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 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.

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.

Public lecture by J. C.Saut, Amphi Hermite

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.

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 (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 (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 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.

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.

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.