Trimester on nonlinear waves

Over the course of the trimester, the programme brought together 80 leading scientists in the field, together with many students during the summer school. It was made possible with the support of the European Commission via an ERC Advanced Grant (Principal Investigator: F. Merle): “Blow up, dispersion and solitons (Blowdisol)” hosted by Université de Cergy-Pontoise. Contributions from Société Générale and the Clay Mathematics Institute were also essential to the organisation of the summer school which concluded this wonderful trimester.

Two of the IHES trimester organisers had already organised a thematic semester in the spring of 2009 called “Nonlinear waves and dispersion”.The objective of this previous programme was to take stock, after almost twenty years of developments linked to the “model” nonlinear dispersive equations, from Korteweg-de-Vries to non-linear Schrödinger, as well as wave equations in their various forms.This work, initiated mostly in the United States by researchers with a background in harmonic analysis (C.Kenig,G.Ponce,L.Vega, J. Bourgain), naturally came across the pioneering efforts in Europe of J. Ginibre, G.Velo, J.-C. Saut, then H. Bahouri, J.-Y. Chemin, P. Gérard, F. Merle and many others after them.

The 2009 programme had been very successful, with a high participation rate from many high-calibre researchers invited from abroad, sometimes for the entire programme, and at least for a month.
In a way, the programme had marked the end point of a cycle of activity in the field of dispersive equations, centred around Cauchy problems for the various model equations. It had at the same time enabled a number of the then latest developments to be presented: the analysis of blow-up models such as focusing Schrödinger, together with concentration- compactness-rigidity methods (which have spread beyond dispersive models).

In some respects, current scientific activity in the field is much more varied than four or five years ago, as can be seen in the scientific activity undertaken during the thematic trimester at IHES: the analysis of the main dispersive toy models has shifted to tricky points relating to the asymptotic behaviour of solutions, be that in the precise dynamical description of blow-up models,in the progress made towards the soliton resolution conjecture, with the analysisofcollisionsbetweenmultiplesolitons,orin studying the stability of breather-type solutions.The emergence of a corpus of clearly identified tools has also highlighted their versatility, their deployment on models other than dispersive models having proved productive, in particular on geometric dispersive equations that were out of reach until recently (for example, the analysis of blow-up dynamics for Schrödinger maps) and also, as already mentioned, to revisit parabolic equations. More generally, classification theorems on the behaviour of solutions of various nonlinear systems are now within reach; there was a full programme of presentations during the trimester (given by C. Kenig during seminars, Hadamard Foundation lectures and the summer school) on the recent results achieved by T. Duyckaerts, C. Kenig and F. Merle on the soliton resolution of solutions for energy-critical focusing nonlinear wave equation.

At the same time, a great deal of development on dispersive effects is being done on much more sophisticated models,oftenclosertophysicalreality. Even simpler yet “physical” models such as the (nonlinear) Dirac equation present difficulties not encountered to date with waves or Schrödinger. The IHES trimester enabled many researchers from different backgrounds to interact on fluid models such as water waves and more generally on wave/dispersive-type models, which appear in many “asymptotic” derivations of fluid phenomena. Dispersive effects have played a key role in the latest existence and asymptotic behaviour results, which were presented during seminars, the two conferences and the summer school. In the context of water waves, asymptotic behaviour is the area of chief interest, as are models that are increasingly sophisticated and close to physical reality (surface tension, finished depth, etc.)

New developments are occurring, in particular in the case of 2D, which is the most difficult: there is a phase shift phenomenon in the study of scattering which had not to date been studied on a quasilinear problem. All these results are promising, because they introduce new tools, liable to being applied in various ways in the field of dispersive fluid models and beyond.

Interesting developments are also to be found in the study of vortex filaments, in connection with geometric dispersive equations such as “Schrödinger maps” or mKdV, which were presented during the trimester. Even in a vacuum, understanding the geometry of space-time requires the analysis of sophisticated quasilinear wave equations; many achievements in the field of general relativity, some of which were directly influenced by the rapid progress in the analysis of dispersive models, were presented and discussed during the various trimester events.

Let us recall that this area of research has seen a number of major advances in recent years, including “the L2 curvature conjecture” (S. Klainerman, I. Rodnianski et J. Szeftel) and the linear stability of the Kerr family of metrics (M. Dafermos, I. Rodnianski, D. Tataru, etc.), two issues where an understanding of dispersive phenomena plays a key part.The formation of trapped surfaces (D. Christodoulou, S. Klainerman, I. Rodnianski) also forms part of asymptotic analysis, with tools very similar to microlocal analysis, and the trimester provided a forum for presenting the latest developments on these very active topics.

In addition, linear equations continue to generate significant work, in contexts closer to realistic physical models:“sophisticated” geometry (variable metrics, possibly not very regular, existence of boundary conditions, influence of the environment’s geometry on propagation and dispersion).

A growing area of research should also be mentioned here, in a field of investigation that is at the interface of dispersive PDEs and probability; it has seen rapid development with the study of nonlinear dispersive equations, where the initial datum is almost certainly chosen in a space that is out of reach of deterministic theories.

The IHES trimester had a number of objectives: taking stock of this new cycle, started a few years ago, paying special attention to young researchers in the field, where the ever-growing complexity of the work is an increasing challenge in the early stages of one’s career. Another objective was to bring together researchers for whom areas of convergence are even more obvious than they were in 2009: let us mention here the large French community engaged in the mathematical study of fluid models, kinetic theory, dynamical systems and partial differential equations linked to infinite-dimensional Hamiltonian systems. In the United States, the work around dispersive modems, fluid and nonlinear wave mechanics (in particular when linked to general relativity and mathematical physics) is expanding rapidly (as evidenced by the success of the large-scale thematic programme at MSRI in the autumn of 2015).There are now recognised areas of overlap among all these communities and the scientific activity of the IHES trimester has made it possible to bring together the various strands of research, by providing opportunities for productive interactions among scientists from different backgrounds.

The IHES trimester was a success, as evidenced by the large number of invited professors who stayed at the Institute for several weeks, several months even, for some of them (more than 80 invited professors over the course of the three months), and by the very high demand for the summer school which concluded the scientific programme. Although it is too early to assess the scientific impact of the trimester itself – beyond noting that the continuum of work from autumn at the MSRI to July in Bois-Marie seems to have accelerated a number of developments – a number of trends can be mentioned: the objective of classifying the behaviour of Hamiltonian partial differential equation models no longer seems unattainable, a complete solution to the soliton resolution conjecture for nonintegrable models has never been so close to being found; techniques arising from dispersive models are now prevalent in most of the work relating to mathematical physics equations, general relativity and fluid mechanics. It is striking to note that an increasing number of researchers are working indifferently on these various themes and that progress in one quickly spreads to related themes.

Yvan Martel, Frank Merle & Fabrice Planchon

> Find the videos of the May conference and of the June conference on the IHES YouTube channel.

> More details on the summer school on “nonlinear waves”