Linear instability of nonlinear Dirac equation in 1D with higher order nonlinearity

2012/03/17 by Andrew Comech, Comech, Andrew · 7 citations
Mathematics · Physics and Astronomy · #35B35 #35C08 #35P99 #35Q41 #37K40 #37K45 #81Q05 #Advanced Mathematical Physics Problems #Amplitude #Analysis of PDEs (math.AP) #Dirac equation #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Hermitian matrix #Instability #Limit (mathematics) #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Schrödinger equation #Nonlinear Waves and Solitons #Nonlinear system #Order (exchange) #Pattern Formation and Solitons (nlin.PS) #Physics #Quantum mechanics #Spectral Theory (math.SP) #math-ph #math.AP #math.MP #math.SP #msc:35B35 #msc:35C08 #msc:35P99 #msc:35Q41 #msc:37K40 #msc:37K45 #msc:81Q05 #nlin.PS

paper · pdf · doi:10.48550/arxiv.1203.3859

published in arXiv (Cornell University) (Cornell University) · 15 pages

openalex publication_date 2012/03/17 · arxiv created 2012/07/13 · arxiv updated 2012/07/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider the nonlinear Dirac equation in one dimension, also known as the Soler model in (1+1) dimensions, or the massive Gross-Neveu model: i∂_tψ=-iα∂_xψ+mβψ-f(ψ^∗βψ)βψ, ψ(x,t)∈\C², x∈\R, f∈ C^∞(\R), m>0, where α, β are 2× 2 hermitian matrices which satisfy α²=β²=1, αβ+βα=0. We study the spectral stability of solitary wave solutions ϕ_ω(x)e^-iωt. More precisely, we study the presence of point eigenvalues in the spectra of linearizations at solitary waves of arbitrarily small amplitude, in the limit ω→ m. We prove that if f(s)=s^k+O(s^k+1), k∈\N, with k≥ 3, then one positive and one negative eigenvalue are present in the spectrum of linearizations at all solitary waves with ω sufficiently close to m. This shows that all solitary waves of sufficiently small amplitude are linearly unstable. The approach is based on applying the Rayleigh-Schrödinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov-Kolokolov stability criterion. Let us mention a similar independent result [Guan-Gustafson] on linear instability for the nonlinear Dirac equation in three dimensions, with cubic nonlinearity (this result is also in formal agreement with the Vakhitov-Kolokolov stability criterion).

Citations

• The quantum theory of the electron

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• On linear instability of solitary waves for the nonlinear Dirac equation

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