NONLINEAR MODELS IN PHYSICS: perspectives for the XXI Century

June 7-8, 2002

 

An international Seminar organized by the Department of Theoretical Physics II of the Universidad Complutense.

 

 

 

Introduction

 

Since the discovery of the soliton by Zabusky and Kruskal in 1965, the progress in the study of nonlinear phenomena in Physics has been spectacular, from both the theoretical and applied points of view. This fundamental discovery was the starting point of a new discipline, currently known as Theory of Integrable Systems, which has led to the creation of many sophisticated mathematical tools to deal with this type of phenomena. The soliton is nowadays a fundamental concept in Physics, appearing in such diverse fields as materials science, nonlinear optics, fluid dynamics, particle Physics and Cosmology. Recently, one of the best known integrable equations, the nonlinear Schrödinger equation, was recognized to play a central rôle in the description of Bose-Einstein condensates, whose experimental realization was awarded the Nobel prize in Physics.

The year 2002 is precisely the 30th anniversary of the solution of the nonlinear Schrödinger equation by Zhakharov and Shabat, a result that constitutes a landmark in the XXth century mathematics. The purpose of this meeting is to bring together some of the leading experts in this field with the aim of analyzing the state of the art and future perspectives in this discipline. The invited speakers will deliver a one-hour talk followed by a brief discussion. We encourage any interested researchers to attend the lectures, whose summary is provided below.

Lecturers

 

M. J. Ablowitz, Department of Applied Mathematics, University of Colorado, Boulder, USA

F. Calogero, Dipartimento di Fisica, Universitŕ di Roma "La Sapienza", Italy

A.S. Fokas, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK

M. Kruskal, Department of Mathematics, Rutgers University, New Jersey, USA

D. Maison, Max-Planck Institüt für Physik, Werner Heisenberg Institüt, Munich, Germany

G. Neugebauer, Theoretish Physikalisches Institüt, Friedrich Schiller Univeristat, Jena, Germany

A.B. Shabat, L.D.Landau Institute for Theoretical Physics, Russian Academy of Sciences, Moscow, Russia

P. Winternitz, Centre de Recherches Mathématiques, Université de Montréal, Canada

Seminars (abstracts)

 

M. J. Ablowitz, Integrability and Nonlinear Waves --- 150+ Years and counting

Historically, the study of nonlinear waves has benefited greatly from the interaction between asymptotic and exact methods of analysis. Asymptotic approximations of fundamental governing equations has led to reduced models which have, in important cases, been integrated by the Inverse Scattering Transform. Integrable equations are known in 0+1, 1+1, 2+1 dimensions. Interestingly most (all?) can also be obtained as reductions of the four dimensional self dual Yang-Mills system. Related issues, physical applications, and some open problems will also be discussed.

F. Calogero, Nonlinear harmonic oscillators

Results obtained recently with V. I. Inozemtsev will be reported. Specifically, the existence will be pointed out of assemblies of an arbitrary number of complex oscillators, or equivalently of an arbitrary even number of real oscillators, characterized by Newtonian equations of motion ("acceleration equal force") with one-body velocity-dependent linear forces and many-body velocity-independent cubic forces, all the nonsingular solutions of which are isochronous (completely periodic with the same period). As for the singular solutions, they emerge, in the context of the initial-value problem, from a closed domain in phase space having lower dimensionality.

            After the presentation of these results -- and compatibly with the time available -- a survey will be given of other results originating from the same "trick" and pointing out the existence of several nonlinear systems featuring lots of completely periodic solutions.

A.S. Fokas, Differential forms, spectral theory and boundary value problems

A new method for analyzing boundary value problems for linear and integrable nonlinear PDEs will be reviewed. This involves three steps: (a) Formulating the PDE as the closure condition of a differential form. (b) Performing the spectral analysis of this form. (c) Analyzing a certain global relation. For linear PDEs relations of this method with Ehrenpreis-Palamodov principle and with applied techniques such as the WienerĐHopf factorization will be discussed. For nonlinear PDEs, the advantage of this formulation for obtaining asymptotic results will be elucidated.

M. Kruskal, TBA

D. Maison, Regular and black hole solutions of Einstein-Yang-Mills theory

First I will present a survey of the many analytical and numerical results on the regular and black hole solutions of the EYM theory. Particular emphasis will be laid on the `critical' solutions on the boundary of the domain of existence in parameter space. In the case of self-gravitating magnetic monopoles these critical solutions yield interesting extremal black holes with `YM hair'.

G. Neugebauer, Rotating bodies as boundary value problems in general relativity

Several rotating body problems in astrophysics (black holes,disklike matter configurations) can be described by boundary value problems and solved by means of the Inverse Scattering Method. Physical properties of the solutions, in particular the interrelationship between angular momentum and black hole formation, are discussed.

A. B. Shabat, A hydrodynamic type model of solitonic theory

We discuss weakly nonlinear hierarchy of hydrodynamic type as the model of mathematical theory of solitons. Simplest reductions of this hierarchy describe Burgers type equations. 

P. Winternitz, Point symmetries and commuting flows for difference equations and lattices

A review is given of different methods for finding point symmetries and generalized symmetries of difference equations. Applications include symmetry reduction for partial difference equations, the classification of difference equations according to their symmetries, and the solution of difference equations with two and three dimensional symmetry groups.

Program

 

Friday, June 7
Saturday, June 8
09:30-10:30
Ablowitz

Shabat

10:30-10:45
Discussion
Discussion
10:45-11:45

Calogero

Winternitz
11:45-12:00
Discussion

Discussion

12:00-13:00

Lunch

Kruskal

13:00-13:15

Discussion

13:15-15:00

15:00-16:00

Fokas

16:00-16:15
Discussion
16:15-17:15
Maison

17:15-17:30

Discussion
17:30-18:30
Neugebauer

18:30-18:45

Discussion

 

Location and directions

The Seminar will take place in the Faculty of Physical Sciences (Facultad de Ciencias Físicas) of the Universidad Complutense. A location map can be found here.

 

Organizing comitee

Javier Chinea Trujillo

Federico Finkel Morgenstern

Artemio González López

Luis Manuel González Romero

Rafael Hernández Heredero

Manuel Mańas Baena

Luis Martínez Alonso

Miguel Ángel Rodríguez González