  We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a
box of volume $\Lambda$ in physical space. The atoms interact with each other
through a pair potential whose strength is inversely proportional to the
density, $\rho=\frac{N}{\Lambda}$, of the gas. We study the time evolution of
coherent excitations above the ground state of the gas in a regime of large
volume $\Lambda$ and small ratio $\frac{\Lambda}{\rho}$. The initial state of
the gas is assumed to be close to a \textit{product state} of one-particle wave
functions that are approximately constant throughout the box. The initial
one-particle wave function of an excitation is assumed to have a compact
support independent of $\Lambda$. We derive an effective non-linear equation
for the time evolution of the one-particle wave function of an excitation and
establish an explicit error bound tracking the accuracy of the effective
non-linear dynamics in terms of the ratio $\frac{\Lambda}{\rho}$. We conclude
with a discussion of the dispersion law of low-energy excitations, recovering
Bogolyubov's well-known formula for the speed of sound in the gas, and a
dynamical instability for attractive two-body potentials.
