Hamiltonian PDEs
Walter Craig, McMaster University
Many partial differential equations of physical relevance to nonlinear wave evolution can be formulated as Hamiltonian dynamical systems with infinitely many degrees of freedom. Examples include the Euler equations for free surfaces and for interfacial waves in incompressible fluids, the nonlinear Schroedinger equations of modulational perturbation regimes and of many-body quantum theory, the Gross -- Pitaevski equation of Bose -- Einstein condensates, and the Fermi -- Pasta -- Ulam system of interacting particles. In these lectures we will introduce the general picture of the phase space, the flow for PDEs and conserved integrals of motion, discuss normal forms transformations and the concept of Nekhoroshev stability.
Wave Turbulence and Complete Integrability
Patrick Gérard, Université Paris-Sud
The most famous completely integrable PDEs --- KdV and 1D cubic NLS --- do not display any kind of wave turbulence, since the conservation laws control high regularity, making impossible the long time appearance of small scales. In this course, I will discuss a new type of integrable infinite-dimensional system, posed on functions of one variable, allowing dramatic growth of high Sobolev norms. The analysis is connected to the solution of an inverse spectral problem for Hankel operators from classical analysis. I will also discuss how this phenomenon can be exported to some Hamiltonian PDEs which can be seen as perturbations of this new type of integrable system.
Riemann-Hilbert Methods
Peter Miller, University of Michigan
Many dispersive equations in one space and one-time dimension that are completely integrable admit the careful analysis of various singular limits through a study of their associated inverse-scattering transforms. As the inverse problem in question is usually a Riemann-Hilbert problem, one needs a toolbox of asymptotic methods for such problems that are nonlinear or noncommutative analogues of well-known asymptotic methods for evaluation of oscillatory or exponential integrals.
In this mini-course, we will show how the inverse-scattering transform solution of the initial-value problem for the defocusing nonlinear Schrödinger equation leads to a Riemann-Hilbert problem of complex function theory. We will discuss the solution of Riemann-Hilbert problems via singular integral equations and their deformations. Then we will consider the use of suitable deformations to study the defocusing nonlinear Schrödinger equation in the long-time limit. Piecewise analytic deformations suffice in the presence of exponential decay of the initial data, but otherwise one can use "nearly analytic'' deformations at the cost of converting the Riemann-Hilbert problem into an $\overline\partial$ problem.
A Functional Analyst Looks at Inverse Scattering
Peter Perry, University of Kentucky
The goal of these lectures is to give a rigorous analysis of the inverse scattering solutions to (1) the defocussing cubic nonlinear Schrodinger equation (dNLS) in one space dimension and (2) the Davey-Stewartson II equation (DS II) in two dimensions. Our discussion of dNLS will follow the seminal paper of Deift and Zhou in Comm. Pure Appl. Math. 56 (2003), no. 8, 1029–1077, while our discussion of DS II will follow the paper of Perry in J. Spectr. Theory 6 (2016), no. 3, 429–481. Our discussion of dNLS will discuss Riemann-Hilbert methods and complement the lectures of Peter Miller, while our discussion of DS II will introduce the $\overline{\partial}$-methods commonly used for integrable dispersive PDE's in two space dimensions.
Comparison of IST and PDE Techniques
Jean-Claude Saut, Université Paris-Sud
Most of the classical nonlinear dispersive equations are derived as asymptotic models (in suitable regimes of amplitudes, wave lengths...) from more complex systems (water waves, nonlinear optics, plasma waves,..).. It turns out that some of them (actually quite a few) are completely integrable systems, allowing one to describe completely (although not always!) the long-time behavior of solutions to the Cauchy problem. We will survey and compare the results obtained by IST and PDE techniques for those integrable equations or systems and also comment on what can be predicted on the dynamics of non-integrable equations from that of the integrable ones, in connection with the so-called "soliton resolution conjecture.''
Presentation Slides
Some summer school speakers have shared their lecture slides, which can be viewed here.
