by @LinShu1964. However, the physical origin and dynamical evolution of spiral arms in disk galaxies have thus far not been fully understood. In this study, we focus on the dynamics of stellar spiral arms and stars around them in a disk galaxy without external perturbations and a bar structure.

The Lin-Shu theory posits that the stellar spiral arms can be interpreted as (quasi-) stationary density waves with a constant pattern speed [@LinShu1964; @BertinLin1996]; the theory has gained wide acceptance as providing a clear explanation regarding the dynamics of spiral arms. However, as pointed out in the study by @Toomre1969, the quasi-stationary hypothesis has a serious limitation because of a dispersal nature of the tight winding spiral waves, they radially propagate with the group velocity, and are consequently absorbed at the Lindblad resonances [@LyndenBellKalnajs1972]. Therefore, the stationary waves that last for a long duration require some amplification mechanisms such as WASER [@Mark1976] and a feedback cycle that involves the reflection of the inward propagating wave into an outward propagating one at the $Q$-barrier [@Bertin+1989a; @Bertin+1989b].

However, nearly all the previous time-dependent simulations that have been executed thus far have been unable to prove the existence of stationary density waves in a 