Oberwolfach Reports

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Volume 1, Issue 4, 2004, pp. 2653–2728
DOI: 10.4171/OWR/2004/50

Published online: 2005-09-30

Nonlinear Waves and Dispersive Equations

Carlos E. Kenig[1], Herbert Koch[2] and Daniel Tataru[3]

(1) University of Chicago, USA
(2) Universit├Ąt Bonn, Germany
(3) University of California, Berkeley, USA

Nonlinear dispersive equations are models for nonlinear waves in a wide range of situations. Mathematically they display an interplay between linear dispersion and nonlinear focusing or defocusing effects. They are linked to diverse areas of mathematics and physics, ranging from nonlinear optics over oscillatory integrals and integrable systems to algebraic geometry. The conference did focus on the analytic (PDE) aspects with a view towards applications. Major results and areas are: \begin{enumerate} \item The introduction of Bourgain-type spaces, which are a tool to describe the $L^2$ information of the linear equation and link it to geometry of the characteristic set and bilinear/multilinear estimates. \item The study of problems with rough initial data. This includes the work on semilinear problems as well as substantial progress in understanding the interaction of nonlinear waves. Also one should add the recent ill-posedness results above the scaling exponents. \item There has been considerable improvement in our understanding of the blow up of solutions to critical dispersive equations. \item The long time and global behavior of solutions to dispersive equations. There are two aspects of this; for small data the dispersive efffects are dominant, while for large data a (largely missing) understanding of the nonlinear focusing/defocusing effects becomes essential. \item Dispersive equations occur in some formal expansion. Typically the first terms of the expansion lead to some universal problems like third order nonlinear Schr\"odinger or Korteweg-de-Vries equations. There has been progress in the understanding of 'better' approximations and higher dimensional waves. \item There has been important work in bridging the gap between the asymptotic dispersive equations and the full problem. \end{enumerate} Motivated by recent developments there has been a series of lectures by T. Kappeler and P. Topalov on their application of the inverse scattering transform to rough initial data for the Korteweg-de-Vries equation. This meeting was attended by 45 participants. The organizers made an effort to include young mathematicians and to give them the opportunity of a shorter talk.

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Kenig Carlos, Koch Herbert, Tataru Daniel: Nonlinear Waves and Dispersive Equations. Oberwolfach Rep. 1 (2004), 2653-2728. doi: 10.4171/OWR/2004/50