On linear instability of solitary waves for the nonlinear Dirac equation

الأصل · EN

We consider the nonlinear Dirac equation, also known as the Soler model: i tψ=-iα· ∇ ψ+m βψ-f(ψ βψ) βψ, ψ(x,t)ⁿ, xⁿ, n≤ 3, f∈ C 2(), where αⱼ, j = 1,...,n, and β are N × N Hermitian matrices which satisfy αⱼ²=β²=Iₙ, αⱼ β+βαⱼ=0, αⱼ αₖ + αₖ αⱼ =2 δjk Iₙ. We study the spectral stability of solitary wave solutions ϕ(x)e⁻ⁱωᵗ. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit ω→ m, proving that if k>2/n, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with ω sufficiently close to m, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion.