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We consider singular real second order 1D Schrodinger operators such that all local solutions to the eigenvalue problems are x-meromorphic for all λ. All algebrogeometrical potentials (i.e. “singular finite-gap” and “singular solitons”) satisfy to this condition. A Spectral Theory is constructed for the periodic and rapidly decreasing potentials in the classes of functionswith singularities: The corresponding operators are symmetric with respect to some natural indefinite inner product as it was discovered by the present authors. It has a finite number of negative squares in the both (periodic and rapidly decreasing) cases. The time dynamics provided by the KdV hierarchy preserves this number. The right analog of Fourier Transform on Riemann Surfaces with good multiplicative properties (the R-Fourier Transform) is a partial case of this theory. The potential has a pole in this case at x = 0 with asymptotics u ∼ g(g + 1)/x 2. Here g is the genus of spectral curve.

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).

An effective description of the inverse spectral data corresponding to the real periodic and quasi-periodic finite-gap (algebro-geometrical) solutions for the sine-Gordon equation is obtained. In particular, the explicit formula for the so-called topological charge of the solutions is found and proven. As was already understood 20 years ago, it is very hard to extract any formula for this quantity from the Θ-functional expressions. A new method was developed by the authors for this goal. In Appendix C an analogue of the Fourier-Laurent integral transform on the Riemann surface is defined, based on the constructions of soliton theory. It is a natural continuous analogue of the discrete Krichever-Novikov bases developed in the late 1980s for the needs of the operator quantization of string theory. © 2003 Wiley Periodicals, Inc.

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.

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.

Mathematical problems cannot always be solved. Sometimes they turn out to be too hard, and their solutions stretch out over decades and even centuries; sometimes they prove to be insoluble “in principle.” Alas, we mathematicians are not like those Soviet giants of the thirties who could raise hundreds of calves in one year from two cows or mine a mountain of coal overnight.

Integrable systems which do not have an “obvious“ group symmetry, beginning with the results of Poincare and Bruns at the end of the last century, have been perceived as something exotic. The very insignificant list of such examples practically did not change until the 1960’s. Although a number of fundamental methods of mathematical physics were based essentially on the perturbation-theory analysis of the simplest integrable examples, ideas about the structure of nontrivial integrable systems did not exert any real influence on the development of physics.

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 ).

In this section we consider the nonholonomic variational problem on the minimum of length on a Riemannian manifold with constraints given by a nonholonomic distribution. The solutions to this problem, the nonholonomic geodesics, satisfy the Euler-Lagrange equations of a conditional problem. They generate a nonholonomic geodesic flow defined on the mixed bundle which is the direct sum of the distribution and its annihilator in the cotangent bundle (see Section 1.3). This flow allows to extend to the nonholonomic case the notion of exponential mapping.

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.

Let us turn to the subject matter of geometry. Our first acquaintance with geometry goes back to school years. School geometry (the geometry of the ancient Greeks) studies the various metrical properties of the simplest geometric figures, that is, basically finds relationships between lengths and angles in triangles and other polygons. Such relationships provide the basis for the calculations of the surface areas and volumes of solids. We would like to pay attention to the fact that the central concepts underlying school geometry are the following: the length of a straight line (or a curve) segment and the angle between two intersecting straight lines (or curves). The angle was always measured at the point of intersection of these lines.

We have already got used to the fact that many quantities are given as numerical functions of a point in space. For example, the distance from a point to a certain fixed centre, etc. If we have several such quantities at our disposal, we already have several functions of a point (or, so to say, the vector function of this point). In three-dimensional space, for a complete characteristic of the poisition of a point in space it is ncecessary, as is well-known, to know the values of at least three numerical functions called coordinates of the point (x 1, x 2, x 3): each of the coordinates x i is a function of the point, and the set (x 1, x 2, x 3) completely determines the point. The reader has already met with different types of coordinates, for example, in a plane there exist Cartesian coordinates x 1, x 2 and polar coordinates r, ф), where x 1 = r cosф), x 2 = r sinф); in space there exist Cartesian, cylindrical r,z, ф or spherical r,θ,ф) coordinates.

Publisher Summary This chapter presents a complex analysis on Riemann surfaces motivated by the operatorial string theory. Any multistring diagram determines one and only one meromorphic differential form. All associated algebras are isomorphic to the ordinary Heisenberg and Virasoro algebra. The Riemann analogs of the Fock spaces and Verma modules are isomorphic to the ordinary ones after topological closure. Therefore, there are two filtering structures and one almost-grading structure on the Riemann analogs of Heisenberg and Virasoro algebras. The chapter also presents the realization of the Fock spaces and Verma modules in semi-infinite exterior forms.

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 a previous paper [1], the author and Shmel’tser started the construction of an extended Lyusternik–Shnirelman–Morse theory for the study of single-valued and multivalued functionals on the space Ω(M) of losed directed curves in a manifold M. The authors applied these methods to the classical problem (Kirchhoffs problem) about the free motion of a rigid body in an ideal incompressible liquid, where the fluid flows potentially and is at rest at infinity. From a modern point of view, Kirchhoff-type equations are defined by a Hamiltonian H (coinciding with the classical energy) and Poisson brackets { , } for functions in phase space (the dual space L∗ of the algebra L of the group E(3) of motions of the Euclidean space R). In the classical Kirchhoff case, we have

In particular, the vacuum vector in the Fok representation is an example of a vector Φ0, although in quantum theory there arises a quite complicated algebraic aggregate composed of different Verma modules (cf. [1, 2]). The geometric approach of Polyakov et al., to the introduction of interactions in the theory of a string necessarily leads to complicated problems of the algebraic geometry of Riemann surfaces [3, 4]. However, the role of the Virasoro algebra in this approach is absolutely not apparent. The goal of the present paper is

A class of problems connected with the description of the motion of an attracted quantum particle in possibly time-dependent, periodic, external fields is studied on the basis of a development of the method of the inverse problem.

The present paper is the direct continuation of the authors’ preceding papers [1, 2], in which the realization of the program of successive operator quantization of “multiloop diagrams” in the theory of boson strings was started. At the base of all our constructions lies the following function theoretic construction on a nonsingular Riemann surface Γ of genus g 1, if the triple Γ, P± is typical; and if g = 1, then this assertion is true for all n 6= 1/2. The definition of the tensors f n given generalizes to half-integral λ, where it depends in addition on the spinor structure. The cases λ = −1, 0, 1/2, 1, 2 (of vector fields, functions, spinors, differentials, quadratic differentials) are the most important. For them we use the special notation f−1 n = en, f 0 n = An, f 1/2 n = Φn, f n = dω−n, f 2 n = d Ω−n. The tensors f n have the following important multiplicative property of “almost gradedness

dk with two simple poles at the points P±, with residues ±1 and purely imaginary periods with respect to all contours on . The real part of the corresponding integral (z) = Rek(z) is single-valued on and represents “time.” The level lines (z) = const represent the positions of the string at the present time. To the collection m = m+ + m of strings corresponds a Riemann surface with two collections of points P+,i,P ,j, i = 1,...,m+, j = 1,...,m with a dierential dk, with real residues c+i,c j at all points P±, c+ > 0, c < 0, and purely imaginary periods on . In exactly the same way the function (z) = Rek(z) is single-valued and plays the role of “time”. As ! ±1 the contours T = const split into free strings. The connected components concentrated near the points P± play the role of asymptotically free “in” and “out” strings. In [1] a rich collection of algebraic objects connected with this picture for m = 1 was constructed, which for genus g = 0 reduce to the theory of the Virasoro algebra and its representations. In the present paper we demonstrate that these algebraic forms arise in the process of quantization of strings on such algebrogeometric models, “diagrams”. To the asymptotic “in” and “out” states correspond the ordinary Fock spaces of free strings—small contours T = const near the points P±. The global algebrogeometric objects on a surface with distinguished points P±. permit one in principle to trace the whole course of the interaction. The algebrogeometric objects in the theory of Polyakov, Belavin, Knizhnik, etc., of strings in Euclidean space-time, as is known, lead to problems on the space of moduli of Riemann surfaces. The algebraic forms we introduce do not appear in this theory. But to the minds of the authors the approach we develop relates the algebrogeometric theory of strings with traditional ideas of operator quantum theory of strings in Minkowsky space and lets us use the mathematical techniques of the method of finite-zone integration in the theory of solitons.

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 develops an algebrotopological approach to the problem of effective selection of real finite-zone solutions of the sine-Gordon equation, which uses the socalledγ-representation on the Riemann surface, in which “action” variables can be computed explicitly. This approach is general and applies to many systems for which the reality problem has not yet been solved.

The Kadomtsev–Petviashvili (KP) equation was first derived in [5] as a physically natural two-dimensional analog of the well-known KdV equation; it arises in the study of solitons and other KdV solutions which are subject to slow perturbations in the direction transverse to that of the main wave. As a physical model, the KP equation has the same degree of universality as the KdV. The KP equation has a Lax commutator representation (see 3, 4])

A surface in a 3-dimensional space is the simplest object having its own, internal or “intrinsic,” geometry. What exactly do we mean by this?

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

Let OD denote a region with piecewise smooth boundary D, of the Euclidean space I1’ with Euclidean co-ordinates x1,..., xn. Consider the linear space F of smooth vector-functions f (x1,..., xn) = (f1,..., fk) defined on D, i.e. with domain D. Let L(x β;p j ;q a i )be a smooth real-valued function of the three arguments xβ, 1 ≤β ≤ n; p j , 1 ≤ j ≤ k; q a i , 1 ≤ i ≤ k, 1 ≤ a ≤n (making altogether n + k + nk real arguments); we call such a function a Lagrangian, and from a given such Lagrangian we construct a functional I[f] defined on F, as follows: $$I\left[ f \right] = \int_D {L\left( {{x^\beta };{f^j}\left( {{x^\beta }} \right);f_{{x^\alpha }}^i\left( {{x^\beta }} \right)} \right)} d{x^1} \wedge \cdots \wedge d{x^n}$$ where the integral is the multiple integral (see §26) over the region D (which we shall later assume to be bounded), and where \(f_{{x^\alpha }}^i\left( {{x^\beta }} \right) = \left( {\partial /\partial {x^\alpha }} \right){f^i}\left( {{x^\beta }} \right)\).

We consider a translation invariant evolution equation of the form $$ \dot u = K[u],({u^1}; = \frac{{\partial u}}{{\partial x}},\dot u = \frac{{\partial u}}{{\partial x}})$$ (10.1) where u is a vector-function of a continuous variable x or of a discrete variable n; the equation (10.1) is invariant with respect to translations in x (or n).

Let M and N be two smooth (for simplicity of class C∞) manifolds and let f: M → N, be a smooth map between them.

According to the simplest of the definitions of an orientation on a manifold given above (see Definition 1.1.3), a manifold M is oriented if the local coordinate systems x α j given on the members U j of a covering collection of local co-ordinate neighbourhoods (or charts) for M, are such that the transition functions from one local co-ordinate system to another on the regions of overlap U j ∩ U k , have positive Jacobian: $$\det \left( {\frac{{\partial x_j^\alpha }}{{\partial x_k^\beta }}} \right) > 0.$$