Descriptions of real life processes often lead to the formulation of evolution equations of various types among which are nonlinear dispersive equations. The problems considered have their origins mostly in physics in subjects such as general relativity, quantum mechanics, water waves, nonlinear elasticity, various field theories, etc. In some evolution processes, a prescribed restriction (or boundedness) over time is given and the question is then whether the boundedness property remains forever, or a certain « unboundedness » can occur (e.g., a freak wave in the ocean or a self-focusing burn in laser optics). These evolution processes are so fundamental in nature, yet we are still in the infancy of their full analytical description.

The field of nonlinear dispersive equations has been experiencing a dramatic growth over the last twenty years. Many new ideas and techniques emerged, enabling mathematicians to work on problems which until quite recently seemed untouchable. One of the key concepts at the heart of the field is that of nonlinear wave interactions. These interactions occur when waves, moving possibly with different speeds and in different directions, intersect. While in the linear theory such waves obey the principle of superposition, in nonlinear equations there is always some interaction. Initial research was concerned with interactions of small amplitude waves, which leads to small perturbations of linear regimes. If the interactions are strong enough, then they may create new waves or even lead to blow-up, or singularity formation.

The goal of this program is three-fold:
1)  to survey recent developments in the field
2) to bring together mathematicians who study the field from different angles. Some develop analytical methods, some create numerical approaches, while others study physical or experimental aspects of such nonlinear evolution equations
3) to bring young researchers together and help establish networking connections between the US and French researchers.        

We plan to have several keynote talks given by mathematicians who are leaders in the field, then complement these talks with a number of more specialized one-hour talks in targeted areas, and also incorporate half-hour talks by the younger researchers. We also plan to have an evening event on careers in math worldwide.

Plenary Speakers
​

• Hajer Bahouri (Université Paris-Est Créteil)

​Dispersion phenomena on the Heisenberg group  when the vertical frequency tends to 0   (pdf)

• Valeria Banica (Université d’Évry Val d’Essonne)

Dynamics of almost parallel vortex filaments​

• Christophe Besse (Université Paul Sabatier Toulouse 3)

​Numerical schemes for nonlinear Schrödinger equation

• Thomas Duyckaerts (Université  Paris 13)

​Bound from below of the exterior energy for the wave equation and
applications   (pdf)

• Stephen Gustafson (University of British Columbia)

​Ground states and dynamics for perturbed critical NLS   (pdf)

• Gadi Fibich (Tel Aviv University)

Continuations beyond the singularity, loss of phase, stochastic interactions, and universality   (pdf)

• Christian Klein (Université de Dijon)

​​Numerical study of solitons, dispersive shock waves and blow-up

• Evgenii Kuznetsov (Landau Institute for Theoretical Physics, RAS)

​​Solitons vs Collapses   (pdf)

• Felipe Linares (IMPA, Rio de Janeiro)

​On the periodic Zakharov-Kuznetsov equation   (pdf)

• Yvan Martel (Université de Versailles-St Quentin)

​Strongly interacting solitary waves for NLS

• Pavel Lushnikov (University of New Mexico)

Stokes wave and dynamics of complex singularities in 2D hydrodynamics with free surface

• Luc Molinet (Université de Tours)

A rigidity result for the Camassa-Holm equation​ and applications   (pdf)

• Andrea Nahmod (University of Massachusetts Amherst)

​Randomization and dynamics in nonlinear PDE: an overview

• Nataša Pavlović (University of Texas at Austin) 

Infinitely many conserved quantities for the cubic Gross-Pitaevskii
hierarchy in 1D​   (pdf)

• Fabrice Planchon (Université Nice Sophia-Antipolis)

The wave equation on a model convex domain revisited​   

• ​Benjamin Schlein (Universität Zürich)

​Dynamical and Spectral Properties of Bose-Einstein Condensates   (pdf)

• Catherine Sulem (University of Toronto)

Soliton Resolution for Derivative NLS equation​

• Nicola Visciglia (Università Degli Studi di Pisa)

On the Growth of Sobolev Norms in a compact setting​   (pdf)

• Michael Weinstein (Columbia University)

Dispersive waves in novel 2d media; Honeycomb structures, Edge States and the Strong Binding Regime

• Vladimir Zakharov (University of Arizona)

Unresolved problems in the theory of integrable systems (pdf)
Rayleigh statistics of waves in integrable nonlinear systems   (pdf)
Integrability of Deep Water Equations   (pdf)
(image 1)   (image 2)   (image 3) 
(surface.mp4)   (main-poles.mp4)   (pirate-gravity.mp4)