Spring 2016
Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.
Date: February 12, 2016
Speaker: Lev Ostrovsky
NOAA Earth Science Laboratory,
Boulder, CO
Title: "Asymptotic Perturbation Theory for Solitons"
Abstract:
The term “soliton” is now widely popular even among the non-specialists. A solitary water wave has been discovered by a ship engineer J. Scott Russell in 1834 in a shallow channel in Scotland. For a long time such a wave was considered as a something exotic, mostly mathematical object. Since the 1960s, the success of the mathematical theory of solitons was largely associated with exact methods such as the inverse scattering method in the theory of integrable nonlinear wave equations. From the practical viewpoint, however, completely integrable equations are often too strong an idealization in which such important factors as dissipation, inhomogeneity, and many others, are ignored. Moreover, even when the exact methods work, the resulting solution can be too cumbersome for practical use, and their physical interpretation may not be simple. Thus, perturbation methods can be important even in these cases. Various approximate approaches for linear and nonlinear waves have been used since very early; it suffices to mention the use of the ray concept in optics and acoustics. The general “theory of waves” understood in the same sense as earlier the “theory of oscillations” was formulated in the mid-1900s. This presentation deals with slowly varying and interacting solitons and their ensembles, treated with the asymptotic perturbation theory modified for localized waves, solitons and fronts (kinks). It considers, among others:
Attenuation and amplification of solitons in a Korteweg-de Vries (KdV) equation modified by different perturbations; Types of interaction of solitons as classical particles: repulsing and attracting solitons and stationary multisolitons; Interaction of compound solitons which consist of two kinks; Soliton ensembles and their stability.
Attenuation and amplification of solitons in a Korteweg-de Vries (KdV) equation modified by different perturbations; Types of interaction of solitons as classical particles: repulsing and attracting solitons and stationary multisolitons; Interaction of compound solitons which consist of two kinks; Soliton ensembles and their stability.