论文标题
具有浓缩非线性的dirac方程孤立波的光谱稳定性和不稳定性
Spectral stability and instability of solitary waves of the Dirac equation with concentrated nonlinearity
论文作者
论文摘要
我们认为，非线性狄拉克方程与soler型非线性集中在一个点上，并详细研究了孤立波的线性化光谱。然后，我们考虑非线性的两种不同的扰动，它们破坏了$ \ Mathbf {su}（1,1）$ - 对称性：第一个保留和第二个打破奇偶校验对称性的。我们表明，打破$ \ mathbf {su}（1,1）$ - 对称性而不是平等对称性的扰动也保留了单个波浪的频谱稳定性。然后，我们考虑一种打破$ \ mathbf {su}（1,1）$ - 对称性和奇偶对称性的扰动，并表明这种扰动破坏了弱相对主义孤立的波的稳定性。发展中的不稳定性是由于嵌入式特征值$ \ pm2Ω\ mathrm {i} $的正面零件特征值的分叉所致。
We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetry. We show that a perturbation which breaks the $\mathbf{SU}(1,1)$-symmetry but not the parity symmetry also preserves the spectral stability of solitary waves. Then we consider a perturbation which breaks both the $\mathbf{SU}(1,1)$-symmetry and the parity symmetry and show that this perturbation destroys the stability of weakly relativistic solitary waves. The developing instability is due to the bifurcations of positive-real-part eigenvalues from the embedded eigenvalues $\pm 2ω\mathrm{i}$.
