Nonlinear water waves

A breaking water wave is one of the most common and yet most complex phenomena in nature. Describing and understanding water waves mathematically, even when smooth and periodic, has posed one of the most long-standing problems in modern mathematics. A pa rticular challenge is to understand what is known as the non-local properties of water waves, namely, that the behaviour and evolution of a wave is influenced, not only by the situation at a particular point in the wave, but by information from all parts of the fluid domain. Mathematically, the analysis of such nonlocal dependence typically involves integrals called Fourier or pseudo-differential operators, the study of which is very different from classical differential equations, about which by now very much is known. This project aims at studying certain classes of non-local equations which describe water waves in a canal or out at sea, with a particular focus on waves that allow for stagnation, like interior vortices, wave-breaking, or peaks and cusp s at the surface. For many equations it is not even known whether certain solutions--i.e., waves--exist, or how they might behave or look like. A particular focus will be put on the Euler equations for waves under a free surface, and a class of model equa tions named after the American mathematician Gerald B. Whitham. In 2015 we managed to prove that the Whitham equation has a highest, cusped, travelling-wave solution. This was a conjecture by Whitham from 1967, and one of the main objectives of the project. During 2016 work on a general theory for highest wave has begun, and a bi-directional equation har been completely analysed with respect to its largest solutions. 2017 has seen the completion of several projects: asymptotic properties of solitary waves, local existence theory, symmetry properties for solutions to a general class of equations, rotational solutions to the Euler equations. The work on a general theory for high-amplitude waves has during 2018 progressed in several subprojects, whereof several have been completed.

Det er oppnått uforventede virkninger av prosjektet: Takk være prosjektet har mange grupper i både Europa og USA begynt forskning på disse likningene. Flere av arbeidene om Whitham har, for matematikk, allerede etter kort tid høye siteringstall. En gruppe ved Princeton og Madrid kommit opp med verdens første datorbaserte PDE-bevis for en formodning fra oss. Ved enkelte konferenser, som IMACS Georgia 2017, har hele sessioner tilegnets bare Whithamlikninger. Intresset for såkalte fullt dispersive likninger har skyvet i høyden. Som en følge av den aktivitet prosjektet skapet har gruppen fått flere medarbeider som bidratt til prosjektet. Dette gjelder stillinger finansiert av instituttet og eksternt finansierte stillinger (ERCIM). Mye av det vi lært i prosjektet har gått inn i både søknaden til og arbeidet med Toppforskprosjektet 250070. Det er i dagene fem postdocs som ansøker om stillig ved instituttet med prosjekt som fortsetter arbeidet fra Forskertalenter-prosjektet.

A crucial and challenging part in the study of water waves is their inherent non-local nature: if reduced to the surface any known exact model of water waves will include non-local operators. In the original setting this is manifested through nonlinear bo undary conditions on the free surface, which appear as Fourier or pseudo-differential operators when the equations are transformed to, or approximated by, equations on the real line. This has important implications for the study of waves with singularitie s in the form of cusps, peaks or vortices, since these do not arise as local phenomena, but are related to the global behaviour of a solution. The aim of this project is to bring further understanding of those phenomena in settings which are largely une xplored. While much is known about peakons and cuspons and related singular waves within the theory of solitons and integrable systems, very little is understood in the context of inherently non-local equations. In particular, we take an interest in waves allowing for breaking or stagnation, i.e. waves that either blow up in norm while remaining bounded, or that exhibit vortices (interior stagnation), peaks or cusps (surface stagnation). To further limit the scope we focus on two settings that have recent ly drawn a lot of attention and also shown some interesting qualitative features, namely (i) exact traveling waves, and (ii) equations of Whitham type (sharing important features with the non-local Whitham wave equation introduced by Whitham in 1967). In strumental goals include existence and properties of waves as mentioned above. In particular, waves with non-constant vorticity and multiple critical layers (induced by multiple stagnation points) are only partially understood; and after half a century it is still not known whether the Whitham equation (or any of its close non-local kin) admits a highest cusped wave.