Scattering Theory and Integrable Systems / Diffusion inverse
et systèmes intégrables
(Jacek Szmigielski, Organizer)
- MARK ADLER, Department of Mathematics, Brandeis University, Waltham,
Massachusetts 02454, USA
Generalized orthogonal polynomials, integral mechanics and
Darboux transforms on band matrices and weights
Classically, a single weight on an interval of the real line leads to
moments, orthogonal polynomials and tridiagonal matrices.
Appropriatedly deforming this weight with times t=(t1,t2,...),
leads to standard Toda lattice and t-functions, expressed as
hermitian matrix integrals. This talk is concerned with a sequence of
t-perturbed weights, rather than one single weight.One example is
m-periodic sequences of weights leading to 2m+1 band matrices of
recurrence relations among generalized orthogonal polynomials and we
use the associated integrable system, 2-Toda, to perform a Darboux
transform on the matrix and the weights.
- RICHARD BEALS, Yale University, New Haven, Connecticut 06520-8283, USA
Inverse scattering, strings, and moments
Work with David Sattinger and Jacek Szmigielski is described.
A number of finite-dimensional integrable systems can be integrated
explicitly by converting them to problems for (possibly indefinite)
string densities. These include the multi-peakon/antipeakon cases of
the Camassa-Holm equation, the finite-dimensional reduction of the
Hunter-Saxton equation, and flows of Jacobi matrices, such as the finite
Toda lattice. In each case the Weyl function for the string problem
has a partial fraction expansion that incorporates the scattering data
and a continued fraction expansion that incorporates the string data.
A classical result of Sttieltjes recovers the latter data from the former.
In particular, asymptotic phase shifts are easily computed.
- YURI BEREST - I, Department of Mathematics, Cornell University, Ithaca,
New York 14853-4201, USA
Huygens' operators and Hadamard's conjecture
The question of characterizing linear differential systems which
propagate waves without diffusion (known also as the problem of
lacunas) goes back to the classical work of Hadamard and Petrovsky. In
early 70s Atiyah, Bott and Garding developed a profound and nearly
complete theory of lacunas for hyperbolic operators with constant
coefficients. Extending this theory to operators with variable
coefficients presents a major challenge, and the problem of lacunas
still remains largely open in this general case. The talk will review
some recent results and ideas in this direction.
- YURI BEREST - II, Department of Mathematics, Cornell University, Ithaca,
New York 14853-4201, USA
Geometry without points
In this talk I will explain connections between four apparently
>From the perspective of integrable systems, an (apparently new)
observation is that the space of algebraic solutions to the KP
hierarchy can be identified with the moduli spaces of framed line
bundles (or, more precisely, rank one torsion-free coherent sheaves) on
M. Artin's quantum projective plane (somewhat in parallel with the
well-known ADHM construction of instanton solutions to the classical
(1) Hilbert schemes of points on the complex affine plane,
(2) Ideal class spaces of the first complex Weyl algebra,
(3) Calogero-Moser Hamiltonian systems,
(4) The Adelic Grassmannian introduced by G. Wilson.
- PERCY DEIFT, Courant Institute and the University of Pennsylvania
On the integrable method and some recent examples of its
Recent applications of the ``Integrable Method'' to fields as far apart
as random matrix theory, orthogonal polynomials, combinatorics and
statistical growth models, are rooted in the ideas introduced by
Gardner, Greene Kruskal and Miura in their famous solution of the KdV
equation in 1967. The speaker plans to trace these developments and to
discuss some of the above applications.
- MISHA GEKHTMAN, University of Notre Dame, Notre Dame, Indiana 46556-5683, USA
Inverse moment problem for elementary co-adjoint orbits
We give a solution to the inverse moment problem for a class of
Hessenberg and symmetric matrices related to integrable lattices of
Toda type. (Joint work with L. Faybusovich)
- MARIA GOLENISHCHEVA-KUTUZOVA, University of Florida, Department of Mathematics, Gainesville,
Florida 32611-8105, USA
Soliton equations in terms of intertwining operator
The integrability of a large classes of nonlinear systems and even of
the hierarchies of integrable systems (including the famous KP, KdV and
NLS hierarchies) is related to a big group of hidden symmetries. First
this approach to the KP hierarchy appeared in the paper of M. Sato
and then was developed in the papers of Kyoto school, using the
boson-fermion correspondence in 2-dim QFT. They interpret the set of
solutions of the KP-hierarschy as an orbit of the vacuum vector of the
group GLinfty on the infinite-dimentional Grassmanian. We
propose a unified method of constructing the hierarchies of integrable
system with a given group of hidden symmetries in terms of intertwining
operators. This is a very beautiful application of representation
theory of infinite dimensional Lie groups.
- JACQUES HURTUBISE, Department of Mathematics and Statistics, McGill University,
Montréal, Québec H3A 2K6
Multi-Hamiltonian structures, classical r-matrix systems
spectral transforms and separation of variables
Classical r-matrices of rational, trigonometric and elliptic type may
be used to define polynomial families of multi-Hamiltonian structures
on loop algebras and loop groups corresponding, respectively, to linear
(Lie-Poisson) and quadratic (Poisson-Lie/Sklyanin) brackets. The
standard r-matrix theory implies commutativity and complete
integrability of the flows generated by spectral invariants on finite
dimensional Poisson submanifolds consisting of meromorphic Lax matrices
over the base curve (of genus 0 or 1) with given pole divisor. The
spectral transform yields an identification with the space consisting
of pairs of: spectral curves and sheaves supported on them, on which a
natural family of algebro-geometric Poisson structures is defined. A
third way to view such Poisson spaces is by identification with
symmetric products of a holomorphic Poisson surface with itself,
leading to a separation of variables of the flows generated by spectral
invariants in the associated ``spectral Darboux coordinates''. It is
shown that the generalized Gel'fand-Zakharevich commuting invariants
associated to the multi-Hamiltonian structures are the same as the
spectral invariants derived from the r-matrix theory, and the
``Nijenhuis-Darboux'' coordinates given by the eigenvalues and
eigenvectors of the Nijenhuis tensor coincide with the spectral Darboux
coordinates. Examples illustrating these results include: constrained
oscillators in n-dimensions, classical spin systems, Toda and
Volterra lattices and reduced systems associated with commuting flows
on the stationary manifolds of the NLS, KdV and Boussinesq
- GERARD MISIOLEK, University of Notre Dame
Well-posedness results for an integrable evolution equation
I will discuss recent results on the periodic initial value problem for
the Camassa-Holm equation. I will review the connection with the
Bott-Virasoro group and describe how it can be used to derive sharp
results on local well-posedness of the corresponding Cauchy problem.
- YOSHIMASA NAKAMURA, Graduate School of Informatics, Kyoto University, Kyoto
City 606-8501, Japan
Solving integrable systems in terms of continued fractions
In his fundamental paper by J. Moser (1975) an inverse scattering
problem for the finite nonperiodic Toda equation was shown to be solved
by using the Chebyshev continued fraction (C-fraction). The discrete
time Toda equation gives a method for computing the C-fraction from a
given power series. Recently, the C-fraction is successfully applied
to the Cammassa-Holm equation by Beals, Sattinger and Szmigielski. In
this talk I will discuss the Schur flow and the relativistic Toda
equation in terms of the T-fraction which includes the Perron
continued fraction. Integrable discretizations of these integrable
systems help me to design new algorithms for computing the
T-fraction. A joint work with Y. Minesaki and A. Mukaihira.
- BEATRICE PELLONI, Department of Mathematics, University of Reading,
Reading RG6 6AX, United Kingdom
Applications of the Fokas transform method to the study of
boundary value problems for PDEs in two dimensions
We survey some recent results obtained by using the method for solving
initial boundary value problems recently introduced by Fokas; see the
review in .
The Fokas transform method is a general method for analyzing boundary
value problems for integrable PDE's in two dimensions. An
equation in two dimensions is called integrable if it admits a
Lax pair formulation, i.e. if it can be written as the
compatibility condition of two linear eigenvalue equations. Examples of
such equations are linear PDEs with constant coefficients and the usual
nonlinear integrable PDEs such as the nonlinear Schrödinger equation
(NLS). The Fokas method, in the two dimensional case, is based on
performing the simultaneous spectral analysis of these two
Assuming the existence of a solution of a given boundary value problem,
this yields: (1) an integral representation of the solution in
Ehrenpreis form, namely a representation with explicit dependence on
the two equation variables; (2) a global relation between the boundary
values of the solution.
Moreover, by exploiting the invariance properties of the global
relation of (2), it is possible to identify the boundary value problems
that are well posed, and prove rigorous existence results for the
solution of such problems.
This method has been so far extensively applied to polygonal domains.
The cases considered include linear evolution equations with spatial
derivatives of arbitrary order on the half line , on a finite
interval , and in time-dependent domains , the NLS on the half
line  and on a finite interval , other nonlinear integrable
PDEs [7, 13], and several elliptic PDEs [6, 1]. Recently,
progress has been made on extending the method to more dimensions.
The author and Fokas have applied this method in the case of integrable
evolution equations in one space dimension, on a finite interval or in
a time-dependent domain. The aim of this presentation is to discuss the
Fokas transform method in general and to give the precise results
obtained when this method is applied to the above two cases.
1. D. ben-Avraham and A.S. Fokas,
The solution of the modified Helmholtz equation in a
wedge and an application to diffusion-limited coalescence.
Phys. Lett. A 263(1999).
A.S. Fokas, On the integrability of linear and nonlinear PDEs.
J. Math. Phys. 41(2000), 4188.
A.S. Fokas and A.R. Its, The generalised Ehrenpreis-Palamodov
representation and the long time asymptotics of evolution equations.
4. , The Nonlinear Schrödinger equation
on a finite interval, 2000, preprint.
5. A.S. Fokas, A.R. Its and L.Y. Sung, The Nonlinear
Schrödinger equation on the half line. 2000, preprint.
6. A.S. Fokas and A. Kapaev, An integral transform
method for the Laplace equation in a polygon. 2000, preprint.
7. A.S. Fokas and C.R. Menyuk, Integrability and
similarity in transient stimulated Raman scattering.
J. Nonlin. Science 9(1999), 1-31.
8. A.S. Fokas and B. Pelloni, A method of solving
moving boundary value problems for linear evolution equations.
Phys. Rev. Lett. 84(2000), 4785.
9. , Two-point boundary value problems for
linear evolution equations. Math. Proc. Cambridge Philos. Soc.
(2000), in press.
10. , Integrable evolution equations in
time-dependent domains. Inv. Prob. (2000), in press.
11. , Boundary value problems for
linearized Boussinesq type systems. Proceedings of the 5th
International conference on mathematical and numerical aspects of
wave propagation, Santiago de Compostela, Spain, July 10-14, 2000.
12. A.S. Fokas and L.Y. Sung, Initial boundary value
problems for linear dispersive evolution equations on the half-line.
13. A.S. Fokas, D. Tsoubelis and L.Y. Sung, The
inverse spectral method for colliding gravitational waves.
Math. Phys. Anal. Geom. 1(1999), 313-330.
14. P.C. Sabatier, Elbow scattering and inverse
scattering applications to LKdV and KdV. J. Math. Phys.
- ENRIQUE REYES, Department of Mathematics, Yale University, New Haven,
Connecticut 06520, USA
On differential geometry and integrability
We survey some recent developments in the geometrical study of
nonlinear partial differential equations:
We introduce the class of equations of pseudo-spherical type, and show
that their basic properties can be understood in geometrical terms. We
will consider mainly their conservation laws, and the existence of
Backlund/linearizing transformations. We will also define hierarchies
of equations describing pseudo-spherical surfaces, and present
transformations between any two hierarchies of pseudo-spherical type.
Finally, we apply this theory to the study of the Camassa-Holm shallow
water equation. In particular, we will show how to construct its
- DAVID SATTINGER, Department of Mathematics and Statistics, Utah State University,
Logan, Utah 84322, USA
Solitary wave interactions for the Euler-Poisson equations
Experimental studies on interacting solitary waves of the Euler-Poisson
equations for ion acoustic waves in plasmas were carried out at a range
of velocities. It is found that even at high Mach numbers, the
interactions are highly elastic, producing trailing dispersive waves of
very small order. At low amplitudes, the major discrepancy in the
Korteweg-deVries approximation comes in the displacements of the
emerging solitary waves from those of the 3-soliton solution, which
are produced by resonant interactions between the solitary waves.
Resonant interactions manifest themselves only at higher orders, so
that the time of validity of the KdV approximation by itself is
(Joint work with David Nicholls, University of Minnesota, and Mariana
Haragus, University of Bordeaux.)
- BARRY SIMON, Division of Physics, Mathematics and Astronomy, California
Institute of Technology, Pasadena, California 91125-3700, USA
The A-function approach to inverse spectral theory
After reviewing the two approaches to the inverse problem for Jacobi
matrices, I'll discuss their analogs for continuum Schrodinger
operators focusing on an approach I introduced in Annals of Math.
150(1999), 1029-1057. This is the analog of continued fractions for
the moment problem. Some open problems and conjectures will be