2005/12/31 by Nabile Boussaïd, Nabile Boussaid
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Dirac (video compression format) #Dirac algebra #Dirac comb #Dirac equation #Dirac operator #Dirac spinor #Eigenvalues and eigenvectors #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear system #Operator (biology) #Physics #Propagator #Quantum mechanics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math-ph #math.AP #math.MP
paper · pdf · doi:10.1007/s00220-006-0112-3
62 pages
arxiv created 2006/04/27 · openalex publication_date 2006/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove that for a Dirac operator with no resonance at thresholds nor eigenvalue at thresholds the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues close enough, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, we show that these solutions have stable directions which are tangent to the subspaces associated with the continuous spectrum of the Dirac operator. This result is the analogue, in the Dirac case, of a theorem by Tsai and Yau about the Schrödinger equation. To our knowledge, the present work is the first mathematical study of the stability problem for a nonlinear Dirac equation.