Search

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 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

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

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

An updated and detailed survey of basic ideas of the finite-gap theory is presented. That theory, developed to construct periodic and quasi-periodic solutions of the soliton equations, combines the Bloch-Floquet spectral theory of linear periodic operators, the theory of completely integrable Hamiltonian systems, the classical theory of Riemann surfaces, and theta-functions.

In recent articles [1]–[3], and ending with [4], there has been discovered a remarkable circumstance resulting from the combinatorial Kazakov–Migdal–Rostov approach in the continuous limit. In cases of nonperturbative conformal string theories interacting with a “two-dimensional gravitation” according to Polyakov’s scheme, as well as some other ones, when the central charge is c < 1, there appears a simple system of equations for the renormgroup, i.e., a set of Laks type equations with certain ordinary differential operators in x

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components., Comment: LaTeX, 18 pages, 3 figures

A non-traditional approach to the discretization of differential-geometrical connections was suggested by the authors in 1997. At the same time we started studying first order difference ``black and white triangle operators (equations)'' on triangulated surfaces with a black and white coloring or triangles. In this work, we develop the theory of these operators and equations, showing their similarity with complex derivatives., Comment: latex, 23 pages, 10 figures

It is shown that the fourth order real self-adjoint difference operator on the Tivalent Tree admits nontrivial deformations preserving one energy level and therefore defines a nontrinial hierarhy of the completely integrable nonlinear systems representible through the ''L-A-B-triple''. The Laplace transformations for these operators are also constructed. Nothing like that exists for the second order difference operators on this tree., LATEX file, 3 pages

This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic Wronskian is a 1-cycle in the graph. This is a vector-valued symplectic 2-form on the space of solutions. This construction was applied to the Scattering Theory on the graphs with finite number of tails. The asymptotic values of solutions is a Lagrangian Plane of half dimension. This property determines all unitary properties of Scattering Matrix, which is also symmetric. All higher order discrete operators and operators on higher dimensional simplicial complexes are included in this scheme. Nonlinear analog of that was invented by the present author in collaboration with A.S.Schwarz., Comment: 20 pages, LaTeX, to appear in Asian Math. J. in the volume dedicated to the 70th birthday of Mikio Sato

This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of non-compact plane sections of level surfaces of periodic function in 3-dimensional Euclidean space which are the quasi-classical electron orbits in the presence of homogeneous magnetic field. The main result is that the observation of electrical conductivity in strong magnetic fields can reveal such nontrivial topological characteristics of Fermi surface as integral planes, connected with conductivity tensor and locally stable under small rotations of magnetic field. This planes are connected with generic non-closed orbits on the Fermi surface. Some non-generic situations are also discussed., 21 pages, 9 Encapsulated Postscript figures

Theorem 1. 1) Under local changes of the fields u = u(w) the coefficient g(u) in the bracket (2) transforms like a bilinear form (a tensor with upper indices); if det g 6= 0, then the expression b k (u) = gΓ j sk transforms in such a way that the Γjsk are the Christoffel tymbols of a differential-geometric connection. 2) In order that the bracket (2) be skew-symmetric it is necessary and sufficient that the tensor g(u) be symmetric (i.e., that it define a pseudo-Riemannian metric

We are going to present here some brief survey of the results of Theory of Solitons (see [1–3]) from the viewpoint of periodic theory including some new results in the theory of 2-dimensional periodic Schrodinger Operators.

The S. P. Novikov seminar by V. M. Buchstaber and S. P. Novikov Semigroups of maps into groups, operator doubles, and complex cobordisms by V. M. Buchstaber Nonlocal Hamiltonian operators of hydrodynamic type: Differential geometry and applications by E. V. Ferapontov Nonselfintersecting magnetic orbits on the plane. Proof of the overthrowing of cycles principle by P. G. Grinevich and S. P. Novikov Spin generalization of the Calogero-Moser system and the matrix KP equation by I. Krichever, O. Babelon, E. Billey, and M. Talon Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations by O. Mokhov Real nonsingular finite zone solutions of soliton equations by S. M. Natanzon Representations of Krichever-Novikov algebras by O. K. Sheinman Huygens' principle and algebraic Schrodinger operators by A. P. Veselov.

For any Phase space, i.e. a manifold M with the Poisson structure, we have a scew-symmetric Poisson Tensor h ij with 2 upper indices in any local coordinates (x i ), such that the Poisson Bracket {f,g} = h ij f i g j satisfies the Jacoby identity. Here f i means a partial derivative of the function f (x) on the manifold and the standard summation rule is used for tensor indices. The Poisson bracket is well-defined for multivalued functions as well (i.e. closed 1-forms df = f i dx i , dg = g j dx j ).

A pair (M, ω) consisting of a 2n-dimensional manifold M together with a closed 2-form ω is called a symplectic manifold if the form ω is nondegenerate, i.e. if ω n = ω ∧ · ... · ω ≢ 0.

I am going to discuss here some results in the soliton theory of a Moscow group during the last years. The group of people who worked here include B.A.Dubrovin, I.M. Krichever, S.P. Tsarev (and the present author). More details may be found in the survey article [1]. Modern needs in the large new classes of hydrodynamic type systems appear in connection with very interesting asymptotic method - so called “nonlinear analog of WKB-method”, method of the slow modulations of parameters or “Whitham method” in the theory of solitons (see for example the book [2], chapter 4). This method is based on the large family of exact solutions φ0(x,t,u)periodic or quasiperiodic in x and t, of the form which we call “soliton lattice ”: $${{\varphi }_{0}}(x,t;u) = F({{\eta }_{0}} + Ux + Vt;{{u}^{1}}, \ldots ,{{u}^{N}}).$$ (0.1) Here \(F({{\eta }_{1}}, \ldots ,{{\eta }_{m}};{{u}^{1}}, \ldots ,{{u}^{N}})\) is the function, which is 2π-periodic in each variable η j, and depends on the N parameters up. All quantities (aj, bk,up,) are constants and φ0(x,t,u) satisfies some nonlinear P.D. equation $${{\varphi }_{t}}(x,t) = K(\varphi ,{{\varphi }_{x}}, \ldots ,{{\varphi }^{{(s)}}}),$$ (0.2) describing the propagation of solitons or some other nonlinear waves.

In this paper, originating from the results [1] on cyclic chains of Backlund-Darboux transforms in the one-dimensional case, we use a Laplace transform to construct exactly soluble periodic operators 2L = (d + B)(d + A) + 2V. Here A, B, V are functions on the (x,y)-plane, ζ = x + iy, d = dx -idy. We call V the potential and Η = (Bz Ατ)/2 the magnetic field. The Laplace transform associates the solution of the equation Lip = 0 with the solution of a new equation £•0 = 0, where φ = (Θ + Α)ψ, 2L = (d + B)(d + A) + 2V = (d + A)(d +B) + 2V, A = A-(InV)t, Β = B. For the potential and the magnetic field the transform has the form V = V + H, Η = Η + | ( lnV) z y. It appeared in classical geometry at the beginning of the 19th century with aims far removed from ours and where the operator L was always hyperbolic: ζ —¥ ξ, ζ —• η (see [2]). An infinite chain of Laplace transforms Vn = Vn_i + Hn, Hn = i?n-i + \ddfn-i, Vn = exp/n, after transition to functions gn — gn-i = fn, coincides with a two-dimensionalized Toda chain, which is widely known in the theory of solitons. As Ferapontov has informed the authors, this circumstance was known to A. M. Vasil'ev but was not utilized in any way. Cyclic chains of Laplace transforms were considered by Darboux and Tzitzoica [3] who made a number of valuable formal calculations. These authors did not, however, pose global problems.

where I = ∫ L ( u, ∂u ∂x , . . . , ∂u ∂xn ) dx, δI δu(x) = ∂L ∂u+ ∑n k=1(−1) d dxk ∂L ∂u(k) . Indeed, these systems are Hamiltonian on any class of functions in which I makes sense, since the Poisson bracket has the form [I, I1] = ∫ δI δu d dx δI1 δu dx. In [1], Novikov has solved on the basis of the Korteweg–de Vries equation (KdV) and its higher order counterparts u = d dx δIn δu(x) , the stationary problem δIn δu(x) = const. The latter equations are also Hamiltonian. In [1] it is asserted that there exists a relationship between the Hamiltonian formalisms of the stationary and the nonstationary problem, though this was not precisely formulated. In this paper we shall formulate and prove a general theorem on such a relationship. Let us consider two formally Hamiltonian flows in a class of smooth functions on the x axis

In this paper we construct the operator fields of the Riemann surfaces of arbitrary genus. The corresponding operator theory of interacting strings can be considered as the direct development of Virasoro-Mandelstam theory for g ≥ 0 and its unifacation with Polyakov-Belavin-Knizhnik theory.

The paper begins with a short survey of results on non-trivial models (that is, those that are not integrable analytically) in general relativity and gas dynamics. The investigation of these models is carried out by the methods of the qualitative theory of many-dimensional dynamical systems, using geometrical and topological ideas. The first section deals with the results of research on the evolution of homogeneous cosmological models with a hydrodynamic energy tensor - the impulse about a singularity. In the second section similar models are applied to the study of the complex oscillating regimes of a classical ideal compressible fluid. The Appendix contains new, unpublished results due to one of the authors, describing stochastic perturbation of a completely integrable Toda chain.

CONTENTS Introduction § 1. The Hamiltonian formalism. Simplest examples. Systems of Kirchhoff type. Factorization of the Hamiltonian formalism for the B-phase of 3He § 2. The Hamiltonian formalism of systems of hydrodynamic origin § 3. What is Morse (LSM) theory? § 4. Equations of Kirchhoff type and the Dirac monopole § 5. Many-valued functional and an analogue of Morse theory. The periodic problem for equations of Kirchhoff type. Chiral fields in an external field § 6. Many-valued functions on finite-dimensional manifolds. An analogue of Morse theory References

The Ginzburg-Landau equations for planar textures of superfluid3He are proved to be equivalent to a completely integrable Hamiltonian system. General solutions to these equations are obtained by means of hyperelliptic integrals.

Geometrically speaking the fundamental problem of GTR is that of finding 4-dimensional manifolds M4 endowed with a metric g ab of signature (+ − − −), satisfying Einstein’s equation (see Part I, §§37.4, 39) $${R_{ab}} - \frac{1}{2}R{g_b} = \frac{{8\pi G}}{{{c^4}}}{T_{ab,}}$$ (1) where Tab is the energy-momentum tensor of the medium. From the most general point of view there is only one restriction on the tensor Tab, namely that the “energy density” be non-negative, which comes down to the condition that for every time-like vector ξ = (ξa) (i.e. such that gab ξ,a ξb > 0) at each point of M4, we have Tabξa ξb ≥ 0. However in the present chapter we shall for the most part concern ourselves with the so-called “hydrodynamical energy-momentum tensor” of an isotropic dense medium (forming a “perfect fluid”), already introduced in Part I, §39.4, important in the investigation of the gravitational fields of macroscopic bodies: $${T_{ab}} = (p + \varepsilon ){u_a}{u_b} - p{g_{ab}},$$ (2) where u = (u0) is the 4-velocity ( u, u) = 1), p is the (isotropic) pressure, and ɛ is the energy density (see loc. cit.).

CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References