Nonlinear Waves and Coherent Structures Online Colloquium

The "Nonlinear Waves and Coherent Structures Online Colloquium"  is a monthly seminar that will have speakers from a diverse range of disciplines to speak on a broad range of topics connected to nonlinear waves and coherent structures. The organizers hope this series will help maintain a sense of community and provide a broad audience with access to research developments on the forefront of this field.

NWCS Colloquium

David Campbell (Boston University)

Tuesday, October 14, 11:00am (Eastern Time)

Title: Interaction of Coherent Structures in Non-Integrable PDEs

Abstract: 

Nonlinear waves and coherent structures can evolve in remarkably surprising manners. In this

talk, we illustrate two examples, one which is understood theoretically and one which remains a

mystery. The first case involves the scattering of kink-like coherent structures in the 1D \phi^4 theory, in which a series of resonances between trapped and reflected scatterings (see figure below) were observed 50 years ago and have been successfully explained via a “Resonant Energy

Exchange Mechanism” (REEM) involving energy exchange between the center of mass motion of

the kink and a spatio-temporal oscillating mode localized around the kink. The second case

involves the scattering of localized coherent structures in a 2D variant of the nonlinear

Schrödinger equation in which it appears that over a restricted range velocities and impact

parameters, the structures can scatter at 90 degrees. This behavior is presently not understood

and represents an important challenge to the NWCS community. I will discuss some recent

approaches to this problem and will illustrate that, in a somewhat more complicated equation,

such scattering at 90 degrees arises naturally and can be both modeled numerically and

explained theoretically.