for this workshop
New frontiers in dispersive integrable equations
at the
American Institute of Mathematics, Pasadena, California
organized by
Tamara Grava, Peter Miller, and Monica Visan
This workshop, sponsored by AIM and the NSF, will be devoted to the study of integrable models for dispersive Hamiltonian partial differential equations. The landscape of dispersive equations is vast in scope, touching on applications from water wave dynamics to nonlinear optics. The mathematical topics range from well-posedness theory to long-time dynamics and other asymptotic limits, in both deterministic and random scenarios. Since their discovery nearly 60 years ago, integrable equations like the Korteweg-de Vries and nonlinear Schrödinger equations have been useful as guideposts in this landscape, since integrability provides tools not otherwise available, allowing for precise results.
The main topics for the workshop are
- Long-time asymptotic behavior of integrable and near-integrable equations. Many dispersive equations admit exact soliton solutions that are localized, coherent structures. It is of great interest to prove soliton resolution conjectures asserting roughly that most initial conditions resolve over long times into sums of solitons and a decaying radiative component. Can we use new methods to establish such conjectures for integrable equations with more complicated features than the traditional examples (e.g., with nonlocality, higher dimensionality, multiple components)? Can any of the results be carried over to nonintegrable equations? There are examples of initial conditions that do not follow the "solitons + radiation" dichotomy; what is the maximal class of such solutions?
- Explicit formulas for integrable models and their applications. Recently Gérard and others have found a way to explicitly express the solution of certain dispersive integrable equations in terms of the given initial data. This raises many questions. For example, what is the scope of the theory? Can explicit formulas be obtained for a broader class of equations? While there are finite-dimensional reductions of some explicit formulas for dispersive equations on the line, are there analogues also for periodic boundary conditions? Can explicit formulas be adapted to a method for construction of solutions without reference to initial data or boundary conditions as can be done in the setting of the Riemann-Hilbert methodology? How much detail can be obtained from an explicit formula regarding solutions, especially in limits like large time and small dispersion?
- Soliton gases as models for turbulence. In integrable equations, solitons retain their identity after interactions and can be regarded as interacting particles. Solutions composed of many solitons can be viewed as gases and it is of great interest to study their macroscopic properties. This can be formulated as a deterministic problem involving many particles, or as in statistical physics one can introduce randomness and attempt to deduce statistics of the gas. It has been shown that randomness can be introduced on the soliton parameters, but can it also be introduced in a natural way on the initial data directly? If so, does this result in different statistics? Kinetic theory for soliton gases is fundamentally based on the spatial phase shift introduced by soliton collisions. What happens in integrable models for which the leading term of the asymptotic phase shift vanishes?
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
Space and funding is available for a few more participants. If you would like to participate, please apply by filling out the on-line form no later than December 1, 2026.
Before submitting an application, please read the description of the AIM style of workshop.
For more information email workshops@aimath.org
