3. WORK PROGRAMME
3.1 Research Projects
3.1.1 Title
Nonlinear Evolution Equations and Dynamical Systems
3.1.2 Objectives
To combine efforts of experts in nonlinear integrable equations in studying new algebraic and analytic
aspects of integrable models and their physical applications is the main objective of the present
project.
To investigate and classify new classes of integrable lattice models together with their cut-offs and
scalings, which preserve integrability.
To derive and examine similarity solutions for NLS-type systems, in particular optical solitons.
To extend the theory of inverse scattering to multi-dimensional potentials not decaying to zero at large
distances.
To study the connection of the periodic spectral problem with the D-bar problem.
To apply the technique of the theory of integrable systems to several interesting problems in
mathematics (differential geometry, discrete geometry) and physics (topological quantum field theory,
string theory, Einstein equations).
3.1.3 Background & Justification for Undertaking the Project
Nonlinear problems and phenomena constitute one of the most interesting and intensively studied
subjects in modern pure and applied mathematics, theoretical and mathematical physics. Theory of
integrable nonlinear evolution equations represents one of the most bright and important fields in
nonlinear phenomena. Solitons, i.e., the coherent structures described by these equations, are found in
various areas of physics and applied mathematics from plasma physics and nonlinear optics, solid
state physics and hydrodynamics, to elementary particle theories and gravitation. Their technological
applications, for instance in fibre optics, are also very promising.
Nonlinear partial differential equations that possess the soliton properties are of great interest to
mathematics. The inverse scattering (spectral) transform method discovered 32 years ago allows
analysing analytically such equations in details. The famous Korteveg-de Vries, nonlinear
Schrödinger and sine-Gordon equations exemplify the best known soliton equations with a great
variety of applications. Recently the apparatus of the theory of integrable systems has been proved to
be a very effective tool to solve important problems in algebraic geometry (Shottky problem),
classical differential geometry of surfaces (classification of particular classes of surfaces) and discrete
geometry (integrable discrete nets). Interrelation and interinfluence of differential geometry and
theory of integrable systems discovered 25 years ago is nowadays a very important and promising
field of research.
The inverse scattering transform method is applicable to a very broad class of differential and discrete
nonlinear equations in 1+1 and 2+1 dimensions. Various interesting results have been obtained in this
field during the last decade. Construction and study of new interesting multidimensional integrable
equations, development of corresponding methods, generalisation of the symmetry approach are tasks
of great importance. It will allow understanding better the entire structure and properties of integrable
system. Special attention was paid, recently, to integrable equations with one or more discrete spatial
variables, such as Toda lattice. A number of discrete integrable models have been discovered in
quantum field theory (matrix models), statistical physics, cellular automata, discrete geometry,
mathematical ecology and economics. A rigorous mathematical study of these models seems to be a
hard problem since the traditional technique of Inverse Scattering Transform does not suit well to the
discrete space-time case. Already a symmetry approach to the classification of integrable lattices has
been founded, which enables to give complete lists of integrable models in certain classes.
1
Appropriate boundary conditions (cut-offs, periodic closures, etc.) can inherit integrability providing
new classes of solutions. The study of scaling limits and asymptotic solutions can reveal new
phenomena similar to fractal generation or quasi-crystal structures.
Application of methods of soliton theory to studying and solving problems in mathematics and
mathematical physics is also very promising. It can provide significant results in classical geometry of
complex curves and surfaces, theory of discrete surfaces, topological quantum field theory and string
theory. Such a study will allow solving some open problems in mathematics and will reveal deeper
connections between theory of integrable systems and classical problems in geometry.
Different teams of the Project have experts both in the theory of integrable systems, in differential and
algebraic geometry and in quantum field theory. Their collaboration will guarantee an important
output of the research both in theory and its applications.
The University of Lecce is a natural place for coordinating the project since at Lecce it has been
created an international inter-university consortium named EINSTEIN (for European Institute for
Nonlinear Studies via Transnationally Extended Interchanges). The main aim of this non-profiting
consortium is to develop the scientific international cooperation in the field of Nonlinear Science with
special attention to the East-West cooperation. The University of Lecce has successfully coordinated
the INTAS project “Nonlinear evolution equations and dynamical systems” (INTAS-93-166 and
INTAS-93-166 Ext.).
General References:
1. S.P. Novikov, S.V. Manakov, L.P. Pitaevsky and V.E. Zakharov, Theory of Solitons. The Inverse
Scattering Method. Plenum, New-York, 1984.
2. F. Calogero and A. Degasperis, Spectral Transform and Solitons I, Northolland (Amsterdam,
1982)
3. L.D. Faddeev and L.A. Takhtadjan, Hamiltonian Methods in the Theory of Solitons,
Springer-Verlag (Berlin-Heilderberg-New York, 1987)
4. M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse
Scattering. Cambridge University Press, Cambridge, 1991.
5. A. Bobenko and R. Seiler, Eds., Discrete Integrable Geometry and Physics. Oxford University
Press, Oxford, 1998.
3.1.4 Scientific/Technical Description
3.1.4.1 RESEARCH PROGRAMME
The development of methods to construct and solve the multidimensional soliton partial differential
and discrete equations is one of the main goals of the project. These include, in particular, the
extension to multidimensions of the inverse spectral transform, of the dressing method and of the
direct linearization method. Different types of exact solutions, like the exponentially localized
solitons, for the multidimensional integrable equations will be studied, including their applications.
An essential direction in this project is an investigation of the nature of the integrability, its
universality and wide applicability and its connection with the symmetry properties both in the case of
discrete and continuous systems. This investigation will result in a group theoretical classification of
the integrable nonlinear partial differential equations and discrete integrable systems.
Another goal of the project is a study of the interrelations between the multidimensional integrable
equations and the theory of surfaces, their integrable deformations and the integrable motions of
curves, modeling variety of nonlinear phenomena which involve interfaces, boundaries and lines. The
project includes also the study of discrete systems and cellular automata and their application in
information theory. Applications of the integrable systems in modern quantum field theory and
statistical physics will be studied as well.
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Task 1. Development of the methods to construct and solve new classes of multidimensional
integrable equations.
Development of different versions of the inverse spectral transform method will be pursued. Particular
attention will be paid to the generalized resolvent approach and to the D-bar dressing method. Their
use will give new classes of solutions of some multidimensional equations and will allow
understanding better general structure and properties of infinite hierarchies of integrable equations.
This task will be carried out in cooperation by the teams named Landau, Steklov, Alma-Ata, Lecce,
Madrid and Rome. The specific subtasks allocated to the different teams are explicitly indicated
below.
Task 1.1 Multidimensional inverse spectral transform will be used to study and, in particular, to get
broad classes of explicit solutions of some reductions of (2+1)-dimensional integrable systems (Lecce,
Landau, Steklov).
Task 1.2 To study integrability properties of multidimensional lattices and associated transformations
(Madrid, Rome).
Task 1.3 To extend and generalize the D-bar dressing method to discrete variables and q-difference
and to develop and apply the analytical-bilinear approach for integrable hierarchies to
multicomponent KP hierarchy and to the hierarchy of two-dimensional Toda lattice (Lecce, Landau).
Task 1.4 To continue the development and set up of a new approach to inverse spectral (or
scattering) transform which could be considered, more naturally than in the traditional formulation, as
an extension of Fourier transform to nonlinear evolution equations. This approach makes use of and
extends the theory of resolvent and has been called the resolvent approach (Lecce, Steklov).
Task 1.5 To analyze the structure and the properties of hidden integrable hierarchies within different
approaches (Madrid, Lecce).
Task 1.6 The development of special methods of constructing solutions of multidimensional
integrable equations and the analysis of associated algebraic and geometric structures (Alma-Ata,
Lecce).
Task 1.7 Further development of the method of extended resolvent and investigation in this
framework of many-dimensional integrable nonlinear evolution equations, in particular their soliton
solutions. Generalization of the inverse scattering method for potentials with non-trivial asymptotic
behavior. Description of integrable hierarchies, of their Poisson and symplectic structures (Lecce,
Steklov)
Task 2. Investigation and classification of new classes of integrable lattice models.
Integrable equations with one or more discrete spatial variables, such as Toda lattice, play an
important role in the modern nonlinear science. A number of discrete integrable models have been
discovered recently in quantum field theory (matrix models), statistical physics, cellular automata,
discrete geometry, mathematical ecology and economics. A rigorous mathematical study of these
models seems to be a hard problem since the traditional technique of Inverse Scattering Transform
does not suit well to the discrete space-time case. Already a symmetry approach to the classification
of integrable lattices has been founded, which enables to give complete lists of integrable models in
certain classes. Appropriate boundary conditions (cut-offs, periodic closures, etc.) can inherit
integrability providing new classes of solutions. The study of scaling limits and asymptotic solutions
can reveal new phenomena similar to fractal generation or quasi-crystal structures
This task will be carried out in cooperation by the teams named Landau, Ufa, ITEP, Rome, Leeds.
Loughborough and Montpellier. The specific subtasks allocated to the different teams are explicitly
indicated below.
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Task 2.1 A generalization of integrability to the classes of chains explicitly depending on the discrete
variable will be considered. Chains, possessing higher symmetries and conservation laws, will be
classified (Landau, Ufa).
Task 2.2 Integrable rebounding reflection systems (mathematical billiards) on two-dimensional
constant curvature surfaces and stability of periodic billiard trajectories on that surfaces will be
studied (Leeds, Landau).
Task 2.3 Investigation and classification of several classes of integrable lattice models, including
Toda lattice, relativistic Toda lattice and their discrete analogues. This research will be based on the
hamiltonian representation for these systems and the fact the Backlund transformations are canonical
(Landau, ITEP, Ufa).
Task 2.4 The new general simple scheme of the discretization of the linear spectral problems with
special applications of lattice equations in the spectral theory will be investigated (Landau,
Loughborough).
Task 2.5 The reduction problem in dressing chain theory (Landau).
Task 2.6 Integrable boundary conditions for the lattice equations. Classification of the boundary
conditions for integrable PDE compatible with integrability. This leads to a study of Backlund
transformations keeping the specified boundary condition, which in its turn, form an integrable lattice
(Ufa).
Task 2.7 Singular scaling limits of the integrable lattices. A number of physically interesting solutions
of integrable PDE's arise from iterative procedures, generating appropriate integrable lattices
("dressing chains”). The study of various asymptotic limits with respect to scaling parameters of the
lattice will give a new description of infinite-soliton solutions of integrable PDE's (Ufa).
Task 2.8 Construction of discrete-time integrable systems, or "integrable maps" via BTs of systems
with finitely many degrees of freedom and its use to derive separation coordinates, both at the
classical and at the quantum level (Rome, Leeds, Loughborough, Landau, Ufa)
Task 2.9 Construction of new differential difference systems associated with more than two linear
problems and study of their algebraic properties (Rome, Landau).
Task 2.10 Construction of a perturbation scheme for differential - difference systems in which the
discrete character of the underlying space is preserved and which give rise to asymptotic integrability
(Rome, Montpellier, Landau).
Task 3. Development of symmetry type classification of non-linear equations.
Classification of nonlinear differential equations according to their symmetries is a very important
field of research. A basic goal consists in extending the detailed symmetry classification of
1+1-dimensional differential equation to the 2+1-dimensional case. Symmetry approach to certain
non-abelian ordinary differential equations and some particular classes of transformations will be
studied.
This task will be carried out in cooperation by the teams named Landau, Ufa, Leeds and Rome. The
specific subtasks allocated to the different teams are explicitly indicated below.
Task 3.1 A complete classification of scalar integrable equations of the third order in 2+1 dimensions
in a rather wide class will be studied as well as a preliminary classification of integrable
multicomponent polynomial equations (Ufa, Landau, Leeds).
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Task 3.2 Symmetry classification of some classes of non-abelian ODE. Generalization of the main
concepts to the case of non-abelian PDE. Graded Lie algebras and Lax representation for these
equations (Landau, Ufa).
Task 3.3 Study of dilation symmetries and one-parametric groups of transformations for scalar
evolution lattice equations. It is planned to develop a theory of such transformations, which generalize
classical point groups of auto-transfromations, and apply that theory to most important integrable
equations (Ufa, Rome)
Task 4. Analysis of concrete integrable equations important for physics and mathematics.
Nonlinear differential equations important in various fields of classical and quantum physics will be
studied.
This task will be carried out in cooperation by the teams named Landau, ITEP, Novosibirsk, Lecce,
Loughborough, Montpellier, Madrid, SISSA, Saclay and Rome. The specific subtasks allocated to the
different teams are explicitly indicated below.
Task 4.1 Integrable systems associated with the Hamiltonian structures on loop spaces of Riemann
manifolds and of the multidimensional Poisson brackets of hydrodynamic type will be studied
(Landau).
Task 4.2 To study applications of integrable systems in nonlinear physics, field theory, optical fibers,
free electron laser, multidimensional ferromagnets, dynamics of interfaces (Lecce, Montpellier,
Rome, Saclay).
Task 4.3 Investigation of the generalized WDVV equations and corresponding geometrical
configurations and Frobenius structures (Loughborough, Landau, ITEP, SISSA, Madrid).
Task 4.4 Similarity solutions for NLS – type systems. Connection with Painlevé
transcedents.Physical application of such solutions to optical solitons and spin dynamics (Landau,
Lecce, Saclay).
Task 4.5 Method of isoperiodical deformations. Finite gap periodic discrete surfaces. Transition
from the scattering problem to the periodic one for 2-dimensional Schrödinger operator (Landau)
Task 4.6 Study of possible relations between factorizability and integrability of nonlinear ordinary
and partial differential equations (Novosibirsk)
Task 4.7 Possible relations to the recently discovered connections to the Bieberbach Conjecture
proved in 1984 will be investigated (Novosibirsk).
Task 4.8 It was recently shown that to almost every one-dimensional quasi-exactly solvable quantum
Hamiltonian one can associate a family of (weakly) orthogonal polynomials. It is planned to continue
the study of the properties of these polynomial systems, with special emphasis on quasi-exactly
solvable periodic potentials, like for instance the Lamé potential (Madrid)
Task 4.9 Following some recent work of Shifman and Ushveridze it is intended to develop
systematically new techniques for generating exactly and quasi-exactly solvable quantum many-body
potentials from known one-dimensional quasi-exactly solvable systems (Madrid)
Task 4.10 It is planned to continue recent work on different transformations available for generating
new quasi-exactly solvable systems from known ones (Madrid)
Task 4.11 To extend recent classification of finite-dimensional Lie superalgebras of 2x2 matrix
differential operators in several promising new directions. These results could then be applied to
generating new examples of quasi-exactly solvable matrix Hamiltonians (Madrid)
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Task 4.12 Study of the generalization of the N-body Calogero system having the four types of
interactions depending on the four constants (the so-called the Calogero-Inozemtsev system) (ITEP).
Task 4.13 Describing PVI with four arbitrary constants as a reduced Hamiltonian system (ITEP).
Task 4.14 Analytical and asymptotic properties of the most generic sixth Painleve transcendent are
still studied insufficiently. Therefore it would be important to investigate PVI applying the averaging
and IDM methods. The goal is to obtain new formulas describing the solution behavior at infinity, 0
and 1. An additional goal is to find new integrable discretizations of the Painleve equations
(Novosibirsk, Saclay).
Task 4.15 The study of short wave asymptotic limit of models of fluid dynamics is a totally new field
of investigation very rich both on the mathematical side, with e.g method of multiple scales, the
integrability of reduced models, and on the physical side with e.g. the properties of short waves in
various situations, the search of explicit solutions and their implications, the onset of instability
(Montpellier).
Task 4.16 The scattering of classical waves in nonlinear, discrete or continuous, medium is a field of
growing interest particularly relevant for the study of light absorption in nonlinear regime like
stimulated Raman and Brillouin scattering, pulse propagation in fibers, infra-red anomalous
absorption in cristals (Montpellier).
Task 4.17 To develop methods such as Painlevé analysis, reduction procedure and separation of
variables to perform the explicit integration, whether full or partial, of all kinds of nonlinear equations
(differential, partial differential, discrete), preferably the ones encountered in physics (Novosibirsk,
Saclay).
Task 5. Application of the technique of soliton equations for the study of problems of
differential geometry of continuous and discrete surfaces.
The principal goal of this task is to reveal and study new promising interrelations between differential
geometry and theory of integrable equations. Weierstrass type formulae for surfaces in
multidimensional spaces and corresponding integrable deformations will be constructed. Discrete
surfaces and nets and their relations with discrete integrable equations will be studied. One of the
goals is to apply ideas of Projective and Finsler geometries to analysis of ordinary differential
equations.
This task will be carried out in cooperation by the teams named Kishinev, Novosibirsk, Lecce, Madrid
and Rome. The specific subtasks allocated to the different teams are explicitly indicated below.
Task 5.1 To study the interrelations between multidimensional integrable equations and theory of
surfaces, their integrable deformations and integrable motions of curves, modeling a variety of
nonlinear phenomena which involve interfaces, boundaries and lines (Lecce, Rome).
Task 5.2 To establish relations between different schemes for solving partial differential equations
with symmetry and geometric methods of integration (Madrid, Rome, Lecce).
Task 5.3 Development of a geometrical approach to the nonlinear dynamical systems with regular and
chaotic behaviour and its use for studying such systems at the change of parameters. These goals are
supposed to be achieved by using the methods of Riemannian, Einsten-Weyl and Finsler geometry,
the theory of Liouville-Tresse-Cartan invariantes for the second order ordinary differential equations
(ODE), the theory of duality for the 2-nd order ODE and the corresponding surfaces of the RP3 and
RP4 projective spaces. Potential results are efficient algorithms for determining the range of
parameters for which the behaviour of the dynamical system is regular or irregular, the existence of
particular integrals, integrability and so on (Kishinev, Lecce).
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Task 5.4 Construction and analysis of the discrete surfaces of revolution which can be related to the
discrete Schrodinger equation (Lecce).
Task 5.5 To study the relation among spectral properties of Dirac operators and related tori in
three-dimensional space (Novosibirsk, Lecce)
Task 5.6 It is planned to look for geometric applications of spectral curve in constructing isothermic
tori and finite gap formulas for them and to study integrable systems such as Hamiltonian stationary
Lagrange tori in complex projective space (Novosibirk, Lecce)
Task 5.7 It is planned to extend previous work on the generation of quadrilateral and circular lattices
within the framework of the multicomponent KP theory, studying a couple of reductions leading to
the Symmetric and Egorov nets. Relevant tools in this study are bilinear identities for Baker and tau
functions as well as the Miwa transformations (Madrid, Lecce)
Task 5.8 Starting from the bilinear formalism will be undertaken the task of obtaining and classifying
classes of solutions of Lamé equations, by using Clifford group dressing transformations and
tau-function techniques. An important question is to find not only expressions for the rotation and
Lamé coefficients but also expressions for the cartesian coordinates of the points of the nets (Madrid)
Task 6. Development of the Hamiltonian formalism for integrable equations.
Bi-hamiltonian character of soliton equations has been one of the important discoveries in this field.
Development of bi-hamiltonian ideas using different formalisms and their applications to various
integrable systems is the principal goal of this task.
This task will be carried out in cooperation by the teams named Landau, Loughborough, Steklov,
Milan, SISSA and Rome. The specific subtasks allocated to the different teams are explicitly
indicated below.
Task 6.1 Geometry of bi-hamiltonian manifolds in se; problems of classification and deformation à la
Gelfand-Zakharevich; comparison with alternative schemes of integrability (Lie-Poisson groups,
dynamical r matrix, algebraic geometry methods); separation of variables and spectral curve method
(Milan, Rome).
Task 6.2 Theory of Frobenius manifolds. The aim is to go into Dubrovin conjecture according to
which semi-simple Frobenius manifolds play the role of spaces of modules of bi-hamiltonian
manifolds (Milan, SISSA).
Task 6.3 To investigate systematically the possibility to construct other examples of integrable
many-body systems of the Calogero and Ruijsenaars type using the recursion operator technique
(Milan, Rome, Steklov).
Task 6.4 Investigation of the Hamiltonian theory of Backlund transformations (Loughborough,
Landau).
Task 6.5 Study of finite-dimensional integrable systems, in particular Ruijsenaars systems, and
application of the reduction procedures to them. Use of the Hamiltonian (Poissonian) reduction
technique to obtain the quantization of the models within the unifying quantum R-matrix formalism
(Milan)
Task 6.6 Investigation of compatible Poisson brackets of hydrodynamic type and their deformations
(Landau, SISSA, Milan)
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Task 7. Study of integrable structures which arise in quantum field theory, string theory, matrix
models and general relativity
Deeper and wider application of integrability ideas in matrix models associated with two-dimensional
quantum gravity, in models of quantum field theory, in different versions of string theory and in
Einstein equations of classical gravity is a goal of great importance. Different techniques to reach
such a goal will be used.
This task will be carried out in cooperation by the teams named Steklov, ITEP, Loughborough and
Lecce. The specific subtasks allocated to the different teams are explicitly indicated below.
Task 7.1 To analyze the correspondence between phenomenological models of matter fields coupled
with Chern-Simons fields, and models derived from a purely topological non-abelian theory (Lecce).
Task 7.2 Study of quantum integrable systems and their relation with the classical ones. Development
of an effective approach for the evaluation of the quantum correlation functions (Steklov, ITEP)
Task 7.3 Study of the integrability property for quantum moduli spaces via the method of matrix
models in external field (Steklov. ITEP)
Task 7.4 The analysis of integrability properties of the Einstein and Einstein-Maxwell field equations
with isometries based on the so called monodromy transform approach (Steklov, Loughborough)
Task 7.5 Study of the integrability property for quantum moduli spaces using the method of matrix
models in external field and the quantum tau function technique (Steklov)
Task 7.6 Study of the Bäcklund transformations for the RTC in the fermionic approach (ITEP)
Task 7.7 To develop the quantum method of separation of variables using the solution of the auxiliary
spectral problem which is easily described in terms of the group elements. To apply these methods
developed in the study of Toda chain to more complicated quantum problems, for example, to
XXX-model and relativistic Toda chain (ITEP).
Task 7.8 Search for extending the Hitchin-Mukai systems to the case of moduli spaces of vector
bundles over certain complex surfaces other than K3 (ITEP).
Task 7.9 To study the cohomological properties of the holomorphic symplectic 2-form on moduli
spaces with the aim of understanding the proper holomorphic prequantization (ITEP).
3.1.4.2 DELIVERABLES, EXPLOITATION & DISSEMINATION OF RESULTS
The standard yearly report with reprints of published or accepted for publication papers will be sent to
INTAS. The dissemination of the project results is guaranteed by publications in the most diffused
international scientific journals and by lectures given at international Conferences and Workshops. A
special event in the six month program "Integrable Equations" in 2001 at the Newton Institute of
Cambridge (Principal Organizer A.V. Mikhailov member of the team of Leeds) will be dedicated to
dissemination of scientific results obtained in the framework of this project.
3.1.5 Description of the Consortium
3.1.5.1 RESEARCH TEAMS
The tasks assigned to each team are listed in the above RESEARCH PROGRAM section. There the specific
scientific cooperations foreseen among teams are also listed.
Team (Landau)
The team works at the Landau Institute for Theoretical Physics of the Russian Academy of Sciences (Moscow,
Russia). They are experts on the integrable equations including Inverse Scattering Transform Method, Bäcklund
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transformations and dressing chains. The team leader A. Shabat is well known for his dressing method and his
study of integrable systems.
List of the scientists:
1
Prof. A.B. Shabat (leader)
2
Prof. S.V. Manakov
3
Dr. L.V. Bogdanov
4
Prof. P.G. Grinevich
5
Prof. V.V. Sokolov
6
Prof. O.I. Mokov
6
Dr. V.G. Marikhin (32 years old)
7
Mr. V.S. Novikov (23 years old)
Landau Institute, Moscow, Russia
Landau Institute, Moscow, Russia
Landau Institute, Moscow, Russia
Landau Institute, Moscow, Russia
Moscow Inst. for Nonlinear Studies, Moscow, Russia
Moscow Inst. for Nonlinear Studies, Moscow, Russia
Landau Institute, Moscow, Russia
Landau Institute, Moscow, Russia
Team (Steklov)
The team works at the Department of Quantum Field Theory of the Steklov Mathematical Institute (Moscow,
Russia). They are experts on the integrable equations including Inverse Scattering Transform Method, both in
the classical and in the quantum case and in field theory including classical gravity. The team leader A.
Pogrebkov is well known for his studies of integrable systems and in field theory and classical scattering theory.
List of the scientists:
1
Prof. A. K. Pogrebkov (leader)
Steklov Mathematical Institute, Moscow, Russia
2
Dr. G. A. Alekseev
Steklov Mathematical Institute, Moscow, Russia
3
Prof. N. A. Slavnov
Steklov Mathematical Institute, Moscow, Russia
4
Dr. L. O. Chekov
Steklov Mathematical Institute, Moscow, Russia
5
Dr. G. E. Arutyunov (31 years old)
Steklov Mathematical Institute, Moscow, Russia
Students involved: K. I. Palmarchuk (Physics Department, Moscow State University, Russia)
Team (ITEP)
The team works at the Theory Department of the Institute of Theoretical and Experimental Physics (Moscow,
Russia). They are experts in classical and quantum integrable systems, topological field theories, matrix models
and string theory.. The team leader M. Olshanetsky is well known for his studies in the theory of integrable
systems and in quantum field theories.
List of the scientists:
1
Prof. M. A. Olshanetsky (leader)
Inst. of Theoretical and Experimental Physics, Moscow, Russia
2
Dr. S. K. Kharchev
Inst. of Theoretical and Experimental Physics, Moscow, Russia
3
Prof. A. V. Marshakov
Inst. of Theoretical and Experimental Physics, Moscow, Russia
4
Dr. A. A. Rosly
Inst. of Theoretical and Experimental Physics, Moscow, Russia
Students involved: I. V. Gordeli, A. V. Zotov (Moscow Institute of Physics and Technology)
Team (Ufa)
The team works at the Institute of Mathematics of the Russian Academy of Sciences (Ufa, Russia). The
members are experts on the integrable equations including Inverse Scattering Transform Method, Bäcklund
transformations and dressing chains. The team leader Prof. V.Novokshenov is well known for his studies of
Painlevé transcendents and asymptotic solutions of integrable PDE’s.
List of the scientists:
1
Prof. V. Yu. Novokshenov (leader)
Institute of Mathematics, Ufa, Russia
2
Prof. I.T. Habibullin
Institute of Mathematics, Ufa, Russia
3
Dr. R.I. Yamilov
Institute of Mathematics, Ufa, Russia
4
Dr. V.E. Adler (33 years old)
Institute of Mathematics, Ufa, Russia
5
Prof. A.B. Borisov
Institute of Metal Physics, Ekaterinburg, Russia
Students involved: V.A.Elichev, N.A.Atnabaev (Ufa Aviation Technical University)
Team (Novosibirsk)
The team members are expert in algebraic geometry, differential geometry, integrable systems and Painlevé
property of ODE and PDE. They are already strictly cooperating scientifically and, therefore, it is considered
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convenient to collect them in one team. The team leader I. Taimonov is a well known expert in algebraic
geometry.
List of the scientists:
1
Prof. I. A. Taimanov (leader)
2
Prof. S. P. Tsarev
3
Dr. V. L. Vereshchagin
Inst. of Math., Siberian Branch of RAS, Novosibirsk, Russia
Krasnoyarsk State Pedagogical University, Krasnoyarsk, Russia
Inst. for System Dynamics & Control Theory,
Siberian Branch of RAS, Irkutsk, Russia
Team (Kishinev)
The team leader V. Driuma works at the Institute of Mathematics of the Academy of Sciences of Moldova,
Kishinev, Moldova. He is a well-known expert in differential geometry, projective geometry and soliton
equations
List of scientists:
1
Prof. V. S. Driuma (leader)
Inst. of Math., Acad. of Sc. Of Moldova, Kishinev, Moldova
Team (Alma-Ata)
The team works at the Laboratory of Relativistic Nuclear Physics of the Institute of Physics and Technology,
Alma-Ata, Kazakhstan. The members of the team are expert in integrable nonlinear evolution equations and
their applications in differential geometry. The team leader R. Myrzakulov is a known expert in soliton systems.
List of scientists:
1
Dr. R. Myrzakulov (leader)
2
Dr. G. N. Nugmanova (33 years old)
3
Dr. A.K. Danlybaeva (26 years old)
4
Dr. R.N. Syzdykova (28 years old)
Lab. of Relativistic Nucl. Phys., Inst. of Phys. & Tech.,
Alma-Ata, Kazakhstan
Lab. of Relativistic Nucl. Phys., Inst. of Phys. & Tech.,
Alma-Ata, Kazakhstan
Lab. of Relativistic Nucl. Phys., Inst. of Phys. & Tech.,
Alma-Ata, Kazakhstan
Lab. of Relativistic Nucl. Phys., Inst. of Phys. & Tech.,
Alma-Ata, Kazakhstan
Team (Leeds)
The team works at the Center for Nonlinear Studies of the Department of Applied Mathematics of the
University of Leeds, Leeds, UK. The members of the team are expert in integrable systems and symmetry
classifications of nonlinear PDE. The team leader A. Fordy is a well-known expert in the Inverse Scattering
Method.
List of scientists:
1 Prof. A.P. Fordy (leader)
2
Prof. A.V. Mikhailov
3
Dr. F.W. Nijhoff
4
Dr. V.B. Kuznetsov
5
Dr. I. Marshall
Department
Leeds
Department
Leeds
Department
Leeds
Department
Leeds
Department
Leeds
of Applied Mathematics, University of Leeds,
of Applied Mathematics, University of Leeds,
of Applied Mathematics, University of Leeds,
of Applied Mathematics, University of Leeds,
of Applied Mathematics, University of Leeds,
Team (Loughborough)
The team works at Department of Mathematical Sciences of the Loughborough University, Loughborough, UK.
The members of the team are expert in discrete dynamical systems, integrable motions on special manifold, and
integrable equations. The team leader J. B. Griffiths is a well-known expert in interaction of gravitational waves.
List of scientists:
10
1
Prof. J. B. Griffiths (leader)
2
Prof. A. P. Veselov
Dept. of Math. Sc.of the Loughborough Univ., Loughborough,
UK
Dept. of Math. Sc.of the Loughborough Univ., Loughborough,
UK
Team (Madrid)
The team works at the Departamento de Fisica Teorica II of the Universidad Complutense, Madrid, Spain. The
members of the team are expert in integrable systems, Hamiltonian structure, tau function approach and
fermionic description of soliton equations and symmetry groups of nonlinear PDE. The team leader L. Martinez
Alonso is a well-known expert in integrable equations, Hamiltonian structure and tau approach of soliton
equations.
List of scientists:
1 Prof. L. Martinez Alonso (leader)
2
Dr. F. Finkel Morgenstern (29 years old)
3
4
Mr. D. Gomez-Ullate Oteiza (25 years
old)
Prof. A. Gonzalez-Lopez
5
Dr. R. Hernandez Heredero (34 years old)
6
Prof. M. Manas Baena
7
Prof. E. Medina Reus (33 years old)
8
Prof. M. A. Rodriguez Gonzalez
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. De Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Depart. de Fis. Teor. II. Univ. Complutense, Madrid,
Spain
Team (Milan)
The team works at the Dipartimento di Matematica ed Applicazioni, Universita' di Milano "Bicocca", Milano,
Italy. The members of the team are expert in Hamiltonian formalism and bi-Hamiltonian structure of integrable
equations. The team leader F. Magri is a well-known expert in in Hamiltonian formalism and bi-Hamiltonian
structure of integrable equations.
List of scientists:
1 Prof. F. Magri (leader)
2 Dr. P. Casati
3 Dr. M. Pedroni
Dept. of Math. & Appl., Univ. of Milan "Bicocca", Milan, Italy
Dept. of Math. & Appl., Univ. of Milan "Bicocca", Milan, Italy
Dept. of Math., Univ. of Genova, Genova, Italy
Team (Rome)
The team works at the Physics Department of the University of Rome III, Rome, Italy. The members of the team
are experts in discrete systems, applications of integrable systems in discrete geometry, Hamiltonian structure
and symmetry groups for discrete and continuous nonlinear equations. The team leader D. Levi is a well known
expert in integrable discrete systems and symmetry group approach to discrete and continuous equations.
List of Scientists:
1 Prof. D. Levi (leader)
Phys. Dep. Fisica "E. Amaldi", Univ. of Roma III, Roma, Italy
2 Prof. O. Ragnisco
Phys. Dep. Fisica "E. Amaldi", Univ. of Roma III, Roma, Italy
3 Prof. P.M. Santini
Phys. Dep. Fisica , Univ. of Roma “La Sapienza”, Roma, Italy
Students involved: Fabio Musso (Università Roma III)
Team (Montpellier)
The team works at the "Laboratoire de Physique Mathématique et Théorique" of the University of Montpellier
2, a laboratory under contract with the CNRS #UMR 5825. The members of the team are expert in the study of
11
nonlinear phenomena by building reduced models (method of multiple scales) and the study of boundary value
problems resulting from the scattering of waves in nonlinear medium. The team leader J. Leon is a well known
expert in nonlinear systems integrable by means of the Inverse Scattering method and in their physical
applications.
List of Scientists:
1 Prof. J. Leon (Leader)
2 Dr. M. Manna
3 Dr. J-G. Caputo
4 Dr. A. Spire (25 year old)
Phys. Math. et Theor., Univ. Montpellier 2, Montpellier, France
Phys. Math. et Theor., Univ. Montpellier 2, Montpellier, France
Phys. Math. et Theor., Univ. Montpellier 2, Montpellier, France
Phys. Math. et Theor., Univ. Montpellier 2, Montpellier, France
Team (Saclay)
The team belongs to the Service de Physique de l'Etat Condensé of Commissariat à l'Energie Atomique, Saclay,
France. Since M. Musette is already strictly cooperating scientifically it is considered convenient to insert her in
this team. The team works on theoretical problems in all kinds of nonlinear differential of difference equations.
The team leader R. Conte is a well known expert in Painlevé analysis of non linear partial differential and
difference equations.
List of scientists:
1 Prof. R. Conte (leader)
Serv. de Phys. De l'Etat Condensé of CAA, Saclay, France
2 Prof. S. Bouquet
Serv. de Phys. De l'Etat Condensé of CAA, Saclay, France
3 Dr. S. Labrunie (27 years old)
Serv. de Phys. De l'Etat Condensé of CAA, Saclay, France
4 Prof. M. Musette
Vrije Universiteit of Brussels, Brussels Belgium
Students involved: C. Verhoeven (from Vrije Universiteit of Brussels, Brussels, Belgium), V. Sciacca (from
University of Palermo, Italy)
Team (SISSA)
The team works at SISSA/ISAS (Scuola Internazionale Superiore per Studi Avanzati/International School for
Advanced Studies), Trieste, Italy. It works on integrable systems and their connection with differential
geometry. The team leader B. Dubrovin is a well known expert in the Inverse Scattering Theory and in algebraic
and differential geometry.
List of scientists:
1 Prof. B. Dubrovin
2 Dr. G. Falqui
SISSA, Trieste, Italy
SISSA, Trieste, Italy
Team (Lecce)
The team works at the Dipartimento di Fisica dell’Università di Lecce, Lecce, Italy. The members of the team
are expert in the Inverse Scattering Method, Bäcklund and Darboux transformation, dressing method, discrete
systems, prolongation structure, and applications of integrable systems to differential geometry. The
Coordinator of the Project and team leader M. Boiti is a well known expert in the Inverse Scattering method,
Bäcklund and Darboux transformation, and integrable discrete systems. The Coordinator is President of the
Consortium named E.I.N.S.T.E.IN. (for European Institute for Nonlinear Studies via Transnationally Extended
Interchanges) created at Lecce with precisely the scope of promoting scientific international cooperation in the
field of Nonlinear Science with special attention to the East-West cooperation.
List of scientists:
1
Prof. M. Boiti (Coordinator)
2
Prof. F. Pempinelli
3
Prof. B. Konopelchenko
4
Prof. G. Soliani
5
Prof. G. Solombrino
6
Prof. A.R. Leo
7
Prof. M. Leo
8
Dr. L. Martina
9
Dr. C. Corianò
Consortium E.I.N.S.T.E.IN., Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
12
10
11
12
13
Dr. G. Landolfi (30 years old)
Dr. V. Grassi (31 years old)
Dr. B. Prinari (27 years old)
Dr. P. Tempesta (27 years old)
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
Dept. of Phys., Univ. of Lecce, Lecce, Italy
3.1.5.2 SCIENTIFIC REFERENCES
Team (Landau)
1.
A.V. Mikhailov, A.B. Shabat, and R.I. Yamilov, Extension of the module of invertible transformations.
Classification of integrable systems, Commun. Math. Phys. 115 (1988) 1.
2. 2 V.G. Marikhin, A.B. Shabat, Integrable lattices, Teor. Mat. Fiz., 118 (1999) 217.
3. A.P.Veselov and A.B.Shabat, Dressing chain and spectral theory of Schrodinger operator, Funkts. Anal.
Prilozen., 27(1993) 1-21.
4. P.G. Grinevich, M.U. Schmidt, Conformal invariant functionals of tori into R 3; Journal of Geometry and
Physics, 26 (1998) 51-78.
5. L.V. Bogdanov and B. G. Konopelchenko, J. Math. Phys., 39 (1998) 4683.
6. V.E. Zakharov, S.V. Manakov, About the reductions in the systems integrable by
Inverse Scattering Problem, Dokladi Akademii Nauk (in russian), 360 (1998) 324.
7. V.S. Novikov, On the equations invariant under the differential autosubstitutions, Teor. Mat. Fiz., 118
(1998) 2.
8. S.P. Balandin, V.V. Sokolov, On the Painleve test for non-Abelian equations,
Phys. Lett. A, 246 (1998) 267-272.
9. P.J. Olver, V.V. Sokolov,
Integrable evolution equations on associative algebras,
Comm. In Math. Phys., 193 (1998) 245-268
10. V.G. Marikhin, Hamiltonian theory for integrable NLS – generalizations. JETP Letters, 66 (1997)
673-678.
Team (Steklov)
1.
M.Boiti, F.Pempinelli, A.K.Pogrebkov. Solving the Kadomtsev--Petviashvili equation with initial data not
vanishing at large distance. Inverse problems 1997, v. 1, pp. L7--L10
2. M.Boiti, F.Pempinelli, A.K.Pogrebkov, B.Prinari Towards an inverse scattering theory for two-dimensional
nondecaying potentials, Theor. Math. Phys. 1998, vol. 116, pp. 741--781.
3. Korepin V. E., Slavnov N. A. The Riemann-Hilbert problem associated with the quantum nonlinear
Schrodinger equation. Journ. Phys. A: Math. Gen. 1997 vol. 30 pp. 8241-8255
4. A. R. Its, N. A. Slavnov On the Riemann-Hilbert approach for asymptotic analysis of correlation functions
of the Quantum Nonlinear Schrodinger equation. Interactiong fermion case. Theor. Math. Phys., 1999 vol.
119, pp. 179-230; math-ph/9811009
5. G. A. Alekseev, "Exact solutions in the General Theory of Relativity" Proc. Steklov Inst. Maths. 3,
215 (1988).
6. G. A. Alekseev and J. B. Griffiths"Exact solutions for gravitational waves with cylindrical, spherical and
toroidal wavefronts", Class. Quantum Grav., 13, 2191-2209, (1996).
7. Chekhov L. Matrix model tools and geometry of moduli spaces. Acta Appl. Math. 1997 v.48, pp. 33-90.
8. Chekhov L., Fock V. Quantum Teichmuller spaces, Theor. Math. Phys., Sept. 1999
9. G.E.Arutyunov, S.A.Frolov, Quantum dynamical R-matrices and quantum Frobenius group, q-alg/9610009,
Comm.Math.Phys. 191 (1998) 15-29
10. G.E.Arutyunov, L.O.Chekhov, S.A.Frolov, $R$-matrix quantization of the elliptic Ruijsenaars-Schneider
model, Comm.Math.Phys. 192 (1998) 405-432.
Team (ITEP)
1.
2.
3.
4.
A. Gerasimov, S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, M. Olshanetsky, Liouville Type
Models in Group Theory Framework. I. Finite-Dimensional Algebras, Int.J.Mod.Phys. A12, (1997),
2523-2584
A. M. Levin, M. A. Olshanetsky, Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as
hierarchy of isomonodromic deformations. Free fields approach, hep-th/9709207
A. M. Levin, M. A. Olshanetsky, Double coset construction of moduli space of holomorphic bundles and
Hitchin systems, Commun.Math.Phys. 188 (1997) 449-466, alg-geom/9605005
S. Kharchev, A. Mironov, A. Zhedanov, Faces of Relativistic Toda Chain, Int.J.Mod.Phys. A12, 1997),
2675-2724
13
5.
6.
S.Kharchev, Kadomtsev-Petviashvili hierarchy and generalized Kontsevich model, hep-th/9810091
A.Marshakov, A.Mironov and A.Morozov, WDVV-like equations in N=2 SUSY Yang-Mills Theory,
phys.Lett., B389 (1996) 43, hep-th/9607109
7. A.Marshakov, A.Mironov and A.Morozov, More Evidence for the WDVV Equations in N=2 SUSY
Yang-Mills Theories, hep-th/9701123
8. A.Marshakov, A.Mironov and A.Morozov, WDVV Equations from Algebra of Forms, Mod.Phys.Lett.,
A12 (1997) 773, hep-th/9701014
9. V. V. Fock, A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix,
math/9802054
10. A. Keurentjes, A. Rosly, A. V. Smilga, Isolated vacua in supersymmetric Yang-Mills theories, Phys.Rev.
D58 (1998) 081701, hep-th/9805183
Team (Ufa)
1.
V.Yu.Novokshenov, A.G.Shagalov, Bound states for the elliptic sine-Gordon equation, Physica D, v.106
(1997) p.81-94.
2. V.Yu.Novokshenov, Reflectionless potentials and soliton series of the nonlinear Schrodinger equation,
Physica D, v.87 (1995) p.101-106.
3. A.R.Its, V.Yu.Novokshenov, The Isomonodromic Deformation Method in the Theory of Painleve
equations, Lecture Notes in Math., Vol. 1191, 313p. Springer-Verlag (1986).
4. A.B.Borisov, S.A.Zykov, Dressing chain of discrete symmetries and breeding of nonlinear equations,
Teor.Mat.Phys. v.115, N 2, (1998) 199
5. V.E. Adler, S.I. Svinolupov, R.I. Yamilov, Multi-component Volterra and Toda type integrable equations,
Phys. Lett. A 254 (1999) 24.
6. A.V. Mikhailov, A.B. Shabat, and R.I. Yamilov, Extension of the module of invertible transformations.
Classification of integrable systems, Commun. Math. Phys. 115 (1988) 1.
7. D. Levi, R.I. Yamilov, Conditions for the existence of higher symmetries of evolutionary equations on the
lattice, J. Math. Phys. 38:12 (1997) 6648.
8. A.V. Mikhailov, R.I. Yamilov, Towards classification of (2+1)-dimensional integrable equations.
Integrability conditions I, J. Phys. A: Math. Gen. 31 (1998) 6707.
9. V.E.Adler, I.T.Habibullin, Integrable boundary conditions for the Toda lattice, J. Physics A: Math. Gen. 28
(1995) 6717.
10. I.T.Habibullin, Boundary conditions for integrable chains, Physics Letters A 207 (1995) 263.
Team (Novosibirsk)
1.
2.
3.
4.
5.
6.
7.
8.
9.
I. Taimanov, The Weierstrass representation of closed surfaces in R3. (Russian) Functional Anal. Appl.
32:4 (1998), 49-62.
I. Taimanov, The Weierstrass representation of spheres, the Willmore numbers, and soliton spheres. (to
appear in Trudy Mat. Inst. Steklov), Preprint SFB 288, TU-Berlin, no. 302, 1998
I. Taimanov, Finite gap solutions ot the modified Novikov--Veselov equation, their spectral properties, and
applications, Siberian Math. J., 1999, no. 5
V.L. Vereschagin Asymptotics of solutions of the discrete string equation // Physica D 95. 1996. P.
268-282
V.L. Vereschagin Whitham equations for one-phase solutions of Volterra Lattice // Nonlinear Analysis,
Theory, Methods and Applications. 1992. v.19. N.2. P. 177-185
V.L. Vereschagin Global asymptotics for the fourth Painlevé transcendent, Matem. sbornik. 1997. v.188.
p.335-344.
J.Gibbons, S.Tsarev, Conformal maps and reductions of the Benney equations, Accepted for publication in:
Physics Letters A, 1999
S.Tsarev, On factorization of nonlinear ordinary differential equations, accepted for publication in Proc.
ISSAC'99, 28--31 July 1999, Vancouver, Canada
S.Tsarev, On factorization of linear partial differential operators and the Darboux method of integration of
nonlinear PDEs, accepted for publication in Theoretical and mathematical Physics, 1999
Team (Kishinev)
1.
V. Driuma, Finsler-Geometrical Approach to the Studying of Nonlinear Dynamical systems, Buletinul a
Republicii Moldova, Matematica, v.1(27), 21-36, 1999. (solv-int/9803003).
14
2.
V. Driuma, On integration of the Einstein equations for multidimensional spaces, Proceedengs of
International Congress of Mathematics ICM' 99. Berlin, Germany, 1998
3. V. Driuma, On the Law transformation of affine connection ad its integration. Part 1. Generalization of the
Lame Equations, Buletinul a Republicii Moldova, Matematica, v.1(26) 55-68, 1998. (solv-int/9803004).
4. V. Driuma, Investigation of dynamical systems using tools of the theory of invariants and projective
geometry, Zeitschrift fur Angewandte Matematik und Physik, ZAMP, v.48, 725-743,1997
5. V. Driuma, Geometrical properties of the multidimensional nonlinear equations and the Finsler
metrics of dynamical systems, Teoretitsheskaya i Matematitsheskaya Fizika, v.99, N2, 241--249, 1994
6. V. Driuma, Geometrical properties of dynamical systems with regular and chaotic behaviour,
proceedings of the First Workshop of Nonlinear Physics. Le Sirenuse, Gallipoli (Lecce), Italy, World
Scientific, Singapoor, 83-93, 1996
7. V. Driuma, 3-dim exactly integrable system of nonlinear equations and its application, Mathematical
Research, v.124, 54--68, 1992, Kishinev
8. V. Driuma, On Geometry of the second order ODE's and their applications, in K.V.Frolov (eds.)
Proceedings of Conference "Nonlinear Phenomena" Moskow, Nauka, 1991, 41—47
9. V. Driuma, On integration of cylindrical Kadomtzeev-Petviashvili equation by the method of the inverse
problem of scattering theory, Soviet Math. Dokl.,v.27, N1, 1983, 6—8
10. V.Driuma, On analitical solutions of two-dimensional Korteweg-de Vrise equation, Pisma v JETF, v.19,
No.21, 1974
Team (Alma-Ata)
1.
Myrzakulov R, Danlybaeva A K and Nugmanova G N., Geometry and multidimensional soliton equations,
Theor. Math. Phys.118 (1999) 441
2. Myrzakulov R, Nugmanova G.N and Syzdykova R N. Gauge equivalence between (2+1)-dimensional
continuous Heisenberg ferromagnetic models and nonlinear Schrodinger-type equations, J. Phys. A: Math.
Gen. 31 (1998) 9535
3. Lakshmanan M, Myrzakulov R, Vijayalakshmi S and Danlybaeva A K., Motion of curves and surfaces and
nonlinear evolution equations in (2+1) dimensions, J.Math.Phys. 39 (1998) 3765
4. Myrzakulov R, Vijayalakshmi S, Syzdykova R N and Lakshmanan M., On the simplest (2+1) dimensional
integrable spin systems and their equivalent nonlinear Schrodinger equations, J.Math.Phys. 39 (1998) 2122
5. Myrzakulov R, Vijayalakshmi S, Nugmanova G N and Lakshmanan M., A (2+1)-dimensional integrable
spin model: Geometrical and gauge equivalent counterpart, solitons and localized coherent structures,
Phys.Lett. 233A (1997) 391
6. Myrzakulov R, Daniel M and Amuda R., Nonlinear spin-phonon excitations in an inhomogeneous
compressible biquadratic Heisenberg spin chain, Physica A 234 (1997) 715
7. Makhankov V G, Myrzakulov R and Pashaev O K., Gauge equivalence, supersymetry and classical solution
of OSPU(1,1|1)-Heisenberg model and nonlinear Schrodinger equation, Lett. in Math. Phys. 16 (1989) 83
8. Makhankov V G, Myrzakulov R and Katyshev Ji V, The vector generalization of the set of equations of
long and shot waves, Theor. Math. Phys. 72 (1987) 32
9. Makhankov V G, Myrzakulov R and Makhankov A V., Generalized coherent states and the continuous
Heisenberg XYZ model with one-ion anisotropy, Physica Scripta 35 (1987) 233
10. Myrzakulov R, Pashaev O K and Kholmurodov Kh T., Particle-like excitations in multicomponent
magnon-phonon systems, Physica Scripta 33 (1986) 378
Team (Leeds)
1.
2.
3.
4.
5.
6.
E.V. Ferapontov and A.P. Fordy, Separable Hamiltonians and integrable systems of hydrodynamic type, J.
Geom. and Phys., 21, 169-82, 1997
A.P. Fordy and S.D. Harris, Hamiltonian flows on stationary manifolds, Methods and Applications of
Analysis, 4, 212-238, 1997
E.V. Ferapontov and A.P. Fordy, Nonhomogeneous systems of hydrodynamic type, related to quadratic
Hamiltonians with electromagnetic term, Physica D, 108, 350-64, 1997
Kuznetsov, V.B., Sklyanin E.K.: “On Backlund transformations for many-body systems'', J. Phys. A:
Math. Gen. 31 (1998), 2241-2251
F. W. Nijhoff, O. Ragnisco and V.B. Kuznetsov, Integrable time-discretization of the
Ruijsenaars-Schneider model, CMP 176 (1996) 681-700
V.B. Kuznetsov, F.W. Nijhoff and E.K. Sklyanin, Separation of variables for the Ruijsenaars system, CMP
189 (1997) 855-877
15
7.
F.W. Nijhoff, A. Ramani, B. Grammaticos and Y. Ohta, On discrete Painlevé equations associated with the
lattice KdV systems and the Painleve VI equation, solv-int/9812011, (submitted to Stud. Appl. Math.)
8. A.V. Mikhailov and R.I. Yamilov, "Towards Classification of 2+1 dimensional Integrable Equations.
Integrability conditions I", Journal of Physics A, 31, pp. 6707-6715, 1998
9. J. Leon and A.V. Mikhailov, "Raman Soliton Generation from Laser Inputs in SRS", Physics Letters A,
253, pp, 33-40, 1999
10. A.V. Mikhailov and V.V. Sokolov, "Integrable evolutionary equations on free associative algebras",
(Accepted by Journal of Theoretical and Mathematical Physics (TMF), 1999).
Team (Loughborough)
1.
Fordy A.P., Shabat A.B. and Veselov A.P. "Factorization and Poisson correspondences", Theor. and Math.
Phys., V.105, n.2, 1995, 225-245
2. Chalykh, O.A., Feigin, M.V., Veselov, A.P., "New integrable generalizations of quantum Calogero-Moser
problem", J. Math. Phys., 39 (2), 1998, 695-703
3. Berest, Yu.Yu. and Veselov, A.P., "On the singularities of the potentials of exactly solvable Schroedinger
equations and Hadamard's problem", Uspekhi Mat. Nauk, 53 (1), 1998, 211-212
4. Veselov A.P. "Deformations of the root systems and new solutions to generalised WDVV equations",
hep-th/9902142. Submitted to Phys.Lett.A
5. Veselov A.P. "On geometry of a special class of solutions to generalised WDVV equations" To appear in
the Procedings of the Conference on Seiberg-Witten theory and Integrability (ICMS, Edinburgh, September
1998)
6. J. B. Griffiths, "Colliding Plane Waves in General Relativity", Oxford University Press, 1991
7. G. A. Alekseev and J. B. Griffiths, "Propagation and interaction of gravitational waves in some expanding
backgrounds", Phys. Rev. D, 52, 4497-4502, (1995)
8. G. A. Alekseev and J. B. Griffiths, "Exact solutions for gravitational waves with cylindrical, spherical and
toroidal wavefronts", Class. Quantum Grav., 13, 2192-2209, (1996)
9. G. A. Alekseev and J. B. Griffiths, "Gravitational waves with distinct wavefronts", Class. Quantum Grav.,
14, 2869-2880, (1997).
10. J. B. Griffiths and S. Micciche, "The extensions of gravitational soliton solutions with real poles", Gen. Rel.
Grav. 31, 869-887, (1999)
Team (Madrid)
1.
J. A. Calzada, M. A. del Olmo and M. A. Rodriguez; Classical Superintegrable SO(p,q) Hamiltonian
Systems, J. Geom. Phys. 23, 14--30 (1997).
2. A. Doliwa, M. Manas, L. Martinez Alonso, E. Medina and P. M. Santini, Charged Free Fermions, Vertex
Operators and Classical Theory of Conjugate Nets, J. Phys. A: Math. Gen. 32, 1197-1216 (1999)
3. F. Finkel, A. Gonzalez Lopez and M. A. Rodriguez, Quasi-exactly solvable Lie superalgebras of differential
operators, J. Phys. A: Math. Gen. 30, 6879-6892 (1997).
4. F. Finkel, A. Gonzalez Lopez and M. A. Rodriguez, Quasi-exactly solvable spin 1/2 Schrodinger operators,
Math. Phys. 38, 2795-2811 (1997).
5. D. Gomez-Ullate, S. Lafortune and P. Winternitz, Symmetries of discrete dynamical systems involving two
species, J. Math. Phys. 40, 2782-2804 (1999).
6. A. Gonzalez-Lopez, R. Hernandez Heredero and G. Mari Beffa, Invariant differential equations and the
Adler--Gel'fand--Dikii bracket, J. Math. Phys. 38, 5720-5738 (1997).
7. R. H. Heredero and P. J. Olver, Classification of invariant wave equations, J. Math. Phys. 37, 6414--6438
(1996)
8. B. Konopelchenko and L. Martinez, Integrable systems for d-bar operators of non-zero index, Phys. Lett.
A236, 431-438, (1997).
9. M. Manas, L. Martinez and E. Medina, Zero sets of tau-functions and hidden hierarchies of KdV type, J.
Phys. A: Math. Gen. 30, 4815-4824, (1997)
10. M. Manas and L. Martinez Alonso, From Ramond fermions to Lame equations for orthogonal curvilinear
coordinates, Phys. Lett. B436, 316-322, (1998).
Team (Milan)
1.
Y. Kosmann--Schwarzbach, F. Magri, Lax-Nijenhuis operators for integrable systems, J. Math. Phys. 37
(1996).
16
2.
P. Casati, G. Falqui, F. Magri, M. Pedroni, A note on fractional KdV hierarchies, J. Math. Phys. 38 (1997),
4606--4628.
3. P. Casati, G. Falqui, F. Magri, M. Pedroni, Darboux coverings and rational reduction of the KP hierarchy,
Lett. Math. Phys. 41 (1997), 291—305
4. F. Magri, J.P. Zubelli, Bihamiltonian formalism and the Darboux-Crum method. Part I: From the KP to the
mKP hierarchy, Inverse Problems 13, (1997), 755-780
5. F. Magri, M. Pedroni, J.P. Zubelli, On the Geometry of Darboux transformations for the KP hierarchy and
its Connection with the discrete KP hierarchy, Comm. Math. Phys. 188 (1997), 305--325.
6. P. Casati, G. Falqui, F. Magri, M. Pedroni, Bihamiltonian Reductions and ${Cal W_n$--Algebras, Journal
of Geometry and Physics 26 (1998), 291-310
7. G. Falqui, F. Magri, M. Pedroni, Bihamiltonian Geometry, Darboux Coverings, and Linearization of the
KP Hierarchy, Commun. Math. Phys. 197 (1998), 303-324
8. F. Magri, The bihamiltonian route to Sato Grassmannian, in “The bispectral problem'' (Montreal, PQ,
1997), pp. 203--209, CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, RI, 1998
9. F. Magri, Eight lectures on integrable systems, in the Proceedings of the CIMPA Summer School on
Nonlinear Systems (B. Grammaticos and K. Tamizhmani, eds.), Pondi-cherry, India, January 1996, Lecture
Notes in Physics 495, pp. 256—296
10. P. Casati, G. Falqui, F. Magri, M. Pedroni, Soliton equations, bihamiltonian manifolds, and integrability,
Notes of a course held at the 21st Brazilian Mathematical Colloquium (July 1997), IMPA, Rio de Janeiro,
1997
.
Team (Rome)
1.
F.Nijhoff O.Ragnisco & V.Kusznetsov: "Integrable Discretisation of the Rujisenaars-Schneider System",
Comm.Math.Phys. 176 (1996), 681-700
2. S.Rauch, O.Ragnisco: "Integrable Maps for the Garnier and the Neumann Systems", J.Phys.A 29
(1996),1115-1124
3. Yu.A.Suris, O.Ragnisco, "What is the relativistic Volterra lattice?", Comm.Math.Phys.200 (1999) 445-485
4. A.Hone, V.Kuznetsov, O.Ragnisco: "Backlund Transformations for Many-Body Systems related to KdV",
solv-int/9904003
5. D.Levi;L.Vinet ;P.Winternitz, Lie Group formalism for equations on lattices J. Phys. A: Math. Gen 30
(1997) 633-649
6. D.Levi ;P.Winternitz, Classification of dynamical systems on the lattice J. Math. Phys. 37 (1996)
5551-5576
7. D. Levi and O. Ragnisco, The inhomogeneous Toda lattice: its hierarchy and Darboux-Backlund
transformations, J. Phys. A: Math. Gen. 24 (1991) 1729-1739
8. D. Levi, O. Ragnisco and A.B. Shabat, Construction of higher local (2+1) dimensional exponential lattice
equations, Can. J. Phys. 72 (1994) 439-441
9. A. Doliwa, S.V. Manakov, P. M. Santini, d-bar Reductions of the Multidimensional Quadrilateral lattice:
the Multidimensional Circular lattice, Comm. Math. Phys. 196(1998)1-18
10. A. Doliwa and P.M. Santini, Multidimensional Quadrilateral Lattices are integrable, Phys. Lett. A
233(1997)365-372
Team (Montpellier)
1.
2.
3.
4.
5.
6.
7.
M. Boiti, J. Leon. F. Pempinelli, Nonlinear Spectral Characterization of Discrete Data, Phys Rev E, Vol.
54 , 5739 (1996)
M. Boiti, J. Leon. F. Pempinelli, Solution of the boundary value problem for the integrable discrete SRS
system on the semi-line, J. Phys. A : Math. Gen. Vol. 32 , 927-943 (1999)
M. Barthès, H.N. Bordallo, J. Eckert, O. Maurus, G. de Nunzio, J. Leon, Dynamics of Crystallized
N-Methylacetamide : Temperature dependence of Infrared and Inelastic Neutron Scattering Spectra, J.
Phys. Chem. B Vol. 102 , 6177 (1998)
J. Leon, A.V. Mikhailov, Raman solitons generation from laser inputs in SRS, Phys. Lett. A, Vol. 253
(1999) 33
F-X. Hugot, J. Leon, Solution of the initial-boundary value problem for the Karpman-Kaup equation,
Inverse Problems Vol. 15 (1999) 701
J.Leon, M. Manna, Multiscale Analysis of Discrete Nonlinear Evolution Equations, J. Phys. A : Math. Gen.,
Vol. 32 (1999) 2845
M.A. Manna and V. Merle, Asymptotic dynamics of short-waves in nonlinear dispersive models, Physical
Review E , Vol. 45, 6206 (1998)
17
8.
M.A. Manna and V. Merle, Modified Korteweg-de Vries Hierarchy in Multiple-Times Variables and the
Solutions of Modified Boussinesq Equations, Proc. R. Soc. London. A , Vol. 454, 1 (1998).
9. J. G. Caputo, N. Flytzanis, Y. Gaididei and M. Vavalis, Two-dimensional effects in Josephson junctions: I
static properties, Phys. Rev. E, Vol. 54, 2092 (1996).
10. A. Benabdallah, J. G. Caputo et A.C. Scott, Flux flow in an Eiffel junction, to appear in Phys. Rev. B.
Team (Saclay)
1.
2.
3.
4.
5.
6.
7.
8.
9.
R. Conte and M. Musette, A new method to test discrete Painlevé equations, Phys. Lett. A 223 (1996)
439—448
R. Conte and M. Musette, Rules of discretization for Painlevé equations, Theory of nonlinear special
functions : the Painlevé transcendents, 20 pages, eds. L. Vinet and P. Winternitz (Springer, New York,
2000). Solv-int/9803014. Montréal workshop, 13--17 May 1996
M. Musette and R. Conte, Bäcklund transformation of partial differential equations from the
Painlevé-Gambier classification, I. Kaup-Kupershmidt equation, J. Math. Phys. 39 (1998) 5617—5630
R. Conte, The Painlevé approach to nonlinear ordinary differential equations, The Painlevé property, one
century later, 112 pages, ed. R. Conte, CRM series in mathematical physics (Springer, New York, 1999).
Solv-int/9710020
R. Conte and M. Musette, Comment on Exact periodic solutions of the complex Ginzburg-Landau equation
J. Math. Phys. 40, 884 (1999)
R. Conte, M. Musette, and A. M. Grundland, Bäcklund transformation of partial differential equations from
the Painlevé-Gambier classification, II. Tzitzéica equation, J. Math. Phys. 40 (1999) 2092—2106
M. Musette, Painlevé analysis for nonlinear partial differential equations, The Painlevé property, one
century later, 54 pages, ed. R. Conte, CRM series in mathematical physics (Springer, New York,
(1999).Solv-int/9804003
H. R. Lewis, S. Bouquet, and R. Conte, Integration by quadratures for an N-degree of freedom
nonautonomous Hamiltonian system : application to a Hénon-Heiles system, preprint S98/026 (1999).
R. Conte and M. Musette, Towards second order Lax pairs to discrete Painlevé equations of first degree,
Chaos, solitons and fractals (2000) to appear. Solv-int/9803013
Team (SISSA)
B. Dubrovin, “Geometry of 2D topological field theories” in Integrable Systems and Quantum Groups, Eds.
M. Francaviglia, S. Greco Springer Lecture Notes in Math., vol. 1620 (1996), pp. 120-348
2. B. Dubrovin, “Functionals of the Peierls - Frohlich Type and the Variational Principle for the Whitham
Equations” in Amer. Math. Soc. Transl. (2) Vol. 179 (1997) 35-44
3. B. Dubrovin, R.Flickinger and H.Segur, Three-Phase Solutions of the Kadomtsev-Petviashvili Equation,
Studies in Applied Math. 99 (1997) 137-203
4. B. Dubrovin and Y. Zhang, Extended Affine Weyl groups and Frobenius manifolds, Compositio Math. 111
(1998) 167-219.
5. B. Dubrovin, Y. Zhang, Bihamiltonian Hierarchies in 2D Topological Field Theory at One-Loop
Approximation; Commun.Math.Phys. 198 (1998) 311-361
6. B. Dubrovin, Geometry of the spaces of orbits of a Coxeter group, Surv. Diff. Geom., IV (1999), 181 - 212
7. B. Dubrovin, Differential geometry of moduli spaces, Surv. Diff. Geom., IV, (1999), 213 - 238
8. G. Falqui, F. Magri, M. Pedroni, Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP
hierarchy. Commun. Math. Phys. 197 303-324 (1998)
9. G.Falqui, C.Reina, A.Zampa,Krichever Maps, Faa` di Bruno Polynomials and Cohomology in KP Theory,
Lett. Math. Phys. 42 (1997) 349-361
10. P. Casati, G. Falqui, F. Magri, M. Pedroni. Bihamiltonian Reductions and W-Algebras. Jour. Geom. Phys.
26 291-310, (1998)
1.
Team (Lecce)
1.
2.
3.
4.
M. Boiti, F. Pempinelli and A. Pogrebkov: ”Properties of solutions of the KPI equation'', Journal of Math.
Phys. 35, 4683 (1994)
M.Boiti, F.Pempinelli, A.K.Pogrebkov and B.Prinari: ”Towards an Inverse Scattering theory for
bidimensional nondecaying potentials'', Theoretical and Mathematical Physics 116, 741-781 (1998)
M.Boiti, J.Leon and F. Pempinelli: ”Solution of the boundary value problem for the integrable discrete SRS
system on the semi-line'', Journal of Physics A 32, 927 (1999)
L. Bogdanov and B. Konopelchenko, Analytic bilinear approach to integrable hierarchies.I. Generalized KP
hierarchy, I and II, J. Math. Phys.,39, 4683-4700 and 4701-4728 (1998)
18
5.
V.S. Dryuma and B.G. Konopelchenko, On equation of geodesic deviation and its solutions, Classical and
quantum gravity, Bulletin of the Academy of Sciences of Moldova, Math. Vol 3, pag. 61-73 (1996)
6. B.G.Konopelchenko and G.Landolfi, Generalized Weiestrass representation for surfaces in
multi-dimensional Riemann spaces, Journal of Geometry and Physics, 29, 319-333 (1999)
7. B.Konopelchenko, L.Martinez Alonso and E.Medina, Hidden integrable hierarchies of AKNS type,
J.Phys.A: Math.Gen., 32, 3621-3635 (1999)
8. L. Martina, O.K. Pashaev, G. Soliani, Bright solitons as black holes, Phys. Rev. D 58, 84025 (1998).
9. V. Grassi, R.A. Leo, G. Soliani, L. Solombrino, Continuous approximations of binomial lattices, to be
published in International Journal of Modern Physics A (1999)
10. M. Beccaria, G. Soliani, Mathematical properties of systems of the reaction-diffusion type, Physica A 260,
301 (1998).
3.1.6 Management
Only 10,000 Euro will be used for the management of the project (overheads) and the remaining 110,000 Euro
of the 120,000 assigned will be integrally devoted to support NIS teams. In spite of the large number of teams
involved, total overhead and management costs can be reduced since the Coordinator will take advantage of the
international inter-university non profiting Consortium named EINSTEIN (for European Institute for Nonlinear
Studies via Transnationally Extended Interchanges). This Consortium was created at Lecce with precisely the
scope of promoting scientific international cooperation in the field of Nonlinear Science with special attention to
the East-West cooperation.
A percentage of approximately 30% of the 110.000 Euro assigned to the NIS teams will be used for supporting
visits of scientists from the NIS teams to the Western Institution participating to the project.
Two meetings involving the largest possible number of participants to the project are foreseen. They will be
dedicated to a critical survey of the organization of the project and of the effectiveness of the scientific
cooperation among Western and Eastern participants.
The second meeting will be organized in the framework of the six-month programme "Integrable Equations" in
2001 at the Newton Institute of Cambridge (Principal Organiser A.V. Mikhailov member of the team of Leeds).
3.1.6.1 PLANNING & TASKS ALLOCATION
A table for each of the main tasks listed in 3.1.4.1 is presented. On these tables for each subtask are indicated the
teams or the team who carry out the task as well as the start and duration of the work. When a subtask is
assigned to a team all the members of the team are considered involved and responsible (at different degrees to
be stated during the work by the leader of the team) of the success in completing the task. Each team is
identified by the name indicated in 3.1.5.1
Task 1
Subtasks
Participants
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Lecce, Landau, Steklov
Madrid, Rome
Lecce, Landau
Lecce, Steklov
Madrid, Lecce
Alma-Ata, Lecce
Lecce, Steklov
Months 1-6
Months 7-12
Months 13-18
Month 19-24
Months 1-6
Months 7-12
Months 13-18
Month 19-24
Task 2
Subtasks
Participants
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Landau, Ufa
Leeds, Landau
Landau, ITEP, Ufa
Landau, Loughborough
Landau
Ufa
Ufa
19
2.8
Rome, Leeds, Landau,
Loughborough
2.9
2.10
Rome, Landau
Rome, Montpellier,
Landau
Task 3
Subtasks
Participants
3.1
3.2
3.3
Ufa, Landau, Leeds
Landau, Ufa
Ufa, Rome
Months 1-6
Months 7-12
Months 13-18
Month 19-24
Task 4
Subtasks
Participants
4.1
4.2
4.3
Landau
Montpellier, Rome, Saclay
ITEP, Landau, SISSA,
Loughborough, Madrid
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
Landau, Lecce, Saclay
Landau
Novosibirsk
Novosibirsk
Madris
Madrid
Madrid
Madrid
ITEP
ITEP
Novosibirsk, Saclay
Montpellier
Montpellier
Novosibirsk, Saclay
Months
1-6
Months 7-12
Months 13-18
Month 19-24
Task 5
Subtasks
Participants
5.1
5.2
5.3
Lecce, Rome
Madrid, Rome, Lecce
Kishinev, Lecce
5.4
5.5
5.6
5.7
5.8
Lecce
Lecce
Novosibirsk, Lecce
Novosibirsk, Lecce
Madrid, Lecce
Madrid
Months 1-6
Months 7-12
Months 13-18
Month 19-24
Task 6
Subtasks
Participants
6.1
6.2
6.3
6.4
6.5
6.6
Milan, Rome
Milan, SISSA
Milano, Rome, Steklov
Lougborough, Landau
Milan
Landau, SISSA, Milan
Months 1-6
Months 7-12
Months 13-18
Month 19-24
Task 7
20
Subtasks
Participants
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Months 1-6
Lecce
ITEP, Steklov
Steklov
Steklov, Loughborough
Steklov
ITEP
ITEP
ITEP
ITEP
Months 7-12
Months 13-18
Month 19-24
3.1.6.2 COST TABLE
MAIN COST TABLE
TEAM NAME
INTAS MEMBER TEAMS
COST CATEGORIES
STATU
S
Labour
Costs
Consortium
Einstein
All other
SUBTOTAL
TEAM NAME
CO
CR
( Euro )
Overhea Travel and Consumabl Equipment Other costs
ds
subsistence
es
10000
0
0
0
0
10000
0
0
0
0
0
0
STATU
S
Landau
Steklov
ITEP
Ufa
Novosibirsk
Kishinev
Alma-Ata
SUBTOTAL
CR
CR
CR
CR
CR
CR
CR
( Euro )
TOTAL
( Euro )
76764
TOTAL
Overhea Travel and Consumabl Equipment Other costs
ds
subsistence
es
0
8245
0
0
0
0
6185
0
0
0
0
5245
0
0
0
0
7000
0
0
0
0
3000
0
0
0
0
1103
0
0
0
0
2458
0
0
0
0
33236
0
0
0
10000
33236
0
NIS Labour Cost Summary Table
Number of individual.
Cost/month (Euro)
Grants
Total
Euro
10000
0
10000
NIS TEAMS
COST CATEGORIES
Labour
Costs
21019
12718
11107
14214
8780
3346
5580
76764
Team Name
TOTAL
0
0
Number of Months
Total
Euro
29264
18903
16352
21214
11780
4449
8038
110000
120000
Total cost
(Euro)
21
Prof. A.B. Shabat
239
14
3346
Prof. S.V. Manakov
239
13
3107
Dr. L.V. Bogdanov
179
13
2327
Prof. P.G. Grinevich
239
13
3107
Prof. V.V. Sokolov
239
13
3107
Prof. O.I. Mokov
239
9
2151
Dr. V.G. Marikhin
179
13
2327
Mr. V.S. Novikov
119
13
1547
Prof. A. K. Pogrebkov
239
14
3346
Dr. G. A. Alekseev
179
13
2327
Prof. N. A. Slavnov
239
13
3107
Dr. L. O. Chekov
179
13
2327
Dr. G. E. Arutyunov
179
9
1611
Prof. M. A. Olshanetsky
239
14
3346
Dr. S. K. Kharchev
179
13
2327
Prof. A. V. Marshakov
239
13
3107
Dr. A. A. Rosly
179
13
2327
Prof. V. Yu. Novokshenov
239
14
3346
Prof. I.T. Habibullin
239
13
3107
Dr. R.I. Yamilov
179
13
2327
Dr. V.E. Adler
179
13
2327
Prof. A.B. Borisov
239
13
3107
Prof. I. A. Taimanov
239
14
3346
Prof. S. P. Tsarev
239
13
3107
Dr. V. L. Vereshchagin
179
13
2327
Kishinev
Prof. V. Driuma
239
14
3346
Alma-Ata
Dr. R. Myrzakulov
133
14
1862
Dr. G. N. Nugmanova
126
13
1638
Dr. A.K. Danlybaeva
80
13
1040
Dr. R.N. Syzdykova
80
13
1040
389
76764
Landau
Steklov
ITEP
Ufa
Novosibirsk
Totals
22