  We investigate the nonequilibrium dynamics of the one-dimension
Aubry-Andr\'{e}-Harper model with $p$-wave superconductivity by changing the
potential strength with slow and sudden quench. Firstly, we study the slow
quench dynamics from localized phase to critical phase by linearly decreasing
the potential strength $V$. The localization length is finite and its scaling
obeys the Kibble-Zurek mechanism. The results show that the second-order phase
transition line shares the same critical exponent $z\nu$, giving the
correlation length $\nu=0.997$ and dynamical exponent $z=1.373$, which are
different from the Aubry-Andr\'{e} model. Secondly, we also study the sudden
quench dynamics between three different phases: localized phase, critical
phase, and extended phase. In the limit of $V=0$ and $V=\infty$, we
analytically study the sudden quench dynamics via the Loschmidt echo. The
results suggest that, if the initial state and the post-quench Hamiltonian are
in different phases, the Loschmidt echo vanishes at some time intervals.
Furthermore, we found that, if the initial value is in the critical phase, the
direction of the quench is the same as one of the two limits mentioned before,
and similar behaviors will occur.
