DISPERSIVE INTEGRABLE EQUATIONS: PATHFINDERS IN HAMILTONIAN PDE

15 juin 2026, 09:30 → 3 juil. 2026, 18:00 Europe/Paris

Institut Henri Poincaré

Thematic 3-weeks programme at the Institut Henri Poincaré, Paris, June 15th to July 3rd, 2026.

Programme talks and conference are in amphithéâtre Yvonne Choquet-Bruhat in the new IHP Perrin building.

Beware: It was reported to us that scammers are sending to participants fraudulent e-mails about accomodation/fees. Please be particularly cautious about e-mails not coming from the organisers nor from an @ihp.fr address. 

Presentation of the programme

Completely integrable systems have long served as pathfinders in mathematical physics.
Integrable PDE are regularly used as effective models for a wide array of phenomena seen in nonlinear optics, magnetohydrodynamics, Bose–Einstein condensates, and for both surface and internal waves in fluid mechanics. That such a seemingly narrow class of equations should attract such enduring attention from generations of mathematiciansand physicists is indicative of several factors: these equations exhibit myriad physical behaviors, including the elastic interaction of solitary waves, the soliton resolution phenomenon, but also blowup, turbulence, and ergodicity. More strikingly, in the completely integrable context, it is sometimes possible to describe such phenomena with explicit formulae! Furthermore, these behaviours that were first witnessed in the completely integrable setting are robust enough to be observed not only in non-integrable analogues, but even in experiments.
In hindsight, it is not surprising that these phenomena were first described mathematically in the completely integrable setting. Indeed, the rich algebraic and analytic structure of these equations renders them amenable to treatment via tools and techniques from across a wide spectrum of mathematics, such as harmonic analysis, Lie theory, algebraic geometry, inverse scattering, partial differential equations, random matrices, etc.
This proposal focuses on analytical tools used in the study of completely integrable systems. Even in this realm there are a multitude of competing technologies, such as Riemann–Hilbert methods, the theory of Hankel and Toeplitz operators, the method of commuting flows, dispersive PDE techniques, and harmonic analysis tools. The goal
of the proposed program is to bring together experts on these multifaceted approaches, with a view toward creating a new generation of researchers that are multilingual and can seamlessly glide between the vocabulary and tools that are currently endemic to
each research group.

 

The program unfolds over three weeks: lectures and a workshop.

Introductory lectures: from June 15 to 19, IHP, Amphithéâtre Choquet-Bruhat - Perrin building

Workshop: "Modern methods, techniques & results in dispersive integrable equation": from June 22 to 26, IHP, Amphithéâtre Choquet-Bruhat - Perrin building

Invited lectures & talks: from June 29 to July 3, IHP, Amphithéâtre Choquet-Bruhat - Perrin building

 

 

Registration is free but mandatory.
Deadline for asking financial support: January 15th, 2026. Financial support is limited.

Deadline for registration : January 31st, 2026. Places are limited.

 

Organising committee:

• Patrick Gérard (Laboratoire de Mathématique d’Orsay) 
• Tamara Grava (SISSA) 
• Peter Miller (University of Michigan) 
• Monica Visan (University of California)
• Nicolas Burq (Laboratoire de Mathématique d’Orsay)

 

Scientific committee:

• Hajer Bahouri (Laboratoire Jacques-Louis Lions - Sorbonne Université)
• Rowan Killip (University of California)
• Catherine Sulem (University of Toronto)
• Jean-Claude Saut (Université Paris-Saclay / CNRS Laboratoire de Mathématiques d'Orsay)

 

Fundings: 

The program receives also support from

 

lun. 15 juin

1 Patrick Gérard (Orsay) and Enno Lenzmann (Basel): Explicit formulae for nonlocal integrable PDEs and applications.

This mini-course provides a systematic introduction to explicit formulae, which have recently found a broad range of applications in the analysis of nonlocal completely integrable PDEs. Central examples for this approach via explicit formulae arise for the Benjamin-Ono equation (BO), Calogero-Moser(-Sutherland) derivative NLS (CM-DNLS), the cubic Szegö equation, and the Half-Wave Maps equation (HWM). A unifying feature of these completely integrable nonlocal PDEs is a Lax pair structure on Hardy spaces. The first part of this course will highlight the operator-theoretic analysis, posed on the torus as well as the real-line case. In the second part of the mini-course, we discuss some fundamental applications covering scaling-critical global well-posedness, finite-time blowup, and soliton resolution.

2 Patrick Gérard (Orsay) and Enno Lenzmann (Basel): Explicit formulae for nonlocal integrable PDEs and applications.

13:00 Lunch break

3 David Damanik (Rice) and Milivoje Lukıc (Emory): Integrable systems with ergodic initial data.

We begin by introducing central examples of Schr¨odinger operators and
Jacobi matrices with ergodic coefficients. As we will see, such operators open a Pandora’s
box of spectral theory: The spectrum can be a Cantor set and the spectral type can be
anything, absolutely continuous, singular continuous, or even a dense set of eigenvalues.
Our model classes of operators are singled out for their relevance to the Korteweg–de
Vries and Toda evolutions. After a brief review of the analysis of the periodic problem,
we will demonstrate how one goes about solving these evolutionary PDEs using the
spectral theory of ergodic operators.

4 David Damanik (Rice) and Milivoje Lukıc (Emory): Integrable systems with ergodic initial data.

We begin by introducing central examples of Schr¨odinger operators and
Jacobi matrices with ergodic coefficients. As we will see, such operators open a Pandora’s
box of spectral theory: The spectrum can be a Cantor set and the spectral type can be
anything, absolutely continuous, singular continuous, or even a dense set of eigenvalues.
Our model classes of operators are singled out for their relevance to the Korteweg–de
Vries and Toda evolutions. After a brief review of the analysis of the periodic problem,
we will demonstrate how one goes about solving these evolutionary PDEs using the
spectral theory of ergodic operators.

mar. 16 juin

5 Deniz Bilman and Ken McLaughlin: The Riemann–Hilbert method

We first explain how the inverse scattering approach to integrable systems can
be reformulated in terms of a Riemann–Hilbert problem and then discuss early methods
for the solution of such problems using the theory of singular integrals. We then describe
the important role played by deformations of a Riemann–Hilbert problem and how this
has led both to detailed information on the long-time asymptotics of integrable PDE
and to the incorporation of ever more singular initial data.

6 Patrick Gérard (Orsay) and Enno Lenzmann (Basel): Explicit formulae for nonlocal integrable PDEs and applications.

This mini-course provides a systematic introduction to explicit formulae, which have recently found a broad range of applications in the analysis of nonlocal completely integrable PDEs. Central examples for this approach via explicit formulae arise for the Benjamin-Ono equation (BO), Calogero-Moser(-Sutherland) derivative NLS (CM-DNLS), the cubic Szegö equation, and the Half-Wave Maps equation (HWM). A unifying feature of these completely integrable nonlocal PDEs is a Lax pair structure on Hardy spaces. The first part of this course will highlight the operator-theoretic analysis, posed on the torus as well as the real-line case. In the second part of the mini-course, we discuss some fundamental applications covering scaling-critical global well-posedness, finite-time blowup, and soliton resolution.

13:00 Lunch break

7 David Damanik (Rice) and Milivoje Lukıc (Emory): Integrable systems with ergodic initial data

We begin by introducing central examples of Schr¨odinger operators and
Jacobi matrices with ergodic coefficients. As we will see, such operators open a Pandora’s box of spectral theory: The spectrum can be a Cantor set and the spectral type can be anything, absolutely continuous, singular continuous, or even a dense set of eigenvalues.
Our model classes of operators are singled out for their relevance to the Korteweg–de Vries and Toda evolutions. After a brief review of the analysis of the periodic problem, we will demonstrate how one goes about solving these evolutionary PDEs using the spectral theory of ergodic operators.

8 Public lecture by J. C.Saut, Amphi Hermite

18:15 Reception, Espace Emmy Noether

mer. 17 juin

9 Deniz Bilman and Ken McLaughlin: The Riemann–Hilbert method

We first explain how the inverse scattering approach to integrable systems can be reformulated in terms of a Riemann–Hilbert problem and then discuss early methods for the solution of such problems using the theory of singular integrals. We then describe the important role played by deformations of a Riemann–Hilbert problem and how this
has led both to detailed information on the long-time asymptotics of integrable PDE and to the incorporation of ever more singular initial data.

10 Ben Harrop-Griffiths (Georgetown) and Maria Ntekoume (Concordia): The method of commuting flows and its applications to optimal well-posedness.

We begin with the notion of Hs-equicontinuity of orbits, how it is proved, and why it is important. We then move to the role of commuting flows beginning with a simple example. Lastly, we describe the increasingly sophisticated techniques that have been required in order to
achieve sharp results across a spectrum of integrable models.

13:00 Lunch Break

11 Manuela Girotti (Emory) and Bob Jenkins (Florida), Soliton gases.

The concept of a soliton gas was introduced by V. Zakharov
in 1971 and further extended by G. El. The physical intuition is that the dispersive dynamic of a strongly nonlinear and integrable random field is dominated by solitons interactions. A deterministic model for such fields involves the idea of a primitive potential as a condensation of many solitons. On the other hand, the
kinetic theory of solitons is rapidly booming and makes strong connections with the theory of dispersive hydrodynamics, originally developed by Whitham and the theory of generalized hydrodynamics that has emerged recently to describe the non-equilibrium physics of integrable systems where the extensive amount of ballistic transport renders conventional theories inapplicable. These lectures will survey both the deterministic and kinetic approaches to soliton gases.

jeu. 18 juin

12 Ben Harrop-Griffiths (Georgetown) and Maria Ntekoume (Concordia): The method of commuting flows and its applications to optimal well-posedness

We begin with the notion of Hs-equicontinuity of orbits, how it is proved, and why it is important. We then move to the role of commuting flows beginning with a simple example. Lastly, we describe the increasingly sophisticated techniques that have been required in order to
achieve sharp results across a spectrum of integrable models.

13 Manuela Girotti (Emory) and Bob Jenkins (Florida): Soliton gases.

The concept of a soliton gas was introduced by V. Zakharov
in 1971 and further extended by G. El. The physical intuition is that the dispersive dynamic of a strongly nonlinear and integrable random field is dominated by solitons interactions. A deterministic model for such fields involves the idea of a primitive potential as a condensation of many solitons. On the other hand, the kinetic theory of solitons is rapidly booming and makes strong connections with the
theory of dispersive hydrodynamics, originally developed by Whitham and the theory of generalized hydrodynamics that has emerged recently to describe the non-equilibrium physics of integrable systems where the extensive amount of ballistic transport renders
conventional theories inapplicable. These lectures will survey both the deterministic and
kinetic approaches to soliton gases.

13:00 Lunch break

14 Deniz Bilman and Ken McLaughlin: The Riemann–Hilbert method

We first explain how the inverse scattering approach to integrable systems can be reformulated in terms of a Riemann–Hilbert problem and then discuss early methods for the solution of such problems using the theory of singular integrals. We then describe the important role played by deformations of a Riemann–Hilbert problem and how this
has led both to detailed information on the long-time asymptotics of integrable PDE and to the incorporation of ever more singular initial data.

ven. 19 juin

15 Manuela Girotti (Emory) and Bob Jenkins (Florida): Soliton gases.

The concept of a soliton gas was introduced by V. Zakharov
in 1971 and further extended by G. El. The physical intuition is that the dispersive dynamic of a strongly nonlinear and integrable random field is dominated by solitons interactions. A deterministic model for such fields involves the idea of a primitive potential as a condensation of many solitons. On the other hand, the kinetic theory of solitons is rapidly booming and makes strong connections with the theory of dispersive hydrodynamics, originally developed by Whitham and the theory of generalized hydrodynamics that has emerged recently to describe the non-equilibrium physics of integrable systems where the extensive amount of ballistic transport renders conventional theories inapplicable. These lectures will survey both the deterministic and
kinetic approaches to soliton gases.

16 Ben Harrop-Griffiths (Georgetown) and Maria Ntekoume (Concordia): The method of commuting flows and its applications to optimal well-posedness

We begin with the notion of Hs-equicontinuity of orbits, how it is proved, and why it is important. We then move to the role of commuting flows beginning with a simple example. Lastly, we describe the increasingly sophisticated techniques that have been required in order to
achieve sharp results across a spectrum of integrable models.