  We study the dynamics of the matter-wave soliton interacting with a vibrating
mirror created by an evanescent light and provide a quantum scattering picture
for the time-domain diffraction of the matter-wave soliton. Under
Kramers-Henneberger (KH) transformation, i.e., in a vibrating coordinate, the
vibration of the mirror can be cast to an effective gauge field. We then can
exploit Dyson series and the quantum scattering theory to investigate the
dynamics of the soliton that moves in the effective gauge field and is
reflected by a static mirror. Our analytical theory can quantitatively deduce
the locations and the relative weights of the scattered wave packets, which is
consistent with our numerical simulations of directly solving a nonlinear
Schr\"{o}dinger equation. In particular, for a two-frequency vibrating case,
our theory predicts some interesting multi-peak sideband structures in the
diffracted matter-wave distributions, which can be resorted to the resonance of
two frequencies. Underlying mechanisms and possible applications are discussed.

Pattern Formation and Solitons;Quantum Gases