  We present a novel one-dimensional nonlinear evolution equation governing the
dynamics short-wavelength longitudinal waves in a nonrelativistic fully
degenerate quantum plasma using kinetic equation for the Wigner function. The
linear dispersion of the equation has a form of "zero sound" $\sim k\exp
(-k^{2})$, where $k$ is the wave number, and it strongly differs from
previously known nonlinear evolution equations. We numerically find the
corresponding soliton solutions and demonstrate that the collisions between
three solitons turn out to be elastic resulting only in phase shifts.
