  Soliton in the hostile turbulent wave dark matter ($\Psi$DM) halo of a galaxy
agitates with various kinds of excitation, and the soliton even breathes
heavily under great stress. A theory of collective excitation for a $\Psi$DM
soliton is presented. The collective excitation has different degrees of
coupling to negative energy modes, where lower-order excitation generally
necessitates more negative energy coupling. A constrained variational principle
is developed to assess the frequencies and mode structures of small-amplitude
perturbations. The predicted frequencies are in good agreement with those found
in simulations. Soliton breathing at amplitudes on the verge of breakup is also
a highlight of this work. Even in this extreme nonlinear regime, the wave
function perturbation amplitudes are moderate. The simulation data shows a
stable oscillation with frequency weakly dependent on the oscillation
amplitude, and hints a self-consistent quasi-linear model for the wave function
that accounts for modifications in the ground state wave function and the
equilibrium density. The mock solution, constructed from the simulation data,
can shed lights on the dynamics of the large-amplitude breathing soliton and
supports the quasi-linear model, as evidenced by its ability to well predict
the nonlinear eigenfrequency shifts and large-amplitude breathing frequency
observed in simulations.
