JournalsowrVol. 4 , No. 4DOI 10.4171/owr/2007/44

Nonlinear Waves and Dispersive Equations

  • Carlos E. Kenig

    University of Chicago, USA
  • Herbert Koch

    Universität Bonn, Germany
  • Daniel Tataru

    University of California, Berkeley, USA
Nonlinear Waves and Dispersive Equations cover

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Abstract

Dispersive equations occur as asymptotic models for the propagation of linear and nonlinear waves. Mathematically they display an interplay between linear dispersion and nonlinear focusing and defocusing effects. They are linked to diverse areas of mathematics and physics, ranging from nonlinear optics over oscillatory integrals to integrable systems. The workshop focused on a PDE based approach to dispersive equations. Current activities covered by the workshop include: \begin{enumerate} \item Progress in understanding critical nonlinear Schr\"oder equations. Generalized Morawetz estimates and an induction on energy improved our understanding on the interaction of waves for nonlinear Schr\"odinger equations, which led to a treatment of many such critical equations. \item The study of dispersive equations with variable coefficients. During the last few years our understanding of wave propagation in nonhomogeneous situations improved considerably and there are now several interesting results on general relativity linearized at the Schwarzschild metric. This may be seen as a step towards nonlinear stability of the Schwarzschild metric. \item The large time asymptotics of solutions, including asymptotics of multisoliton solutions in the nonintegrable case and sharp asymptotics of singular solutions. \end{enumerate} There is a large number of promising young mathematicians working in this area. The meeting was attented by 45 participants. The organizers gave a strong preference to talks by young researchers.