2009/10/03 by Alexander Komech, Komech, Alexander, Andrew Komech +1
Mathematics · Physics and Astronomy · #35B41 #37K40 #37L30 #37N20 #81Q05 #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35B41 #msc:37K40 #msc:37L30 #msc:37N20 #msc:81Q05
paper · pdf · doi:10.48550/arxiv.0910.0517
5 pages, 1 figure (simplified assumptions on coupling function; now only 3D case is considered)
openalex publication_date 2009/10/03 · arxiv created 2010/02/26 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a U(1)-invariant nonlinear Dirac equation in dimension n=3, interacting with itself via the mean field mechanism. We analyze the long-time asymptotics of solutions and prove that, under certain generic assumptions, each finite charge solution converges as t→±∞ to the two-dimensional set of all "nonlinear eigenfunctions" of the form ϕ(x)e\sp-iωt. This global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. The research is inspired by Bohr's postulate on quantum transitions and Schrödinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schrödinger and Maxwell-Dirac equations.