The 11th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory

By Jeremy Upsal

The 11th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory was held on April 17-19, 2019 at the University of Georgia in Athens, GA. This conference is a biennial meeting held at the University of Georgia. The conference brings together researchers from around the world working on nonlinear evolution equations and wave phenomena. This year there were three keynote speakers and 26 organized sessions. 

The first keynote speaker was David Ambrose from Drexel University. In his talk, Ambrose introduced the Duchon and Robert method for proving ill-posedness of the vortex sheet problem. Ambrose went on to show how this approach can be used to prove ill-posedness of other equations, including some Boussinesq equations. Ambrose made two remarks that stood out to me. The first is that structure does not matter for well-posedness. Many in the field of Nonlinear Waves use integrability properties for their analysis. Ambrose remarked that even though a system may have nice structure, such as integrability, it can still be ill-posed. Another remark that stood out to me is that linear well-posedness and nonlinear well-posedness are distinct. There are many examples for which the linear problem is ill-posed, but the nonlinear problem is well-posed.

The second keynote speaker was Alex Himonas from the University of Notre Dame. Himonas’s talk focused on the well-posedness of equations which were introduced as models for the Euler equations governing the behavior of incompressible fluids. Himonas walked through the history of such models, mentioning why they were introduced and telling us about the well-posedness results for the equations. Two equations of interest were the Camassa-Holm equation and the Novikov Equation (NE). Both equations were introduced as dispersionless models for the Euler equations. The dispersionless models are of interest because they possess peakon solutions. Himonas then described his new result which is the construction of 2-peakon solutions for NE and the proof of ill-posedness of NE in \(H^s\) for \(s<3/2\).

The third keynote speaker was Stefano Trillo from the University of Ferrara. This keynote had a decidedly different flavor from the other keynotes. Trillo’s talk was about comparing water wave experiments with predictions from nonlinear PDEs, especially integrable systems. Trillo and collaborators were able to experimentally recreate the results of the famous Fermi-Pasta-Ulam-Tsingou numerical experiments which showed recurrence of the initial condition for integrable systems. Trillo also demonstrated experiments which show the shedding of solitons off of a cosine initial condition. This experiment is a replication of the the work of Zabusky and Kruskal on the KdV equation. Trillo also compared the results of experiments in the periodic setting with finite-gap integration for the defocusing nonlinear Schrödinger equation. Finally, Trillo connected many of his experimental findings to theoretical results and clearly explained where and why discrepancies may occur. 

There were 26 organized sessions over the span of the 3 day conference. As expected from the name of the conference, these sessions ranged from theory to numerics to experiment with everything in between. Since my interests are in stability of solutions to nonlinear evolution equations, many of the talks I attended were about stability.

There were many talks about different tools used for determining the stability of solutions for nonlinear evolution equations. One of these tools is the Krein signature which gives a necessary condition for a solution to be unstable via a Hamiltonian-Hopf bifurcation. Three talks using the Krein signature stood out to me. The first was Todd Kapitula’s (Calvin College) talk about the Krein Matrix whose derivative gives information about the Krein signature. Ross Parker (Brown University) followed Kapitula’s talk by showing how the Krein matrix can be used to show that spectral stability of multi-pulses for the Chen-Mckena suspension bridge equation. The third talk on Krein signatures that stood out to me was by Richard Kollár (Comenius University). In this talk, Kollár explained the evolution of his understanding of the Krein signature. As a student who is currently trying to come to a robust understanding of the Krein signature, it was relieving to hear how even an expert can go through phases of understanding.

Overall, the conference was an excellent opportunity for researchers in the field to get together to communicate their findings and work on problems together. Having talked to some who have been to many years of this conference, it sounds like this was one of the more lively and well-attended iterations of this conference. If you are interested in learning about others’ work in this field or have something to present, I recommend attending the conference in April 2021!

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