Nonhomogeneous boundary-value problems for nonlinear waves

May 13 to May 17, 2013

at the

American Institute of Mathematics, San Jose, California

organized by

Jerry Bona, Min Chen, Shuming Sun, and Bingyu Zhang

Original Announcement

This workshop will be concerned with boundary-value problems for nonlinear dispersive evolution equations and systems.

Nonlinear, dispersive evolution equations and systems of such equations arise as models for wave motion in a very wide variety of physical, biological and engineering. Since the 1960's, there has been a steady increase of interest in the theory and applications of such equations. On the mathematical side, the pioneering work of Ginibre and Velo and Kenig, Ponce and Vega was followed by the spectacular progress of Bourgain, Tao and their collaborators, as well as many others.

If one tries to use the pure initial-value formulations in practice, one is immediately beset by the difficulty of determining accurately a wave profile in the entire spatial domain of its definition at a single instant of time. Generally speaking, this is not possible to accomplish with any semblance of accuracy. Moreover, when these equations are used in engineering and science, the natural way to pose them is with specified, not necessarily homogeneous boundary conditions. And, problems of control of dispersive equations demand a firm grasp of boundary-value problems as a starting point for developing cogent theory.

By contrast with the initial-value problem, theory for boundary-value problems other than those featuring periodicity has generally lagged behind the developments for the pure initial-value problems. The overall goal of the proposed workshop is to advance the study of boundary-value problems for nonlinear dispersive wave equations. Within this larger framework, there are several specific topics we have in mind.

  1. Investigate the smoothing properties enjoyed by solutions of boundary-value problems and associated well-posedness theory.
  2. Investigate the controllability and stabilizability of solutions of nonlinear, dispersive wave equations. Experience shows that results from the first topic above will be central to such an investigation.
  3. Extend the theory to multi-space dimensional problems arising in geophysical applications such as coastal dynamics and elsewhere.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Stability of solitary-wave solutions of systems of dispersive equations
by  Bona, Chen and Karakashian,  Appl. Math. Optim. 75 (2017), no. 1, 27–53  MR3600389
The KdV equation on the half-line: The Dirichlet to Neumann map
by  Jonatan Lenells,  J. Phys. A 46 (2013), no. 34, 345203, 20 pp  MR3101682
The KdV-equation on the half line: Time-periodicity and mass transport
by  Bona and Lenells
Well-posedness for the BBM-equation in a quarter plane
by  Bona, Chen and Hsia,  Discrete Contin. Dyn. Syst. Ser. S 7 (2014), no. 6, 1149-1163  MR3228871
Interface Problems for Dispersive equations
by  Natalie E Sheils and Bernard Deconinck,  Stud. Appl. Math. 134 (2015), no. 3, 253-275  MR3322696
Non-steady state heat conduction in composite walls
by  Bernard Deconinck, Beatrice Pelloni and Natalie Sheils