A bstract Based on the spectral decomposition technique, we introduce a simple and universal numerical method to analyze the stability of solitons. Adopting this method, the linear dynamical properties of Q -balls are systematically revealed, from the fundamental to the excited states. For the fundamental Q -ball, the well-known stability criterion holds. However, for the excited Q -balls, the situation becomes extremely complicated, in which the stability criterion is violated. The system exhibits dynamical instability to both spherically symmetric and non-spherically symmetric perturbations, manifested in the appearance of complex and imaginary modes. In addition, we observe two interesting phenomena. One is that the oscillation mode and the complex or imaginary mode can transform into each other, marking the transition of the dynamical properties of the system. The other is the existence of excited Q -balls capable of resisting perturbations with low-order spherical harmonics. Such results indicate that the excited Q -balls will exhibit rich dynamical behaviors.
Chen et al. (2026) studied this question.