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Speaker: 

Katrin Grunert, Professor,

Department of Mathematical Sciences, 

Norwegian University of Science and Technology

Title:

The Camassa--Holm equation - peakons and traveling waves

Abstract:

Solutions of the Camassa--Holm (CH) equation might enjoy wave breaking in finite time. This means that even classical solutions, in general, do not exist globally, but only locally in time, since their spatial derivative might become unbounded from below pointwise in finite time, while the solution itself remains bounded. Furthermore, energy concentrates on sets of measure zero when wave breaking occurs. Thus the prolongation of solutions beyond wave breaking is non-unique and depends heavily on how the concentrated energy is manipulated. 

In this talk we will focus on two classes of solutions: peakons and traveling waves. Peakon solutions comprise a class of solutions, which can be computed explicitly, and hence offer a unique possibility to gain insight into the possible behavior of general solutions. In particular, we will use them to illustrate wave breaking and why the continuation thereafter is non-unique. Loosely speaking they are as important for the CH equation as the solitons for the Korteweg--de Vries equation. The traveling wave solutions, on the other hand, are not smooth - only H{\"o}lder continuous. Therefore local, classical traveling waves can serve as building blocks for global traveling waves by gluing them together. We will illustrate how this can be done using ideas from scalar conservation laws.