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

09:30 → 11:00 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.

11:00 → 11:30 Coffee break

11:30 → 13:00 Patrick Gérard (Orsay) and Enno Lenzmann (Basel): Explicit formulae for nonlocal integrable PDEs and applications.

13:00 → 15:00 Lunch break

15:00 → 16:30 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.

16:30 → 17:00 Coffee break

17:00 → 18:30 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

09:30 → 11:00 Deniz Bilman (Cincinnati) and Ken McLaughlin (Tulane): 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.

11:00 → 11:30 Coffee break

11:30 → 13:00 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 → 15:00 Lunch break

15:00 → 16:30 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.

17:15 → 18:15 Public lecture by J. C.Saut, Amphi Hermite

18:15 → 19:15 Reception, Espace Emmy Noether

mer. 17 juin

09:30 → 11:00 Deniz Bilman (Cincinnati) and Ken McLaughlin (Tulane): 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.

11:00 → 11:30 Coffee break

11:30 → 13:00 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 → 15:00 Lunch Break

15:00 → 16:30 Manuela Girotti (Emory) and Bob Jenkins (Central 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

09:30 → 11:00 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.

11:00 → 11:30 Coffee break

11:30 → 13:00 Manuela Girotti (Emory) and Bob Jenkins (Central 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 → 15:00 Lunch break

15:00 → 16:30 Deniz Bilman (Cincinnati) and Ken McLaughlin (Tulane): 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

09:30 → 11:00 Manuela Girotti (Emory) and Bob Jenkins (Central 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.

11:00 → 11:30 Coffee break

11:30 → 13:00 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.

lun. 22 juin

09:00 → 09:30 Opening Remarks

Orateur: Mariana Grana (Institut Henri Poincare)

09:30 → 10:30 An explicit formula for the Benjamin-Ono hierarchy with applications to traveling waves and zero-dispersion limits

In this talk, we demonstrate how the Lax pair structure leads to an explicit formula for the Benjamin–Ono Hierarchy on the line. We then present two main applications of this formula. First, we obtain a complete classification of traveling wave solutions for all higher-order flows within the hierarchy. Second, we investigate the zero-dispersion limit of these flows and provide a precise characterisation of the limit as an alternating sum of branches. This is a joint work with Patrick Gérard.

Orateur: Jiao He (LMO)

10:30 → 11:00 Coffee Break

11:00 → 12:00 The soliton resolution conjecture for the Benjamin-Ono equation

We discuss the soliton resolution conjecture for the Benjamin-Ono equation on the line. More precisely, we show that for a general class of initial data, the solution can be decomposed as a sum of soliton solutions, a radiative term, and a small remainder term, when time goes to infinity. The proof lies on an explicit formula derived by Gérard in 2023. This is a joint work with P. Gérard and P. D. Miller.

Orateur: Louise Gassot (CNRS et Université de Rennes)

12:00 → 15:00 Lunch Break

15:00 → 16:00 The long-period limit of the Benjamin-Ono equation

When numerically approximating a PDE on the line, it is common practice to replace the line by a large circle. At least for short times, this seems like a reasonable substitution, but how reasonable is it really? In this talk, I will address this question for the Benjamin–Ono equation, whose integrable structure leads to quantitative answers. This is ongoing joint work with Yvonne Alama-Bronsard.

Orateur: Ola Maehlen (University of Paris, Saclay)

16:00 → 16:30 Coffee Break

16:30 → 17:30 From explicit formulas to wave-kinetic theory for the Benjamin-Ono equation

In the first part of this talk, we begin by introducing an explicit solution formula for the Benjamin-Ono (BO) equation obtained by Patrick Gérard. We show how building on this representation yields new tools for understanding the long-time dynamics of BO, both theoretically and numerically.

In the second part, we study the weakly nonlinear evolution of BO starting from a randomized initial data on a large torus. By rescaling and iterating this explicit formula, combined with probabilistic arguments, we obtain insight on the wave-kinetic theory for BO, up to the physically relevant kinetic time scale and across all scaling laws in the kinetic regime. To our knowledge, this provides the first rigorous characterization of wave-kinetic dynamics for a one-dimensional integrable PDE up to these physically relevant timescales.

Orateur: Yvonne Alama Bronsard (MIT)

mar. 23 juin

09:30 → 10:30 On global well-posedness of the derivative nonlinear Schrödinger equation on the circle

In this talk, I will discuss a recent joint work with Hajer Bahouri, establishing global well-posedness of the derivative nonlinear Schrödinger equation in $H^1$ on the circle.

Orateur: Galina Perelman (UPEC)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Direct and inverse scattering for the continuum Calogero-Moser equation

The CCM equation (also known as Calogero–Moser derivative nonlinear Schrödinger equation) is a nonlinear dispersive equation in 1+1 dimensions that is completely integrable. The corresponding Lax operator is a first order operator in the Hardy space on the real line. We develop a spectral theory of this operator, building Jost solutions, proving absence of singularly continuous spectrum and introducing scattering coefficients. We also prove trace formulas of Birman-Krein and Faddeev-Zakharov type. Finally, we propose an inverse scattering scheme for the solution of the CCM equation.

The talk does not assume any previous knowledge of the CCM equation. It is based on joint work with Larry Read.

Orateur: Rupert Frank (University of Munich)

12:00 → 15:00 Lunch Break

15:00 → 16:00 The Hamiltonian formulation of the continuum Calogero-Moser model on the torus

The continuum Calogero-Moser equation has received a significant amount of attention from the mathematical community since Gerard and Lenzmann showed it to be a completely integrable system in 2022. The equation was originally derived on the real line, however attention has recently turned to the equation on the torus. A key feature which is missing in the direct generalization to the torus is the Hamiltonian structure. In this talk we present a Hamiltonian formulation for equation on the torus in a moving frame, and use our results to provide a new proof of global well-posedness in the scaling critical Hardy space. The content of this talk is joint work with Rowan Killip and Monica Visan.

Orateur: Katie Marsden (UCLA)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Well-posedness for the intermediate nonlinear Schrödinger equations

The intermediate nonlinear Schrödinger equation was first introduced as a defocusing model to describe the modulation of internal waves in a stratified fluid. However, with either a focusing or defocusing nonlinearity the resulting system is completely integrable. For both models, the shallow-depth limit leads to the cubic NLS equations, while the infinite-depth limit yields the continuum Calogero–Moser equations. In this talk, we will discuss some recent well-posedness results for the intermediate nonlinear Schrödinger equations on both the line and the circle. This is based on joint works with Andreia Chapouto and Justin Forlano.

Orateur: Thierry Laurens (University of Wisconsin–Madison)

17:30 → 18:30 TBA

TBA

Orateur: Andreia Chapouto (CNRS, Laboratoire de Mathematiques de Versailles)

mer. 24 juin

09:30 → 10:30 Asymptotics of Padé and potential theory on Riemann surfaces

I will discuss recent progress in the quest to generalize the Riemann-Hilbert techniques for asymptotics in higher genus Riemann surfaces, discussing in some detail the related potential theoretic side and connection with classical problems of Chebotarev-Polya.

Orateur: Marco Bertola (Concordia university)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Extremal problems related to focusing NLS soliton condensates

Soliton gases for integrable systems is a rapidly developing new area in the
theory of nonlinear waves. For a given spectral support set, a soliton gas of maximal average intensity is called soliton condensate. A spectral support set for a fNLS soliton condensate is often represented by a finite collection of points (anchors) $E$ in the upper half-plane $\mathbb{C}^+$, connected with each other and/or with the real axis by some arcs. Given a set of anchors $E$, a natural question is to find a collection of such arcs that produces fNLS soliton condensate of minimal average intensity. That leads to a modification of the well-known Chebotarev’s continuum problem, which consists in finding a continuum $K\subset\mathbb{C}$ of minimal logarithmic capacity that contains a given set of anchors $E\subset\mathbb{C}$. In this talk we discuss the modified Chebotarev’s problem, where $E$ and $K$ are subsets of $\mathbb{C}^+$ and where we minimize the Dirichlet energy of the Green potential (for $\mathbb{C}^+$) in $\mathbb{C}^+ \setminus K$ instead of the logarithmic capacity of $K$. This is a joint work with M. Bertola.

Orateur: Alexander Tovbis (University of Central Florida)

12:00 → 18:00 Free Afternoon

jeu. 25 juin

09:30 → 10:30 Global well-posedness for the mNV and NV equations

This talk concerns joint work with Adrian Nachman and Daniel Tataru on global well-posedness for two completely integrable, nonlinear dispersive equations in two space dimensions. We'll begin by pointing out some important differences between inverse scattering for equations in 2+1 dimensions as opposed to 1+1 dimensions. We will then describe how inverse scattering techniques combine with nonlinear harmonic analysis of the scattering operator to produce the global well-posedness result. The paper is available at https://arxiv.org/pdf/2511.21564.

Orateur: Peter Perry (University of Kentucky)

Perry-IHP-2026-06.pdf

10:30 → 11:00 Coffee Break

11:00 → 12:00 Infinite-peakon solutions of the Camassa-Holm equation

The aim of this talk is to discuss infinite-peakon solutions to the Camassa-Holm equation on the line, that is, a class of (conservative) low regularity solutions having the form of an infinite superposition of peakons (peaked solitons). Our major tool is the classical moment problem (in the framework of generalized indefinite strings). In particular, we will discuss which solutions are amenable to this approach, determine which part of the solution can be recovered from the moments of the underlying spectral measure and provide explicit formulas.

As an application, our results can be used to investigate the long-time behavior of these solutions. We will demonstrate this by considering several illustrative examples.
The talk is based on joint work with X.-K. Chang (Beijing) and J. Eckhardt (Loughborough).

Orateur: Aleksey Kostenko (University of Ljubljana/TU Graz)

12:00 → 15:00 Lunch Break

15:00 → 16:00 Asymptotic stability of Benjamin-Ono multisolitons in $L^2(\mathbb{R})$

The Benjamin-Ono equation is a canonical, completely integrable model for interface waves in stratified fluids of great depth. Recently, the orbital stability of its multisoliton solutions has been established in $H^s(\mathbb{R})$ for all $s > -1/2$. In this talk, we establish the asymptotic stability of these multisolitons under arbitrary, generic $L^2(\mathbb{R})$ perturbations, completely removing the need for spatial decay hypotheses. Relying on the explicit formula for the Benjamin-Ono flow, we evaluate the long-time limits of the solution along arbitrary spacetime rays to demonstrate that the flow ultimately decouples into individual solitons. This is joint work with Rowan Killip and Monica Vişan.

Orateur: Rana Baddredine (UCLA)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Large amplitude focusing events in the AKNS hierarchy

Typical solutions of small dispersion NLS regularize the shock formation predicted by the dispersionless equation in a universal way. This behavior, described in detail by Bertola and Tovbis, describes solutions which remain bounded in a neighborhood of the dispersionless shock point. In this talk I will discuss a special class of initial data, introduced by Talanov, for which the dispersionless shock generates finite time blow-up. Our results show this blow-up is regularized by dispersion, but have amplitudes inversely proportional to the dispersion. This new universal behavior is described by a particular solution of the Painleve III equation associated with "rogue waves of infinite order". We further show this blow-up behavior extends to the full focusing NLS hierarchy. This is joint work with Peter Miller and Robbie Buckingham.

Orateur: Robert Jenkins (University of Central Florida)

ven. 26 juin

09:30 → 10:30 The rogue waves of NLS and their universality

The study of extreme, or "rogue", waves in physical systems has gained significant attention, with the focusing Nonlinear Schrödinger (NLS) equation serving as a canonical integrable model. While much of the mathematical literature focuses on deterministic initial data, incorporating randomness is crucial to understand the emergence and structure of these extreme events.

In this seminar, I will present different classes of NLS rogue waves with randomness. Specifically, we will focus on extremal $N$-soliton solutions that achieve theoretical maximal amplitudes, where the discrete eigenvalues are randomly drawn from sub-exponential distributions. By analyzing the underlying Riemann-Hilbert problems, I will show that the formation of these rogue waves is a universal phenomenon robust to randomness. We identify two distinct universality classes governed by the Painlevé-III and Painlevé-V equations.

This talk is based on recent joint works with A. Gkogkou, K. D. T-R McLaughlin, T. Grava, R. Jenkins, M. Girotti and M. Yattselev:

• Painlevé Universality classes for the maximal amplitude solution of the Focusing Nonlinear Schrödinger Equation with randomness, Aikaterini Gkogkou, GM, and Kenneth D. T.-R. McLaughlin, 2026, https://arxiv.org/2602.05101.
• Soliton synchronization with randomness: rogue waves and universality, Manuela Girotti, Tamara Grava, Robert Jenkins, GM, Ken McLaughlin and Maxim Yattselev, Nonlinearity, Nov 2025, https://arxiv.org/2507.01253.

Orateur: Guido Mazzuca (Tulane university)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Riemann-Hilbert singularities from a distance: weakly localized data and long-time asymptotics

I will discuss two families of weakly localized solutions of the focusing nonlinear Schrödinger equation, both represented by Riemann-Hilbert problems posed on large circles. In each case, the jump matrix is regular on the contour but encodes a singularity hidden inside: a Blaschke-type/logarithmic singularity in a Painlevé-V-related family, and an essential singularity for general rogue waves of infinite order. Although these solutions belong to $L^2(\mathbb{R})$ but not $L^1(\mathbb{R})$, and hence fall outside the standard inversescattering framework, their Riemann-Hilbert representations still allow for rigorous long-time asymptotic analysis. I will compare the resulting decay rates, $O(t^{-1 / 2})$ in the Painlevé-V case and the anomalously slow $O(t^{-1 / 3})$ rate for the Painlevé-III-related infinite-order rogue waves, and describe a limiting regime connecting the two families. Time permitting, I will describe ongoing work with P. Miller on another class of solutions that are weakly localized on a nonzero background. The Painlevé-V part is joint work with A. Gkogkou, G. Mazzuca, and K. McLaughlin, and the infinite-order rogue-wave part is joint work with L. Ling and P. Miller.

Orateur: Dr Deniz Bilman (University of Cincinnati)

12:00 → 15:00 Lunch Break

15:00 → 16:00 Soliton gas for the focusing nonlinear Schrödinger equation

We consider soliton gas for the focusing nonlinear Schroedinger equation, and study its long-time asymptotic properties. We show that the $x,t>0$ half-plane is divided into several sharply separated regions, where the asymptotics is described in terms of hyperelliptic functions of genus from one to three. This is a joint work with Tamara Grava and Giuseppe Orsatti.

Orateur: Oleksandr Minakov (Charles University, Prague, Czechia)

16:00 → 16:30 Coffee Break

16:30 → 17:30 Semiclassical soliton ensembles for the intermediate long wave and Korteweg-de Vries equations

Semiclassical soliton ensembles (SSE) in the small dispersion limit are initially coherent collections of many solitons that well-approximate some initial profile. Evolving forward in time, the profile will eventually undergo wave breaking, shedding the solitons and generating a dispersive shock wave. We study this phenomenon for two PDE. The first SSE, for the intermediate long wave equation, is constructed to approximate general smooth Klaus-Shaw initial data. We first conduct a heuristic WKB approximation to determine the approximate scattering data and then rigorously study the inverse scattering problem using the methods of Lax and Levermore. We show the initial condition is recovered in the limit and the solution up until wave breaking approaches that of Invicid Burgers' equation in an $L^2$ sense. The second SSE is the $\mathrm{sech}^2$ initial condition for the Korteweg-de Vries equation. Inverse scattering is done via a Riemann-Hilbert problem and the method of nonlinear steepest descent is employed. This project is joint work with K. Schmidt (University of Central Florida) and R. Buckingham (University of Cincinnati).

Orateur: Matt Mitchell (University of Central Florida)