School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China.
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China.
Phys Rev E. 2023 Mar;107(3-1):034210. doi: 10.1103/PhysRevE.107.034210.
We consider the instability and stability of periodic stationary solutions to the classical ϕ^{4} equation numerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figure eight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.
我们通过数值方法研究了经典ϕ^{4}方程的周期定态解的不稳定性和稳定性。在超光速区域，模型具有 dnoidal 和 cnoidal 波。前者是调制不稳定的，谱形成一个在谱平面原点相交的 8 字形。后者可以是调制稳定的，在这种情况下，谱在原点附近由沿着纯虚轴的垂直带表示。在这种情况下，cnoidal 态的不稳定性源于远离谱平面原点的椭圆带的复特征值。在亚光速区域，只有 snoidal 波存在，这些波是调制不稳定的。考虑到次谐波扰动，我们表明，亚光速区域的 snoidal 波对于所有次谐波扰动都是谱不稳定的，而在超光速区域的 dnoidal 和 cnoidal 波中，谱稳定状态和谱不稳定状态之间的转变是通过哈密顿 Hopf 分岔发生的。还考虑了不稳定状态的动力学演化，导致在时空背景上发生了一些有趣的局域事件。
