On linear instability of solitary waves for the nonlinear Dirac equation
On linear instability of solitary waves for the nonlinear Dirac equation
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DOI：
10.1016/j.anihpc.2013.06.001
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发表时间：
2012-09
期刊：
影响因子：
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通讯作者：
A. Comech;M. Guan;S. Gustafson
中科院分区：
文献类型：
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作者：
A. Comech;M. Guan;S. Gustafson
我们考虑非线性狄拉克方程，也称为Soler模型：i <$t <$=− i α <$$><$+ m β <$−（<$$> β <$）k β <$，m> 0，<$（x，t）∈ C N，x∈ Rn，k∈ N。研究了在ω→ m极限下小振幅孤立波线性化的点谱，证明了当k> 2/n时，当ω充分接近m时，这些孤立波线性化的点谱中存在一个正的和一个负的本征值，从而这些孤立波是线性不稳定的.该方法是基于应用瑞利-薛定谔微扰理论的非相对论极限方程。结果与Vakhitov-Kolokolov稳定性判据形式一致。
We consider the nonlinear Dirac equation, also known as the Soler model: i∂ t ψ=− i α⋅∇ ψ+ m β ψ−(ψ⁎ β ψ) k β ψ, m> 0, ψ (x, t)∈ C N, x∈ R n, k∈ N. We study the point spectrum of linearizations at small amplitude 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–Schrödinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov–Kolokolov stability criterion.