The seminar aims at an extensive study of initial- and boundary-value problems, as well as control and stabilization for several interesting classes of nonlinear dispersive wave equations. These include unidirectional models such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the regularized long-wave equation, and systems of equations of Boussinesq-type that model two-way propagation of plane waves. These equations arise as models for wave motion in a variety of important physical contexts, including surface water waves, internal waves in density-stratified fluids, acoustic-gravity waves in compressible heavy fluids, planetary waves in the β-plane approximation, plane compressional waves in effervescent liquids, axisymmetric waves in elastic cords, hydromagnetic waves in cold plasmas, axisymmetric waves in rotating liquids, acoustic waves in anharmonic crystals, and waves on thin films, to mention only some. Considerable progress has been made over the last two decades (and especially recently) in understanding fundamental aspects of both pure and periodic initial-value problems for some of the differential equations considered here, for example in the work of Ginibre and Velo; of Kenig, Ponce, and Vega; and of Bourgain. By contrast, the theory for boundary-value problems other than those imposing periodicity has lagged behind. In particular, the study of control theory for nonlinear dispersive wave equations is still at an early stage. The goal of this seminar is to assess the current state-of-the-art, to master the relevant harmonic analysis tools and techniques, and then to apply them to study initial-boundary-value problems of nonlinear dispersive wave equations, as well as their control and stabilization.

The five themes of the year-long seminar are:

1. Harmonic analysis techniques and their applications to nonlinear dispersive wave equations

2. Well-posedness of initial-boundary-value problems of nonlinear dispersive wave equations in a bounded domain

3. Control and stabilization of nonlinear dispersive wave equations

4. Study of nonlinear dispersive wave equations from the dynamical system point of view

5. Computability and complexity analysis of nonlinear dispersive wave equations

The seminar is anchored by the Taft Visiting Fellows: Professors Shuming Sun, Lionel Rosier, and Vilmos Komornik respectively in the Fall, Winter, and Spring quarters.