of interesting issues that are being actively studied at present.

We consider some general aspects of dependence of magneto-conductivity on a magnetic field in metals having complicated Fermi surfaces. As it is well known, a nontrivial behavior of conductivity in metals in strong magnetic fields is connected usually with appearance of non-closed quasiclassical electron trajectories on the Fermi surface in a magnetic field. The structure of the electron trajectories depends strongly on the direction of the magnetic field and usually the greatest interest is caused by open trajectories that are stable to small rotations of the direction of $\, {\bf B} $. The geometry of the corresponding Stability Zones on the angular diagram in the space of directions of $\, {\bf B} \, $ represents a very important characteristic of the electron spectrum in a metal linking the parameters of the spectrum to the experimental data. Here we will consider some very general features inherent in the angular diagrams of metals with Fermi surfaces of the most arbitrary form. In particular, we will show here that any Stability Zone actually has a second boundary, restricting a larger region with a certain behavior of conductivity. Besides that, we shall discuss here general questions of complexity of the angular diagrams for the conductivity and propose a theoretical scheme for dividing the angular diagrams into "simple" and "complex" diagrams. The proposed scheme will in fact also be closely related to behavior of the Hall conductivity in a metal in strong magnetic fields. In conclusion, we will also discuss the relationship of the questions under consideration to the general features of an (abstract) angular diagram describing the behavior of quasiclassical electron trajectories at all energy levels in the conduction band., Comment: 27 pages, 42 figures, RevTex

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., 14 pages, revtex

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 ωBτ →∞. We describe the history of the question and investigate 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.

We consider angular diagrams describing the dependence of the magnetic conductivity of metals on the direction of the magnetic field in rather strong fields. As it can be shown, all angular conductivity diagrams can be divided into a finite number of classes with different complexity. The greatest interest among such diagrams is represented by diagrams with the maximal complexity, which can occur for metals with rather complicated Fermi surfaces. In describing the structure of complex diagrams, in addition to the description of the conductivity itself, the description of the Hall conductivity for different directions of the magnetic field plays very important role. For the evaluation of the complexity of angular diagrams of the conductivity of metals, it is convenient also to compare such diagrams with the full mathematical diagrams that are defined (formally) for the entire dispersion relation., Comment: 24 pages, 32 figures, revtex

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., Comment: 39 pages, 17 figures, Latex

We consider the problem of exact experimental determination of the boundaries of Stability Zones for magneto-conductivity in normal metals in the space of directions of $\, {\bf B} \, $. As can be shown, this problem turns out to be nontrivial since the exact boundaries of Stability Zones are in fact unobservable in direct measurements of conductivity. However, this problem can be effectively solved with the aid of the study of oscillation phenomena (cyclotron resonance, quantum oscillations) in normal metals, which reveal a singular behavior on the mathematical boundary of a Stability Zone., 10 pages, 14 figures, revtex

We consider the electric conductivity in normal metals in presence of a strong magnetic field. It is assumed here that the Fermi surface of a metal has rather complicated form such that different types of quasiclassical electron trajectories can appear on the Fermi level for different directions of B. The effects we consider are connected with the existence of regular (stable) open electron trajectories which arise in general on complicated Fermi surfaces. The trajectories of this type have a nice geometric description and represent quasiperiodic lines with a fixed mean direction in the p-space. Being stable geometric objects, the trajectories of this kind exist for some open regions in the space of directions of B, which can be represented by "Stability Zones" on the unit sphere. The main goal of the paper is to give a description of the analytical behavior of conductivity in the Stability Zones, which demonstrates in general rather nontrivial properties., 26 pages, 24 figures, Revtex

We consider a deformation of the Whitham system for the nonlinear Klein-Gordon equation. This deformation has a Lorentz-invariant form. Using the Lagrangian formalism of the original system, we obtain the first nontrivial correction to the Whitham system in the Lorentz-invariant approach.

We consider here special Poisson brackets given by the "averaging" of local multi-dimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the "physical meaning" of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of "Hydrodynamic Type" in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a "pseudo-canonical" form under some special ("physical") requirements on the initial bracket., Comment: 33 pages, Latex

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system., Comment: 64 pages, Latex

New observable integer-valued numbers of the topological origin were revealed by the present authors studying the conductivity theory of single crystal 3D normal metals in the reasonably strong magnetic field ($B \leq 10^{3} Tl$). Our investigation is based on the study of dynamical systems on Fermi surfaces for the motion of semi-classical electron in magnetic field. All possible asymptotic regimes are also found for $B \to \infty$ based on the topological classification of trajectories., Comment: Latex, 51 pages, 14 eps figures

This is a survey article dedicated to the study of topological quantities in theory of normal metals discovered in the works of the authors during the last years. Our results are based on the theory of dynamical systems on Fermi surfaces. The physical foundations of this theory (the so-called “Geometric Strong Magnetic Field Limit”) were found by the school of I. M. Lifshitz many years ago. Here the new aspects in the topology of quasiperiodic functions are developed.

In this paper we extend the Balian--Low theorem, which is a version of the uncertainty principle for Gabor (Weyl--Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian--Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian--Low theorem for differential operators associated with a symplectic basis for the symplectic form on ${\mathbb R}^{2d}$.

We study semiclassical dynamics of anisotropic Heisenberg models in two and three dimensions. Such models describe lattice spin systems and hard core bosons in optical lattices. We solve numerically Landau-Lifshitz type equations on a lattice and show that in the phase diagram of magnetization and interaction anisotropy, one can identify several distinct regimes of dynamics. These regions can be distinguished based on the character of one dimensional solitonic excitations, and stability of such solitons to transverse modulation. Small amplitude and long wavelength perturbations can be analyzed analytically using mapping of non-linear hydrodynamic equations to KdV type equations. Numerically we find that properties of solitons and dynamics in general remain similar to our analytical results even for large amplitude and short distance inhomogeneities, which allows us to obtain a universal dynamical phase diagram. As a concrete example we study dynamical evolution of the system starting from a state with magnetization step and show that formation of oscillatory regions and their stability to transverse modulation can be understood from the properties of solitons. In regimes unstable to transverse modulation we observe formation of lump type solutions with modulation in all directions. We discuss implications of our results for experiments with ultracold atoms.

We study in this work the important class of nonlocal Poisson Brackets (PB) which we call weakly nonlocal. They appeared recently in some investigations in the Soliton Theory. However there was no theory of such brackets except very special first order case. Even in this case the theory was not developed enough. In particular, we introduce the Physical forms and find Casimirs, Momentum and Canonical forms for the most important Hydrodynamic type PB of that kind and their dependence on the boundary conditions., Comment: 45 pages, latex

The behavior of the electrical conductivity tensor in strong magnetic fields in the presence of unclosed quasiclassical electron trajectories of complex form near the Fermi surface is considered. It is shown that the asymptotic behavior of the conductivity tensor in the limit B→∞ differs in this case from the picture previously described for trajectories of simpler form. The possibility of blocking the longitudinal conductivity in strong magnetic fields at low temperatures in the case of a Fermi surface of special form is also treated theoretically.

The paper considers the topological characteristics of dispersion functions ɛs(p) in energy bands in single crystals related to classical electron trajectories in uniform magnetic fields. Specifically, the topological properties of open trajectories in p-space on various energy levels within one energy band and related physical effects are described.

It is shown that the investigation of the conductivity in a single crystal of a normal metal with a complicated Fermi surface in strong magnetic fields B can reveal integral topological characteristics which are determined by the topology of open-ended quasiclassical electron trajectories. Specifically, in the case of open-ended trajectories of the general position there always exists a direction η orthogonal to B in which the conductivity approaches zero for large B, and this direction lies in some integral (i.e., generated by two reciprocal-lattice vectors) plane that remains stationary for small variations of the direction of B.

We consider the construction of the deformed Whitham system for the KdV-equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then, that both the Gardner - Zakharov - Faddeev and the Magri brackets give the deformed Dubrovin - Novikov brackets (the brackets of Dubrovin - Zhang type) for the deformed Whitham system constructed by our procedure. The general approach used in the paper gives a scheme for the averaging of the Poisson structures in the general situation., Comment: 54 pages, 2 figures

We consider the questions connected with the Hamiltonian properties of the Whitham equations in case of several spatial dimensions. An essential point of our approach here is a connection of the Hamiltonian structure of the Whitham system with the finite-dimensional Poisson bracket defined on the space of periodic or quasi-periodic solutions. From our point of view, this approach gives a possibility to construct the Hamiltonian structure of the Whitham equations under minimal requirements on the properties of the initial system. The Poisson bracket for the Whitham system can be considered here as a deformation of the finite-dimensional bracket with the aid of the Dubrovin - Novikov procedure of bracket averaging. At the end, we consider the examples where the constructions of the paper play an essential role for the construction of the Poisson bracket for the Whitham system., Comment: 69 pages, 1 figure, latex

In this chapter we give the basic concept of the “topological numbers” in the theory of quasiperiodic functions. Attention is especially paid to appearance of such quantities in transport phenomena, including galvanomagnetic phenomena in normal metals (Sect. 2.1) and the modulations of 2D electron gas (Sect. 2.3). We give a detailed introduction to both these areas and explain in a simple way the appearance of the “integral characteristics” in both these problems. Though this chapter cannot be considered a detailed survey in the area, it explains the main basic features of the corresponding phenomena.

We consider the deformations of Whitham systems including the "dispersion terms" and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with B.A. Dubrovin problem of deformations of Frobenius manifolds corresponding to the Whitham systems of integrable hierarchies. Under some non-degeneracy requirements we suggest a general scheme of the deformation of the hyperbolic Whitham systems using the initial non-linear system. The general form of the deformed Whitham system coincides with the form of the "low-dispersion" asymptotic expansions used by B.A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius manifolds., 27 pages, Latex

We compare Whitham averaging method using Lagrangian and Hamiltonian formalism., 15 pages