Spectral and asymptotic stability in nonlinear Dirac equations (12frg188)


(Université de Franche-Comté)

Andrew Comech (Texas A&M University)

Stephen Gustafson (University of British Columbia)


The "Spectral and asymptotic stability in nonlinear Dirac equations" workshop will be hosted at The Banff International Research Station.

Whenrn the amplitude of waves increases (be they water waves, gravitational rnwaves, waves in plasmas, etc.), their interaction with the medium (or rnself-interaction) becomes important. This interaction may seriously rnchange the behavior of the waves and lead to the appearance of nonlinearrn solitary waves, or solitons. Solitary waves describe numerous natural rnphenomena of purely nonlinear origin. They appear in Ocean Dynamics rn(surface waves, including rogue waves; tidal bores; undersea internal rnwaves), in the Atmosphere (such as the Morning Glory cloud), and in rnQuantum Field Theories. Solitons exist in plasmas and crystal lattices. rnRecently they have come under much scrutiny due to applications of fiberrn optics for data transmission.rnrnWe are interested in the stability properties and large time asymptoticsrn of solitary waves in the Dirac equation, which plays a fundamental rolern in Quantum Physics. Similar equations appear in nonlinear optics and rnphotonics (in particular in manufacturing of photonic crystal fibers). rnStability properties are related to the following most natural rnquestions: How often do solitary waves form? How easily do they dissolvern under small perturbations? rnrnDetailed knowledge of stability properties of solitary waves will help rndevelop optical waveguides and describe quantum effects on the scales rnbeing inexorably approached by today's electronics and chip rnmanufacturers, let alone experimental physics. The relevant mathematics rn-- for example nonlinear analysis and spectral theory -- involves a rnblend of ideas from many different active branches. This blend provides arn fruitful interdisciplinary environment for research, and we intend to rnexploit the involvement of top specialists in several adjacent fields torn make progress on these basic questions.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí255a (CONACYT).