23 Jul - 19 Dec 2001

**Organisers**: Professor JC Eilbeck (*Heriot-Watt*), Professor AV Mikhailov (*Leeds*), Professor PM Santini (*Rome*), Professor VE Zakharov (*Moscow*)

Professor PM Santini

** Scientific Background**

**Structure of the Programme**

**Workshops and Conferences**

**Outcome and Achievements**

Many
natural systems can be modelled by partial differential equations (PDEs),
especially systems exhibiting wave-like phenomena. Such systems often have
quantities that are conserved in time, common examples being energy or momentum.
Often such systems are nonlinear; small changes in input can produce large
changes in output, or vice versa. Mathematically, such nonlinearities make such
systems difficult to study except using computer simulations.

Rather
surprisingly, relatively sizable classes of nonlinear systems are found to have
an extra property, integrability, which changes the picture completely.
Integrable systems have a rich mathematical structure, which means that many
interesting exact solutions to the PDEs can be found. Although important in
their own right, these systems form an archipelago of solvable models in a sea
of unknown, and can be used as stepping stones to investigate properties of
\nearby" non-integrable systems.

A
typical feature of integrable nonlinear wave equation is the existence of multi-soliton
solutions, where a soliton is a stable solitary wave. The velocity of the
soliton depends on its amplitude, so a bigger soliton can overtake a smaller
one: the surprise is that after the collision the two waves separate with
unchanged form, except for a phase shift.

Now
that properties of most of the “standard" physical models (KdV,
sine-Gordon, Non-linear Schr Ä dinger, etc.) exhibiting integrable behaviour
are known, interest has shifted to more abstract mathematical questions, and to
the remarkable connections between integrable system theory and many other core
areas of mathematics: for example algebraic geometry, differential geometry,
group theory, invariant theory, spectral theory, etc.

For a given PDE or discrete system, there are a variety of partially understood methods to deter mine the integrability or otherwise of the system. We would like to understand these methods better, and to understand the deep links that must lie between them, and to fully classify all integrable systems in some sense. Although it would be false to claim that the field was now completely unified and clarified after the meeting, much progress was made during the programme, and a large number of new collaborations were started which will bear fruit over the next few years.

During
the meeting there were a regular series of seminars, at least two per week on
Mondays and Wednesdays. In addition there were a number of workshops and
Conferences with up to a hundred participants for the major meetings.

Several
visitors to the programme have travelled to other UK Universities to give
seminars (Bogdanov, Boiti, Buchstaber, Calogero, Degasperis, Enolskii, Krichever
Kruskal, MatinezAlonso, Tamizhmani, Tsarev, Zakharov).

In
the tradition of the NEEDS meetings, there were an eclectic mix of 78 half-hour
talks plus a poster session in a very crowded but enjoyable schedule. The
meeting covered a range of theory for integrable and near-integrable systems,
with examples drawn from fluid mechanics, plasma, physics, nonlinear optics,
general relativity, and

Particular
highlights included the developmentof a Painlevé test for difference equations
(Costin), the discovery of a new class of nonlinear evolution equations
possessing many periodic trajectories (Calogero), a new theory of separation of
variables for bi-Hamiltonian systems (Pedroni and Falqui), a dbar approach for
the dispersionless KP hierarchy (Matinez Alonso), and a spectral approach to
boundary value problems for linear and nonlinear PDEs (Pelloni).

The
proceedings of this meeting will be published in two special issues of the
Journal ofTheoretical and Mathematical Physics, edited by Mikhailov and Santini.

One
of the main themes of this School was how to test for integrability in ordinary
and partial differential equations. One type of test involves perturbative or asymptotic methods; these were represented in the lectures of Degasperis on
multi-scale expansions, Zakharov on multidimensional perturbbation theory, and
Kodama on normal forms. A second
approach is algebraic, involving the classification of symmetries, described by
Sokolov and Mikhailov, with connections to number theory, as discussed by
Saunders. Yet another method is based on the analytic behaviour of solutions in
the complex domain, the technique of Painlevé analysis were covered in the
lectures by Kruskal.

Other
major aspects of integrability were also treated. Olver lectured on
multi-Hamiltonian structures, Shabat described BÄacklund transformations within
the framework of dressing chains for linear operators, and Hietarinta described
the Hi-

Selected
lecture courses will be published in the book “Integrability" by
Princeton University Press, edited by Mikhailov.

**EuroWorkshop:
Discrete systems and
integrability**

There
were 36 talks, almost all of them one-hour talks, organised in thematic sessions
(most of which lasted a full working day), linked by an organized discussion.
The themes of the sessions could be roughly divided by the following headings:

1.
Discrete Painlevé equations and affne Weyl groups

2.
Integrable lattices (partial difference equations)

3.
Discrete and difference geometry

4.
Algebraic integrability and computational aspects

5.
Integrable mappings

6.
Discrete Painlevé property

7.
Quantum many-body systems and special functions

8.
Inverse problems and solutions

9.
Symmetries of difference equations

10.
Cellular automata and applications

The
talks were all of a very high level, and a large number of new results and ideas
were put in front of the audience. In particular we mention the following
contributions:

·
M. Noumi (Kobe): “q-Painlevé” equations arising
from a q-version of the modified KP hierarchy" (on the similarity reduction
of the KP hierarchy and the emergence of discrete Painlevé equations form it).

·
Veselov (Loughborough): “Discrete hydrodynamics and
Monge-Ampere equations"

·
C. Viallet (Paris VI): \Complexity, singularity and
integrability of maps" (on the algebra-geometric analysis of singularities
in birational mappings and their resolution through the blowing-up procedure)

·
M.D. Kruskal: “Equivalent of the Painlevé property
for difference equations and study of their solvability" (on a new class of
“analysable" functions and their role in the definition of integrability
in the discrete domain).

**Workshop:
Geometrical aspects of integrability
**

** 17 - 18 September 2001
Organisers:
N Manton, L Mason and R Ward**

Approximately
53 participants registered forthis meeting and there were several more attending
many of the lectures. There were 9 one hour lectures over the two days
including a selection from the leaders in the field and from younger up and
coming researchers. One of the main
foci was the interaction with equations coming from particle physics. Nigel
Hitchin gave a new overview of various integrable geometric structures on the
moduli space of Calabi Yau manifolds that have been discovered in the context of
string theory. Nick Manton reviewed the theory of gauged vortices (not, strictly
speaking integrable, but there are nevertheless many exact analytic results).
Roger Bielawski reviewed the theoryof hyperkahler structures as an integrable
system

On
more traditional topics, there were two lectures on Painlevé equations,
Mazzocco's investigating when solutions can be obtained in terms of `classical
functions' and Woodhouse's studying the isomonodromy problem using methods from
twistor theory. Professor Zakharov's lecture concerned the integrability of a
classical problem in geometry: finding metrics with `diagonal' curvature.

The
meeting brought together people on the programme who's interest was from more of
a traditional applied maths background with those whose primary interest was in
geometry and physics, and this led to significant cross-fertilization.

16 - 17 November 2001

The
workshop brought together users of computer algebra programs related to
integrable systems (mainly among the audience), people involved more with the
design of algorithms and others with more emphasis on the implementation of
algorithms.
The scope of talks reflected this wider scope and a number of people were happy
to get in personal contact for the first time with others they knew only through
their published papers. A novel aspect was that many computer packages in this
are were made available during the meeting for participants to experiment with,
also the organisers encouraged participants to suggest suitable computational
challenges which could be addressed during and after the meeting.

26-30 November

One
of the highlights of the Programme was the special week devoted to Algebraic
Aspects of Integrable Systems. The principal speakers were Pavel Etingof (MIT),
Atsushi Matsuo (Tokyo) and Sergei Barannikov (ENS, Paris). They gave series of
review lectures: Etingof on rational Cherednik algebras in relation to quantum
Calogero-Moser systems, Matsuo on vertex operator algebras and the moonshine
module, Barannikov on quantum periods and integrable hierarchies. Other speakers
include Ch. Athorne (Glasgow), T. Brzezinski (Swansea), V.M. Buchstaber
(Moscow), O.A. Chalykh and M.V. Feigin (Loughborough), V.Z. Enolski (Kiev).

The
week culminated in the LMS Spitalfields Day at Loughborough University (November
30) where the most important recent achievements in the area were presented. P.
Etingof gave a talk on his joint results with V. Ginzburg on symplectic
reflection algebras, A. Matsuo on parafermion algebras and the Monster group, G.
Wilson (IC) on his joint work with Yu. Berest in noncommutative projective
geometry and O. Chalykh on his recent proofs of the Macdonald conjectures. These

2 - 8 December 2001

This
was a Satellite Meeting hosted by the International Centre for Mathematical
Sciences, Edinburgh, and took place on the Heriot-Watt University campus, close
to the famous Union canal of soliton fame.

Approximately
70 participant took part in this lively meeting. There were 34 talks. 24 of
these were by the main invited speakers. The remaining10 shorter talks were by
specifically young participants. This mix was very effective in bringing new

In
the short talks category, the presentations by Caux, Castro-Alvaredo and Doikou
notably inspired many scientific conversations. Many of the main talks presented
important new results. The talks by Cardy, McCoy, Smirnov and Shiraishi might
perhaps be particularly singled out. Cardy's talk showed that they are still
many interesting physical systems for which novel exact results can be obtained
via conformal field theory. McCoy presented his recent work in which,
remarkably,

Smirnov
discussed recent important results in his ongoing programme to construct a more
algebraic-geometrical description of quantum integrable systems. Shiraishi
described a recent breakthrough that has enabled him to construct, after 10
years

There
was much animated discussion of these and many other results at the meeting.
Several collaborations certainly grew out of these discussions; two we know of
involve Delius and Nepomechie, and Konno and McCoy.

In
recent years, through the work of Sklyanin, Smirnov and others there have
started to be signs of some convergence of the fields of classical and quantum
integrability. The coming together of the two communities at this meeting, and
the participation

17 -18 December

The
workshop had been planned during the first meeting of the semester dedicated to
Integrable Systems, when various short talks had focussed on boundary value
problems, notably for the nonlinear Schroedinger equation (NLS), and it was felt
that

The possibility of exposition and discussion among few researchers interested in similar problems was very welcome by all, coming as it did at the end of a semester of concentration on integrable systems in general. It also stimulated the possibility of collaboration among different approaches; at least one such collaboration, between Pelloni and Jerome on the sine-Gordon equation, was a direct result of the meeting. More generally, the workshop took advantage and fully exploited the opportunity offered by the presence at the Newton Institute of most of the people involved in this area of research.

The
organisers and participants felt the programme was very successful, in bringing
together a large number of key figures in the area, and in attracting a
promising number of younger researchers. Apart from the hectic workshop
schedule, much progress was made by the long-and short-stay participants in the
invigorating surroundings e Newton Institute. Some long-standing problems were
solved, and other new areas opened up for further investigation.

Martin
Kruskal and KM Tamizhmani and others made progress in towards developing a
“simple"proof of the Painlevé property for the six Painlevé equations.

Inspired
by the close proximity to the haunts of the famous Cambridge Mathematician H. F.
Baker, Chris Athorne, Chris Eilbeck, and Victor Enolskii developed a fully SL(2)
invariant theory of genus 2 hyperelliptic } functions, opening up new areas
inthis field as well as clarifying some rather obscure steps in Baker's original
treatment. Athorne also worked with Sanders and with Hietarinta on
generalizations of Hirota's bilinear derivative.

Tmara
Grava, with Victor Enolskii, found new explicit solutions for the Riemann-Hilbert
problem involving algebro-geometrci integration of Fuchsian systems and the
Schlesinger equations. Mikhailov, Sanders and Wang discovered a O(N) invariant
integrable generalisation of the famous sine-Gordon equation. This equation has
important applications in the Riemanian geometry. Wang has also found an
explicit generalisation of the Hasimoto transformation to the N-dimensional
case.

Harry
Braden made progress on a number of problems, including twistor theoies and the
Calogero-Moser model (with Lionel Mason), StÄ ackel systems (with Alan Fordy),
and Toda theory and Nahm monopoles (with Hermann Flaschka and Nick Ercolani).
Flaschka also worked on combinatorial rules for the decomposition of tensor
products of compact Lie groups, and related problems on the distribution of
eigenvalues of random matrices.

Francesco
Calogero worked on a number of ODE's and also on “cool" irrational
numbers and their approximations. Darryl Holm performed a breakthrough numerical
simulation showing that the NS-alpha model stimulates decay of a turbulent shear
layer at least as well as highly tuned Large Eddy Simulation models.

Using
asymptotic methods and the Inverse Scattering Transform, Mikhailov and
Novokshenov found a new approach to the problem of DM solitons in non-linear
optics. The analytically predicted shape of the soliton fits well with the
correspondingnumerical simulations.

Andrew
Hone, together with Degasperis and Holm, developed the integrability theory for
a new equation proposed by Degasperis at the School. Mikhailov and Novikov
formulated a perturbative version of the Symmetry Approach which is suitable for
non-evolutionary and non-local equations, including multi-dimensional equations,
and proved that the Camassa-Holm equation and a new equation dicovered by
Degasperis are the only integrable equations in a certain class. They made a
complete classification of integrable Benjabin-Ono type equations.

Andrei
Kapeav, with Alexander Its, worked on assymptotics of the second Painlevé
transcendent. Yuri Kodama started a new collaboration on the dispersionless KP
equations, and worked with Mikhailov on normal forms and the symmetry approach
for near integrable systems. Boris Konopelchenko worked with Martinez Alonso on
integrable dispersionless hierarchies, quaso-conformal maps, and -dressing
method. Decio Levi, with Pavel Winternitz and Rafael Heredo, worked on Lie
symmetries of difference equations. Vladimir Matveev worked on topological
results for geodesically equivalent Riemannian metrics. Fank Nijhof and
Hietarinta developed new ideas involving the characterisation of integrable
mappings in one and two dimensions through the commutativity of discrete flows.
Beatrice Pelloni worked on boundary value problems for the sine-Gordon and
Schrodinger equations. Pogrebkov and Fokas studied initial value problems for
the KPI equations with initial conditions given by single soliton solution plus
a rapidly decaying term. Pogrebkov also constructed a hierarchy of quantum
explicitly solvable models which can be considered as a quantum version of the
dispersionless KdV hierarchy. A. B. Shabat studied a discrete version of the
SchrÄ odinger spectral problem, and the connection between certain isospectral
flows and infinite-dimensional hydrodynamic type systems.

Vladimir
Sokolov, with Thomas Wolf, completed an investigation of vector (1 + 1)
dimensional integrable models. With Andrey Tsiganov developed new integrable poynomila deformations of known
integrable models from rigid body dynamics, includinga new integrable case fro
the classical Kirchoff problem of motion of a rigid body.

Sergei Tsarev completed a comprehensive survey of work on integrable exponential systems and a review on algorithmic methods of integration of nonlinear ODEs. Vladimir Zakharov worked with a numer of people: with Tom Bridges on a hamiltonian description of a “renormalized" fluid; with Konopoelchenko and others on the dispersionless KP hierarchy, with Frederick Diaz on one-dimensional turbulence, with Fokas on boundary problems for integrable systems, with Jerry Griffiths on diagonal metrics in general relativity, and in addition found time to serach for new integrable solutions of the Einstein equations.