In this talk, I present an overview of the algebro-geometrical and
dynamical characterization of integrable systems whose generic invariant
manifolds can be completed to nonlinear strata of hyperelliptic Jacobians
(hyperelliptically separable systems).
I consider analogs of the Jacobi-Mumford systems (coordinates and vector
fields) on the natural fibering over the base formed by genus g hyperelliptic
curves G whose fibers are n-dimensional strata Wn of Jac (G) (n £ g). At any point D of Wn, n independent invariant
vector fields can be constructed which span the tangent space of Wn at D.
The vector fields I construct on Wn are restrictions of the
usual Jacobi-Mumford ones on Jac(G). It turns out that
h.s.s. may be viewed as constrained a.c.i. integrable Hamiltonians
according to the Dirac formalism.
I apply the above construction to two families of examples: the generalized
Henon-Heiles and the Neumann hierarchies.
Finally, I recall our results on the Kowalevski-Painlevé analysis for
hyperelliptically separable systems estimating number and leading behaviour of
their formal singular solutions.
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The domain structure of a potential and solenoidal field on a plane is
shown to have a symmetry of the Kleinian group. It allows to build a
classification of domain structures for the superconductors and magnetics
and explain their main properties. Different types of the domain branching
on a plane boundary of superconductors and magnetics are described. Possible
generalization of the theory is considered with an aim to take in account the
more general types of fields, the symmetry of crystal lattice, the
3-dimensionality of space, nonlinearity of a thermodynamical functional
and other types of boundaries.
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I shall explain how the classical phase space M of the rational
(complexified) Calogero-Moser system can be realized as an orbit
in the space of ideal classes of the Weyl algebra A under the
natural action of the automorphism group G = Aut(A). Due to
Shafarevich, one can provide G with several natural structures of
an infinite-dimensional algebraic group. It turns out that a proper
choice of such a structure makes M into a coadjoint orbit in the
dual of (a central extension) of Lie(G). The talk is based on joint
work with G. Wilson and related results of V. Ginzburg.
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Various connections between integrable systems and functional equations
will be surveyed. A new characterisation of the Calogero-Moser models will be given.
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Some recent ideas will be presented including progress
towards a universal approach for demonstrating the
integrability of Calogero-Moser models based on root
systems or, more generally, the Coxeter reflection
groups.
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We construct two basic Q-operators for general non-homogeneous
Heisenberg spin chain as integral operators acting in
the space of polynomials.
We prove the commutativity of obtained Q-operators, the
Wronskian relation and derive the Baxter equation for them.
We derive several different representations for
obtained integral operators and investigate
two limiting cases: the homogeneous spin chain and the limit
to the DST-model.
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An orbit of the Kowalevski top is generic if the
corresponding curve has genus 2 and the corresponding
real torus is nondegenerate, non-resonant, and has twist.
An orbit violating any of these conditions is called
special. We give an overview of all special motions
in the Kowalevski top and discuss some of their
interesting properties.
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The dynamics of this third integrable case of classical rigid body theory
is presented on several levels of abstraction, starting from real motion of
physical model and an analogous computer simulation of the Kowalevski
equations of motion. The first abstraction involves the separation of a
cyclic angular variable, i. e., the transition to a reduced description,
and the introduction of a six-dimensional (gamma-iota)-phase with two
Casimir constants. In the next step, relative equilibria are used to
identify bifurcations between different topological types of three-dimensional
energy surfaces. The third level is concerned with the foliation of these
energy surfaces by invariant tori, and the identification of critical tori
which mark bifurcations in the tpe of foliation. The tori are shown in
various 3D projections and in homeomorphic deformations of the energy
surfaces.
The final step in the analysis uses the technique of Poincaré surfaces of
section. A comprehensive survey on all possible motions is given in terms
of animation series where all Poincaré sections for a given energy are shown
in succession.
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A generalization of the Weiertrass theory of elliptic functions to
the case of higher genera curves of the form yn-xs+ lower
order terms = 0, n,s are coprime is developed. The higher genera
Ã-functions are introduced as second logarithmic derivatives of the
multidimensional s-function, which is automorphic with respect to
the action of symplectic group. The principal result is the construction
of (n+g)×(n+g)-matrix, g = (n-1)(s-1)/2 - genus of the curve
which minors are generating (i) to solve Jacobi inversion pronlem
(ii) to describe Jacobi and Kummer varieties as algebraic varieties
in projective space (iii) to decribe integrable hierarchies associated
with the curve. The theory is examplified by the cases of hyperelliptic
and trigonal curves. Rational analogies of such Abelian functions are
described in terms of Schur functions constructed by the Weierstrass
gap sequence.
V.M.Buchstaber, V.Z.Enolskii, D.V.Leykin, Kleinian function, hyperelliptic
Jacobians and applications, Reviews in Mathematics and Mathematical
Physics, (S.P.Novikov and I.M.Krichever eds.) 10:2, p 1-125, 1997.
V.M.Buchstaber, V.Z.Enolskii, D.V.Leykin, Rational analogues of Abelian functions,
Funct.Anal.Appl. 33:2 (1999), 1-15
J.C.Eilbeck, V.Z.Enolskii, D.V.Leykin, On the Kleinian construction of
Abelian functions of canonical algebraic curves, CMR Proceedings and
Lecture Notes, 1999
V.M.Buchstaber, V.Z.Enolskii, D.V.Leykin, Abelian functions of
(n,s)-curves, in preparation.
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The bundle of conformal blocks is a bundle over the moduli space
of complex curves of arbitrary genus
with marked points. It is endowed with a natural flat connection, the KZB
connection. These objects arise naturally in Conformal Field
Theory.
We present a functional realization of this
bundle, which allows to write explicit formulas for
the KZB connection. We also present integral formulas for flat sections of
this connection.
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We present a geometrical scheme suitable to study the separation
of variables problem for bi-Hamiltonian systems. In particular, we discuss
the case of systems with an arbitrary number of independent Lenard chains.
We show how they can be integrated by separation of variables in
Darboux-Nijenhuis coordinates on the symplectic leaves of a suitable
element of the Poisson pencil.
Finally we give examples of these constructions concerning Toda-like
systems and polynomial pencils of matrices.
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Fritz Kötter, a contemporary of Sophie Kowalevski,
found theta-functional solutions for the Clebsch and
Steklov-Lyapunov integrable cases of dynamics, as well as for the
Kowalevski top. He obtained quadratures containing Abelian integrals on
hyperelliptic curves and, as a matter of fact, Darboux coordinates
represented by divisors on such curves.
On the other hand, according to recent studies, the above systems are
linearized on Prym subvarieties of nonhyperelliptic Jacobians.
The aim of the talk is to highlight the algebro-geometric background of
Kötter's enormous computations for the Clebsch and Steklov systems and
to extend it for solving their higher-dimensional integrable
generalizations.
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We consider an ordinary differential reduction
of the gas-dynamical equations proposed by
Ovsiannikov (1965) and by Dyson (1968),
representing a tri-axial ellipsoidal gas cloud
rotating as it expands into the vacuum.
For a monatomic gas (g = 5/3)
without vorticity, the system has the Painlevé
property and is integrable, at least in cases of rotation
around a fixed axis. We also present preliminary results
concerning fully general states of rotation.
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For a finite reflection group G there is a rich
theory developed by Dunkl, Heckman and Opdam leading
to the notion of a commuting set of Bessel differential
operators. These systems play an important role in the
study of Calogero-Moser systems and other problems
of physical interest.
When G acts on the real line one recovers
the usual Bessel function with a well known power series
expansion at the origin. We obtain some such expansions
in the case of G = A2 acting in the plane and we use these
to produce plots of some of these functions.
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We show that maximal rank 1 commutative rings of difference operators
can be systematically constructed from their differential analogues by
making an appropriate shift in the variables of the tau function of the
KP hierarchy. When the spectrum of the ring is a unicursal curve, the
operators in the ring enjoy a bispectral property reminiscent of the
familiar
bispectral property satisfied by the classical orthogonal polynomials.
As an illustration, the tau functions leading to the rings
with spectral curves y2 = r2K+1(r+1), K = 1, 2, 3, ...
are explicitly characterized.
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In this work we show the essentials of the theory of Lie symmetries
(point and generalized) of discrete equations. We show that, at least
in certain examples, the continuous limit from a difference equation to a
differential one leads to a contraction of the corresponding Lie algebras
of symmetries, L(D) and L(d). A subset S of elements of
L(D) contracts to a subalgebra of point symmetries Lp of L(d).
The subset S Ì L(D) does not form a Lie algebra, and contains
both point symmetries and generalized symmetries of the discrete equation.
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We report on a relationship between the soliton cellular
automata (ultradiscretization of the discrete KP equation)
and the crystal base (ground state of the integrable vertex
model).
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Similarity reductions of the Hirota-Satsuma system and another
gauge-related system yield non-autonomous Hamiltonian systems with
quartic potentials. We present classes of special solutions and
Bäcklund transformations which are interpreted in terms of the
action of an affine Weyl group on the space of parameters. We show how
separation of variables also has an application in this context.
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We exhibit the elliptic
Calogero-Moser system as a Hitchin system of G-principal
Higgs pairs. The group G, though naturally associated to any root
system, is not semi-simple. We then interpret the Lax pairs with spectral
parameter of d'Hoker and Phong and Bordner, Corrigan and
Sasaki in terms of equivariant embeddings of the Hitchin system of G
into that of GL(N).
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Using the Riemann-Hilbert approach, the large x asymptotics of the Fredholm determinant,
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The isomonodromy approach to the scaling limits in Painlevé transcendents
allows us to describe many charming features of these nonlinear special
functions. Among them we mention the discriminant
sets for the parameterized scaling limits, families of reductions
of the Painlevé equations to each other, interference of the
isomonodromy and Hamiltonian structures of the Painlevé equations,
as well as applications of such limits to the random matrix theory.
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We present general results and particular examples of
the non-Schlesinger isomonodromy deformations
for 2×2 and 3×3 matrix Fuchsian ODEs.
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We use the solution of the quantum inverse problem to calculate the quantum
correlation functions of the XXZ spin chain in the framework of the
algebraic Bethe Ansatz. This method permits to obtain multiple integral
representations for the correlation functions of the XXZ model in an
external constant magnetic field in all the regimes.
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We present a review of some results on Kowalevski's top (KT) in classical and quantum mechanics. The following items are considered:
Unsolved questions will be also discussed.
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Invited 30' historical talk on the work and life of
Sophie Kowalevski.
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We propose a formulation of the quantization problem of Poisson homogeneous
spaces by algebra-modules over the Hopf algebra quantizing a Poisson Lie
group. Solutions of this problem are obtained using quasi-Hopf algebras
quantizing Manin pairs.
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An integral representation for the eigenfunctions of the quantum periodic
Toda chain is constructed for the N-particle case. The multiple integral
is calculated using the Cauchy residue formula. This
gives a representation which reproduces the particular results
obtained by Gutzwiller for the N = 2,3 and 4-particle chain. Our
method to solve the problem combines the ideas of Gutzwiller and the
R-matrix approach of Sklyanin with the classical results in the theory
of the Whittaker functions. In particular, we calculate Sklyanin's invariant
scalar product from the Plancherel formula for the Whittaker
functions derived by Semenov-Tian-Shansky, thus obtaining a natural
interpretation of the Sklyanin's measure in terms of the Harish-Chandra
function.
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A general approach to the computation of exact correlation
functions of lattice quantum integrable models proposed recently is
reviewed. It is based on the actual resolution of the quantum inverse
scattering problem that we have achieved for almost all known quantum
integrable lattice models, including the so-called fundamental models,
the fusion models and also models with impurities. The application of
this method to the XXZ Heisenberg spin-1/2 magnet in a magnetic field leads
to multiple integral representations of the n-point correlation functions
in the thermodynamic limit. For zero magnetic field, this result agrees,
in both the massless and massive (anti-ferromagnetic) regimes, with the
one obtained from the q-deformed KZ equations (massless regime) and
the representation theory of the sl2 quantum affine algebra together
with the corner transfer matrix approach (massive regime) by Jimbo, Miwa and
their collaborators.
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Several curves of genus 2 are known such that the equations
of motion of the Kowalevski top are linearized on their Jacobians.
One can expect from transcendental approaches via solutions of equations of
motion in theta-functions that their Jacobians are isogeneous.
The paper focuses on two such curves: Kowalevski's and that
of Bobenko-Reyman-Semenov-Tian-Shansky, the latter arising from
the solution of the problem by the method of spectral curves.
An isogeny is established between the Jacobians of these curves by purely
algebraic means, using Richelot's transformation of a genus 2 curve.
It is shown that this isogeny respects the
Hamiltonian flows. The two curves are completed into an infinite
tower of genus 2 curves with isogeneous Jacobians.
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The Lax pair representation found by Reyman and Semenov-Tian-Shansky
can be obtained directly using the r-matrix method. A standard construction can
then be
used to give a bi-Hamiltonian description of the Kowalevski gyrostat in two
fields.
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Two (Riemannian) metrics on
the same manifold are geodesically equivalent, if
they have the same geodesics, considered as unparameterized curves.
I will show that in this case the Beltrami-Laplace operator of
each metric admits n
commuting differential operators of second order,
where n is the dimension of the manifold.
These "quantum integrals" are independent, if the
metrics are in general position at least at one point of the manifold
general position ( = the eigenvalues of one metric with respect to another
are different).
In particular, let two metrics on one connected manifold
be geodesically equivalent
and be in general position at least at one point of the manifold.
Then the geodesic flows of the metrics are completely
integrable.
The converse is also true. If the Beltrami-Laplace operator of a metric
admits sufficiently many independent commuting differential operators of
some special
form, then the metric
admits a geodesically equivalent one.
As an application, I will show that if
a connected closed manifold admits two geodesically equivalent
metrics (being in general position at least at one point of the
manifold), then the topology of the manifold can not
be very complicated.
The current version of this is that the manifold can be covered by the
product of spheres.
For example, if the dimension is two, and the
manifold is orientable, then it is either the Torus or the Sphere.
These commuting operators are elliptic. If the manifold is closed
then they are self-adjoint. In particular,
if the metrics are in general position, we can globally separate the
variables in the equation on eigenvalues of the Beltrami-Laplace
operator of the metric g: the equation on eigenfunctions of the
Beltrami-Laplace operator is equivalent to a system of 1-dimensional
Schroedinger equations.
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We classify all values of the parameters a,
b, g and d of the Painlevé VI equation such that
there are rational solutions. We prove that all rational solutions belong
to one-parameter families of classical solutions of PVI. Moreover
we characterize geometrically classical and rational solutions of
Painlevé VI equation via their auxiliary linear monodromy data.
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Given a finite matrix ensemble, equipped with a natural
probability distribution, what is the statistics of its spectrum? In this
lecture, I obtain nonlinear PDE's for the probability that the spectral
points all belong to a union of intervals. For a single semi-infinite
interval, the equations reduce to Painlevé-type equations.
The point is that certain matrix integrals (over Lie algebras) form the
tau-functions of certain integrable lattices, arising in the context of
affine Lie algebra splittings, in the same way that the Kowalevski top
appears in such a context. However, the matrix integrals solutions are
quite different from the mechanical solutions; indeed, they satisfy an
extra-set of equations, which forms a subalgebra of an appropriate Virasoro
algebra. Combining the integrable lattice equations with the Virasoro
constraints leads to the PDE's above. Similarly, integrating over Lie
groups leads to interesting questions about random permutations. They again
are intimately related to special solutions to integrable lattices.
The main purpose is to give an overview of this exploding and fascinating
subject.
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The two quadratically-coupled PDEs of optical cascading are known to
possess many classes of explicit travelling wave solutions, both `vector
soliton' solutions expressible in terms of hyperbolic
functions and many classes of periodic solutions expressed in terms of
Jacobian elliptic functions. By seeking instead solutions in terms of
the Weierstrass Ã-function, it is shown that the search procedure
i) may be made more compact, ii) reveals connections between
previously distinct classes and iii) may use an extended ansatz
which reveals new classes of explicit periodic solutions.
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I will discuss how the geometry of certain line bundles with symmetries on
abelian surfaces is used to recover elements of known integrable systems
such as the defining equations in suitable basis and vector fields.
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We obtain an explicit solution of the Boussinesq equation
which evolves according to a 5-particle Calogero-Moser-Krichever
system. By using Halphen's Ansatz and Kleinian abelian
functions, we derive algebraic equations for the
reduction of the time flows. In particular we check a posteriori
that this solution is a co-elliptic (as well as elliptic)
soliton, in the sense that the first two KP time flows remain
tangent to a sub-abelian variety of codimension 1.
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We shall present a new theory of quasipotential Newton equations of the
form: acceleration=(matrix)(gradient of a function). Potential systems
correspond to the identity matrix. Paricularly interesting are
quasipotential equations admitting quadratic in velocities integrals of
motion. Such systems appear to be completely integrable in a somewhat
nonstandard sense (by embedding). All such equations are completely
characterised by a certain Poisson pencil or by a set of second order
linear partial differential equations called Fundamental Equations. A
subclass of triangular systems can be completely solved by quadratures by
separating variables in new type separation variables corresponding to
non-confocal quadric surfaces.
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A characteristic integrable system on a factorizable
Poisson Lie group is a Hamiltonian integrable system whose
phase space is a symplectic leaf of this Poisson Lie group.
The Hamiltonian of such system is given by an adjoint invariant
function on the group. It will be shown that such systems are
integrable for simple Poisson Lie groups with standard Poisson
Lie structure. Toda systems and their relativistic counterparts
are examples of such systems whose integrability is well known.
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The Kowalevski top is certainly one of the most fascinating integrable systems
in classical mechanics, yet it has received conspicuously little attention among physicists.
Why is this so? Four possible reasons will be addressed in this talk.
1. A heavy rigid body with SO(3) configuration space may not even exist in the real world:
some device is needed to keep one point fixed. In the setup that will be presented for demonstration,
this is done with a Cardan suspension. But then the configuration space is T3, and
the bifurcation diagram is seriously altered. How?
2. Any slight deviation from the Kowalevski parameter set (as for example the Cardan frame)
leeds to chaotic rather than integrable motion. This problem has not systematically been attacked.
In the general case with T3 dynamics, the system cannot even be reduced to two degrees
of freedom.
3. The sheer complexity of the Kowalevski system may have put it beyond physicists' grasp.
The best way to alleviate this problem is careful visualization. An attempt in this direction will be
presented in terms of a movie which was produced together with H.R. Dullin and A. Wittek.
4. It has been notoriously difficult to determine the action variable representation of isoenergy
surfaces, as a basis for perturbation analysis and semiclassical quantization. This problem
has essentially been solved recently.
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The correspondence between geometry of surface with
constant curvature and soliton equations has been well
known. Recently, this was generalized to study a discrete
analogue of differential geometry in connection with
discrete soliton equations. Among other things the
conjugate net has been recognized being a key concept
in characterizing discrete integrable systems, hence
the discrete geometry.
On the other hand, the discrete soliton equations are
solved by correlation function of strings in particle
physics. This talk is addressed to the question, what
is the conjugate net of strings. When the space-time
is compactified, the associated momenta of particles
are quantized to discrete numbers. I will explain that
the correlation function turns out to satisfy discrete
Laplace equation and the sum of components of each
particle momentum parametrizes the conjugate net.
Some connection of the discrete geometry with geometric
quantization will be also discussed.
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Model potentials for quantum dots, realistic in the whole range of energy,
are introduced starting from the integrable motion of a particle on a sphere
under the action of an external quadratic field. We show that in the case of
rotational invariant potentials, the associated 2D Schroedinger equation
has exact zero energy eigenfunctions in terms of which the whole discrete
spectrum can be characterized.
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The talk is based on the joint work with A.L. Pirozerski. The universal DS
reduction which was introduced a few years ago by
B. Khesin and F. Malikov is based on the use of the so-called algebra of
complex size matrices defined by B. Feigin. In the present
talk a q-deformed version of this construction is described, which
interpolates between different q-deformed classical W-algebras.
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For a generic quantum integrable sl(2)-invariant magnetic spin
chain a Q-operator is constructed as a trace of a monodromy matrix.
A variety of formulas describing the Q-operator are obtained,
including explicit formulas for the matrix elements of Q in a polynomial
basis. The Q-operator is a quantum analog of Bäcklund transformation.
It shares the common eigenfunctions with the commuting Hamiltonians, and
its eigenvalues satisfy certain finite-difference equation (Baxter's
equation) allowing to determine the spectrum of the Hamiltonians.
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A new third order equation, with unknown function being an N-component vector
on the sphere, is constructed. If N = 3 it coincides with the third order flow for
the anisotropic Landau-Lifschitz model. The spectral parameter in the Lax representation
for the general N-component model belongs to an algebraic curve of genus
g = 1+(N-3)2N-2.
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We discuss the general scheme of appearance
of Euler-Poincare equations on the semidirect product
algebras in the Lagrangian dynamical systems, both with
the continuous and the discrete time. The concepts are
illustrated with the motion of the Lagrange top and a
rigid body in a quadratic potential and with integrable
discrete time versions of these systems.
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It is shown that the Wess-Zumino-Witten model on elliptic
curves degenerates to a system of differential equations on
the Cartan subalgebra, when the level is equal to -(the dual
Coxeter number). This system would be interpreted as the
Gaudin limit for an IRF type lattice model. We construct its
Bethe type eigenvectors, using the bosonisation technique.
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In this talk I want to explain the geometric meaning of the mysterious
notion of ``spectrality'' for Bäcklund transformations, a notion which
was introduced recently by Kuznetsov and Sklyanin. As an application
of our point of view it will be shown that we can actually construct
Bäcklund transformations for a.c.i. systems in a very systematic
way. I will give a different construction for systems of Lax type and
I will show how both methods are related through the method of
separation of variables. Finally it will be pointed out in which way
our Bäcklund transformations lead to discrete a.c.i. systems.
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From the Bäcklund transformation obtained by singularity analysis,
we derive the permutability theorem for the fifth order
Kaup-Kupershmidt equation. This proof allows us to give
a simple explanation to the ``anomalous structure'' of the N-soliton
solution recently discussed by A. Parker [Physica D 137 (2000) 25-33,
34-48].
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The integral formulas and Picard-Fuchs equations for the
actions of the Kowalevski top and a special role
of the Kowalevski variables in this story will be discussed.
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We propose a new star product which interpolates the Berezin
and Moyal quantization. This product is associative only if the
manifold is flat. A multiple of this product is shown to
reduce to a path-integral quantization and associativity in the case
of Kaehler manifold is recovered in the continuous time limit.
In flat space the action becomes the one of free bosonic strings.
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The fundamental theorem of Cauchy and Kowalevski says that any
`formal' power series solution of initial value problems must be
convergent, i.e., the solution is actually a `real' one. In another
fundamental work, Kowalevski used the formal Laurent series analysis to
discover her famous top. The work was later formalized to become the
Painlevé test. However, for a long time there has been no `singular
version' of the theorem of Cauchy and Kowalevski that gurantees the
formal Laurent series is convergent.
In this talk, we explain that validity of the singular version of
Cauchy-Kowalevski theorem is equivalent to passing the Painlevé test.
This is a consequence of the fact that passing the Painlevé test is
equivalent to regularizing the equation at movable singularities. We
will discuss the implications of this fact and also the extra structures
for Hamiltonian systems.
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For a complex n-dimensional locally symmetric orbifold M
uniformized by the complex n-ball Bn, there is a system E of
linear differential equations on M (of rank n+1,
which is called the uniformizing equation) such that a set
u0,...,un of linearly independent solutions
gives the developing map from M to Bn.
For an n-orbifold uniformized by the symmetric domain of type IV (a domain
of a hyperquadric), there is also the uniformizing equation (of rank n+2).
We are interested in examples of such M of which uniformizing equations
E can be explicitly known. I will describe an example which we found
recently; our M is the moduli space (4-dimensional) of cubic surfaces.
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