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

Arriving in Banff, Alberta Sunday, December 2 and departing Wednesday December 12, 2012


(Université de Franche-Comté)

(Texas A & M University)

(University of British Columbia)



The broad goal is to develop an understanding of properties of nonlinear
Dirac solitary waves, especially concerning their stability, by
building upon a few very recent advances, and by developing novel
techniques. The methods we aim to develop and refine should be more
widely applicable to the Dirac-Maxwell system from Quantum
Electrodynamics, and to the coupled mode equations which appear in Solid
State Physics and Nonlinear Optics.

The proposed research lies at the borderline of Mathematics and Physics,
and is directly related to Optics, Waveguide Theory, Field Theories,
Solid State Physics and High Energy Physics. Its mathematical tools are
rooted in Harmonic and Functional Analysis, Spectral Theory, and Partial
Differential Equations. This interconnection of several disciplines
allows us to engage top experts from adjacent fields, who represent the
main body of the group. The focused research group at BIRS would provide
a unique opportunity for this geographically diffuse collection of
specialists to combine forces, without distractions, in an intense
effort to solve a specific set of pressing problems.

Relevance and importance.

The Dirac equation remains of utmost importance in High Energy Physics
and Solid State Physics since its invention in 1928. It is a main
building block of Quantum Electrodynamics. Similar equations (coupled
mode equations) appear in nonlinear Optics. Understanding stability
properties of localized solutions (solitary waves, or gap solitons) is
of critical importance both for practical applications and for the
quantum theory. Furthermore, the related mathematical questions lead
into research areas of wider interest, such as the spectra of
non-selfadjoint operators, the limiting absorption principle,
bifurcation theory, and Hamitonian systems theory.


The stability properties of solitary waves in systems involving the
Dirac equation (nonlinear Dirac, Dirac-Maxwell, etc.)
have remained a complete mystery. In the last three years, the
participants of the proposed focussed research group have
made several breakthroughs which provide an entry point to the rigorous
study of spectral and asymptotic stability. We hope that combining our
efforts will make the nonlinear Dirac equation as clearly accessible to
stability analysis as the nonlinear Schrodinger equation has been since
pivotal classical papers by Vakhitov-Kolokolov, Weinstein, and