非线性Dirac方程Vakhitov-Kolokolov稳定性判据的意义
On the meaning of the Vakhitov-Kolokolov stability criterion for the nonlinear Dirac equation
摘要：
研究了任意维非线性Dirac方程孤波解<$（x）e^{-iωt}的谱稳定性.这个方程被理论物理学家称为Soler模型（或在一维中称为Gross-Neveu模型），多年来一直备受关注。我们发现，一般情况下，在Vakhitov-Kolokolov稳定性准则崩溃的值，一对实本征值（一个正，一个负）出现从原点，导致相应的线性不稳定的孤波。作为一个辅助结果，我们陈述了非线性Dirac方程的维里恒等式（“Pohozhaev定理”）。我们还证明了\pm 2ωi是非线性狄拉克方程在<$（x）e^{-iωt}处线性化的特征值，它们嵌入到连续谱中，|ω|> m/3。这一结果适用于具有Soler形式的任何非线性（“标量-标量相互作用”）和任何维度的非线性狄拉克方程。
英文摘要：
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 \pm 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.
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