Program of the 2nd year of Master for 2005-2006 :
Solitons and Integrable Systems
V.B. Matveev, M. A. Semenov-Tian-Shansky
***********************************************************************************************************************
The theory of
integrable systems
represents a real breakthrough in the development of mathematics at the
end of
the 20th century. After the striking discovery, by Gardner, Green,
Kruskal and
Miura, of an unexpected link between the Korteweg - de Vries equation
in
non-linear hydrodynamics and Scattering Theory in Quantum Mechanics,
the new
approach to the study of partial differential equations, called CISM (Classical
Inverse Scattering Method), enjoyed a rapid expansion and allowed
to discover
new and unexpected connections between different branches of
mathematics. The
theory of integrable systems found various applications in physics,
ranging
from hydrodynamics and non-linear optics to astrophysics and elementary
particle theory. In the last decades it has also brought about many
importantant
discoveries in pure mathematics, such as a solution of Schottki problem
in
algebraic geometry, the creation of quantum group theory, the discovery
of new
knot invariants, or Donaldson's theory in the low dimensional
differential
topology. In its modern version, the theory of integrable system
reunites
powerful analytic methods (Riemann-Hilbert problem, Riemann's theta
functions)
with the algebraic methods of Lie theory (especially in infinite
dimension).
The program for the second year of the Master "Mathématiques
Approfondies" at Burgundy University, for the academic year 2005-2006,
is
conceived as a solid introduction to the theory of integrable systems
which
will allow students not only to learn the basic elements in the field
but also
to be able to tackle open problems. The program might be modified
according to
the mathematical background of the students attending the course.
During the first semester a certain amount of time will be devoted to
revisions
of the basic concepts needed. The first term will consist of three
courses:
I- Révisions et compléments
d'Analyse (revision and
complements of analysis) (30 hours
and 6 CE)
II- Révisions et
compléments de
Mathématiques Générales (revision and complements of general
mathematics)(30 hours and 6 CE).
III- Compléments de théorie
des fonctions
analytiques (Advanced
topics of the theory of analytic functions)
(V.B. Matveev, 20 hours and 8 CE).
***********************************************************************************************************************
The program of the course
Advanced topics of
the theory of analytic
functions
will acquaint the students with the analytic tools
necessary for the second part of the MMA2 and, at the same time, will
be useful
to prepare the "agrégation" exam. The program will include:
Infinite products and the Weierstrass
theorem. Entire functions ; meromorphic
functions and the Mittag-Leffler theorem.
Riemann surfaces and analytic
multi-valued functions.
Conformal maps.
Asymptotical methods: Laplace's method,
stationary phase method, saddle point method.
Linear differential equations with
rational coefficients in the complex plane.
Special functions I: Euler’s Gamma
function; Stirling's formula.
Airy's function.
Theta functions and elliptic functions.
Cauchy integral and the Riemann-Hilbert
problem I.
***********************************************************************************************************************
The course « Solitons and Integrable System » for the second semester will be split into two
distinct
and largely independent parts
***********************************************************************************************************************
The first part UE4 :
Solitons
and Integrable System: Analytic Approach
(V.B. Matveev, 40h
and 10 CE)
will be devoted to the analytic aspects of
the theory,
from the Inverse Scattering Problem to the methods of algebraic
geometry and of
the isomonodromic deformation theory. It will also serve as a brief and
useful
introduction to the geometry of algebraic curves and to the
Riemann-Hilbert
problems. A list of subjects discussed in this first part of the course
is as
follows:
* Integrable systems associated to Schrödinger's and Dirac's
differential
operators on the line. Lax representation for the Korteweg - de Vries
and for
the non-linear Schrödinger equations.
* Elements of the Quantum Scattering
Theory.
Jost solutions. Reflectionless potentials.
* Darboux transformations. Elementary constructions of multisoliton
solutions.
* Integrals of motion for integrable PDE and trace formulae.
* Schrödinger's operator with periodic potential. Finite band
potentials and algebraic
curves.
* KP equation and its solitons.
* Analysis on compact Riemann surfaces; genus, canonical cycles and
meromorphic
differentials ; abelian integrals and their periods; period matrix.
* Riemann-Roch theorem and Riemann's inequality.
* Abstract theta functions. The Riemann theta function and Jacobi
inversion
problem.
* Solutions of the Korteweg de Vries equation and hyperelliptic curves.
Baker–Akhiezer
fonctions. Integration of the Kadomtsev–Petviashvily equation.
* Weierstrass points of algebraic curves and the classes of solutions
of the KP
equation which reduce to solutions of the Boussinesq equation.
* The Schottki problem. Classical Schottki groups and the
parametrisation of
the nonsingular real solutions of the KP equation.
* Matrix Baker–Akhiezer functions. Integration of the Sine-Gordon
equation and
of the non-linear Schrödinger equation.
* Solitons as degenerate algebro-geometric
solutions.
* Lax pairs with variable spectral parameter. Einstein's equation in
vacuum
with axial symmetry. Applications of the algebro-geometric formalism to
General
Relativity.
* Isomonodromic deformations, Riemann-Hilbert problem and applications
to integrable
systems; Schlesinger systems and their connections
with the Riemann-Hilbert problems. Schlesinger systems which are
integrable in
terms of theta functions. Painlevé equation.
***********************************************************************************************************************
The second part of the course UE5:
Solitons
and Integrable System: Algebraic Approach
(M.A.
Semenov-Tian-Shansky, 40h et 10
CE)
will discuss the hidden symmetry of integrable
systems and
their relations with Lie groups and Lie algebras. The basic notions
which yield
a key to the geometric understanding of integrable systems are the
classical r-matrices
and the classical Yang-Baxter equation. These notions are very rich
from an
algebraic point of view and allow to treat in a unified way a vast
number of
examples, ranging from the equations of classical mechanics studied in
the 19th
century to the nonlinear partial differential equations (like
Korteweg-de Vries
or Sine-Gordon equations). They are also fundamental in the
semi-classical
study of quantum groups, thus giving a link between the classical and
quantum
theories of integrable systems.
The program of this part of the course is summarised here:
* Introduction: Symplectic manifolds; Poisson manifolds, symplectic
leaves and
stratification of Poisson manifolds.
* Lie–Poisson brackets and the theory of coadjoint orbits of Lie groups.
* Reduction theory for Poisson manifolds.
* The theory of r-matrices and the factorisation theorem.
* Integrable systemes associated to finite dimensional Lie algebras.
Toda
lattice.
* Loop algebras and Lax representations with spectral parameter.
* Riemann–Hilbert problem and factorisation in loop groups;
Birkhoff–Grothendieck
theorem.
* Lax equations with spectral parameter and algebraic geometry;
linearization
theorem.
* Examples of integrables systems associated to affine Lie algebras;
periodic
Toda lattice; the integrable cases of solid body motion.
* Zero curvature equations and central extensions of loop algebras;
classification of their coadjoint orbits.
* Construction of local integrals of motion.
* Dressing transformations and homogeneus spaces of loop groups
associated to
integrable systems .
* Generalised Korteweg-de Vries equations; Drinfeld-Sokolov theory.
* Poisson–Lie groups and zero curvature equations on lattices. Dressing
transformations as a non-linear version of the theory of coadjoint
orbits.
* Introduction to quantum group theory. Quantum R-matrices; Yang–Baxter
equation; the role of Hopf's axiom ; q-deformed enveloping
algebreas.
The program will give the students an introduction to the current
research
subjects. According to the their taste, students may pursue the study
of the
more abstract aspects of the theory, or choose its more practical
applications,
for instance in non-linear optics, condensed matter physics or
astrophysics.
BIBLIOGRAPHY
* D. Mumford. Tata lectures on thêta, vol 1,2, Basel Birkhauser
(1983-1984).
* V.B. Matveev, M. Salle. Darboux transformations and Solitons.
Springer-Verlag
ser. Studies in Nonlinear Dynamics, 1991.
* E. Belokolos, A. Bobenko, A. Its, V. Enolskij and V.B. Matveev.
Algebro-geometrical Approach to the Nonlinear Integrable Systems,
Springer-Verlag, 1994.
* L. Faddeev and L. Tahtadjian. Hamiltonian Approach in the Theory of
Solitons,
Springer-Verlag, 1986.
* A.G. Reyman and M.A. Semenov-Tian-Shansky. Group-theoretical methods
in the
theory of Integrable systems. II. In : Encyclopaedia of Mathematical
Sciences,
16. Springer-Verlag, Berlin, 1994. vi+341 pp.
* N. Bogolubov, A. Izergin and V. Korepin. Correlation functions of
Integrable
systems and quantum inverse methode. Cambridge Univ. Press 1994.
* S. Novikov, I. Gelfand, I. Krichever, B. Dubrovin and V.B. Matveev.
Integrable systems. London Math. Soc. Lect. notes
v.
61, Cambridge Univ. Press 1981.
* S. Novikov, S. Manakov,V. Zakharov and L.
Pitaevskij. Theory of Solitons. Academic Press 1982.
* E. Lieb and D. Mattice. Mathematical Physics in one dimension. New
York-London : Academic Press, 1966.
* O. Babelon, D. Bernard and M. Talon. Introduction to Classical
Integrable
Systems. Cambridge Monographs of Mathematical Physics, Cambridge
University
Press, Cambridge 2003.
* Yu. B. Suris. The problem of integrable discretization : Hamiltonian
approach. Basel : Birkhäuser, 2003. 1070 pp. Progress in Mathematics,
Vol. 219.
|
|
Reourn to the MMA2 homepage (English version) |
Version |
