2011/07/09 by Andrew Comech, Comech, Andrew · 1 citation
Mathematics · Physics and Astronomy · #35B35 #35C08 #35P99 #35Q41 #37K40 #37K45 #81Q05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS) #hep-th #math-ph #math.AP #math.MP #msc:35B35 #msc:35C08 #msc:35P99 #msc:35Q41 #msc:37K40 #msc:37K45 #msc:81Q05 #nlin.PS
paper · pdf · doi:10.48550/arxiv.1107.1763
13 pages, minor corrections
openalex publication_date 2011/07/09 · arxiv created 2011/08/15 · arxiv updated 2011/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the spectral stability of solitary wave solutions ϕ(x)e-iωt to the nonlinear Dirac equation in any dimension. This equation is well-known to theoretical physicists as the Soler model (or, in one dimension, the Gross-Neveu model), and attracted much attention for many years. We show that, generically, at the values of where the Vakhitov-Kolokolov stability criterion breaks down, a pair of real eigenvalues (one positive, one negative) appears from the origin, leading to the linear instability of corresponding solitary waves. As an auxiliary result, we state the virial identities ("Pohozhaev theorem") for the nonlinear Dirac equation. We also show that ± 2ωi are the eigenvalues of the nonlinear Dirac equation linearized at ϕ(x)e-iωt, which are embedded into the continuous spectrum for |ω| > m/3. This result holds for the nonlinear Dirac equation with any nonlinearity of the Soler form ("scalar-scalar interaction") and in any dimension.