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.

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

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.

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.

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.

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.

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.

On Boussinesq and related systems

• Jean-Claude Saut (Université Paris-Saclay)

Public lecture, in Amphi Hermite, Borel Building

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.

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.

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.

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.

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.

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.

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.

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.

Opening Remarks

• Mariana Grana (Institut Henri Poincare)

An explicit formula for the Benjamin-Ono hierarchy with applications to traveling waves and zero-dispersion limits

• Jiao He (LMO)

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.

The soliton resolution conjecture for the Benjamin-Ono equation

• Louise Gassot (CNRS et Université de Rennes)

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.

The long-period limit of the Benjamin-Ono equation

• Ola Maehlen (University of Paris, Saclay)

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.

From explicit formulas to wave-kinetic theory for the Benjamin-Ono equation

• Yvonne Alama Bronsard (MIT)

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.

The Hamiltonian formulation of the continuum Calogero-Moser model on the torus

• Katie Marsden (UCLA)

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.

Gauge transform for the Korteweg-de Vries equation and well-posedness below the H^{-1}-scale

• Andreia Chapouto (Monash University)

In this talk, we consider the low regularity well-posedness problem for the Korteweg-de Vries equation (KdV) on the real line. Aiming to bridge the regularity gap between the scaling critical space $H^{-3/2}$ and the known optimal well-posedness in $H^{-1}$ in $L^2$-based Sobolev spaces, we consider rough data in Fourier-Lebesgue spaces. Via infinite normal form reductions and exploiting algebraic cancellations, we introduce a new gauged KdV equation, equivalent to the original one at high regularity, but better behaved for rough solutions below the $H^{-1}$-scale. Surprisingly, our method does not rely on the completely integrable structure of KdV and is easily adapted to other equations with quadratic derivative nonlinearities, such as the dispersion-generalized Benjamin-Ono equations. This talk is based on joint work with Simão Correia (IST, U. Lisboa) and João Pedro Ramos (IMPA).

On global well-posedness of the derivative nonlinear Schrödinger equation on the circle

• Galina Perelman (UPEC)

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.

Direct and inverse scattering for the continuum Calogero-Moser equation

• Rupert Frank (University of Munich)

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.

Asymptotics of Padé and potential theory on Riemann surfaces

• Marco Bertola (Concordia university)

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.

Extremal problems related to focusing NLS soliton condensates

• Alexander Tovbis (University of Central Florida)

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.

Well-posedness for the intermediate nonlinear Schrödinger equations

• Thierry Laurens (University of Wisconsin–Madison)

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.

Large amplitude focusing events in the AKNS hierarchy

• Robert Jenkins (University of Central Florida)

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.

Global well-posedness for the mNV and NV equations

• Peter Perry (University of Kentucky)

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.

Infinite-peakon solutions of the Camassa-Holm equation

• Aleksey Kostenko (University of Ljubljana/TU Graz)

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).

Asymptotic stability of Benjamin-Ono multisolitons in $L^2(\mathbb{R})$

• Rana Baddredine (UCLA)

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.

The rogue waves of NLS and their universality

• Guido Mazzuca (Tulane university)

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.

Riemann-Hilbert singularities from a distance: weakly localized data and long-time asymptotics

• Deniz Bilman (University of Cincinnati)

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.

Soliton gas for the focusing nonlinear Schrödinger equation

• Oleksandr Minakov (Charles University, Prague, Czechia)

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.

Semiclassical soliton ensembles for the intermediate long wave and Korteweg-de Vries equations

• Matt Mitchell (University of Central Florida)

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).

Finite-time blow-up solutions for the focusing Calogero–Sutherland derivative nonlinear Schrödinger equation

• Xi Chen (Laboratoire de Mathematiques d'Orsay)

In this talk, we will discuss finite-time blow-up solutions for the focusing Calogero–Sutherland derivative nonlinear Schrödinger equation on the torus. While global well-posedness below the critical mass threshold was previously known, we construct a family of smooth finite-time blow-up solutions with supercritical mass. The construction is based on an analysis of the explicit formula. We consider a class of finite-gap potentials and identify a precise resonant condition under which the corresponding solution blows up in finite time. This resonance is detected through the appearance of a unimodular eigenvalue of the operator arising in the explicit formula. We then use the explicit formula to obtain a complete description of the blow-up dynamics, including the blow-up rate of all Sobolev norms. We will also explain the complementary non-resonant case, where the finite-gap family gives global solutions with uniform-in-time Sobolev bounds. This yields a sharp resonance/non-resonance dichotomy and shows instability of the constructed blow-up solutions. This is joint work with Enno Lenzmann.

Global well-posedness for the intermediate NLS equation with non-vanishing data at infinity

• Slim ibrahim (University of Victoria, Department of Mathematics and Statistics)

The intermediate nonlinear Schrödinger equation (INLS) describes the dynamics of the envelope of weakly nonlinear internal waves in a stratified fluid of finite depth. While the INLS equation is known to admit dark soliton solutions, these solutions possess nonvanishing boundary conditions at spatial infinity and therefore fall outside the scope of existing well-posedness frameworks. This paper establishes the local and global well-posedness of a generalized INLS equation in Zhidkov-type spaces tailored to these nonvanishing boundary conditions. Furthermore, we rigorously justify the deep-water limit, proving that solutions of the generalized INLS converge to those of the generalized Calogero-Moser (CM) derivative NLS equation in Zhidkov-type spaces. Our well-posedness theory relies on the modified energy method combined with frequency envelopes, marking the first application of these techniques to Zhidkov-type spaces. This is a joint work with T. Akahori (Shizuoka), R. Badreddine (UCLA) and K. Nobu (RIMS)

The formation of a soliton gas condensate for the focusing nonlinear Schrödinger equation

• AIKATERINI GKOGKOU (Tulane University)

In this talk, we consider the focusing Nonlinear Schrödinger (NLS) equation and its multi-soliton solution when the number of solitons grows to infinity. We discover configurations of multi-soliton solutions that exhibit the formation of a soliton gas condensate. Specifically, we show that when the associated discrete eigenvalues accumulate on two bounded horizontal segments in the complex plane, and the associated norming constants are bounded away from zero, the solution to the focusing NLS equation is described by a rapidly oscillatory elliptic wave with constant velocity, on compact subsets of the $(x,t)$ domain.

Soliton and breather resolution for the cubic Szegö flow on the line

• Sandrine Grellier (Université d'Orléans/IDP)

We investigate the long–time behaviour of the solutions of the cubic Szegö equation on the line in the Sobolev space of order 1/2. We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi–periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem.

Direct Scattering of the Focusing nonlinear Schrödinger equation with step-like oscillatory initial data and full soliton gas

• Zechuan Zhang (SISSA)

In this talk, I will discuss recent work on the focusing nonlinear Schrödinger equation with step-like elliptic backgrounds. I will first present the direct and inverse scattering theory for step-like traveling wave solutions and then describe a special setting in which the left and right spectral bands coincide, corresponding to the same elliptic background with only a small perturbation. Finally, I will introduce the full soliton gas problem on elliptic backgrounds and discuss its motivation and mathematical formulation.

Developments in the Riemann-Hilbert problem for the defocusing nonlinear Schrödinger equation

• Joanne Dong (University of Michigan)

We discuss some progress in formulating the Riemann-Hilbert problem associated with the Defocusing Nonlinear Schrodinger (dNLS) equation with nonzero boundary conditions. This project builds on the work of Jin, Levermore, and McLaughlin, and aims to describe the solution to dNLS after the shock forms and the Madelung description of the problem breaks down. By strengthening the limit in recovering the solution to dNLS from the Riemann-Hilbert problem, we hope to contribute another verification of the Dubrovin-Grava-Klein-Moro universality conjecture.

Riemann problem for KdV soliton condensates

• Xiaodong Zhu (Beijing Normal Universaty)

We investigate the Riemann problem for KdV soliton condensates, where the left and right states are characterized by distinct densities of states associated with two soliton condensates. At the level of physical initial data, this problem may be viewed as a perturbation of a step-like finite-gap background. However, such perturbations lie beyond the scope of the currently available inverse scattering theory for step-like finite-gap potentials, making a direct IST analysis unavailable. In contrast to previously studied configurations, we allow the spectral bands of the left and right condensates to overlap. The resulting overlap region represents a full soliton gas and produces a new intermediate asymptotic region in which the leading-order term is a genus-one finite-gap solution with fixed spectral data but a nontrivially modulated phase. Such a phenomenon does not occur in the non-overlapping regime.

Infinitely many Calogero-Moser particles

• Matthew Kowalski (University of California, Los Angeles)

We consider infinitely many Calogero-Moser particles on the line. While the system with finitely many particles has been intensively studied, the case of infinitely many particles has so far only been studied in relation to long-wave or continuum limits. In this talk, we establish global well-posedness through integrability, an explicit formula in the sense of Olshanetsky-Perelomov, and construct a non-negative Hamiltonian. [Link to talk notes.][1] [1]: https://www.math.ucla.edu/~mattkowalski//documents/CM_gwp_chalk.pdf