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We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to abox of volume $\Lambda$ in physical space. The atoms interact with each otherthrough a pair potential whose strength is inversely proportional to thedensity, $\rho=\frac{N}{\Lambda}$, of the gas. We study the time evolution ofcoherent excitations above the ground state of the gas in a regime of largevolume $\Lambda$ and small ratio $\frac{\Lambda}{\rho}$. The initial state ofthe gas is assumed to be close to a \textit{product state} of one-particle wavefunctions that are approximately constant throughout the box. The initialone-particle wave function of an excitation is assumed to have a compactsupport independent of $\Lambda$. We derive an effective non-linear equationfor the time evolution of the one-particle wave function of an excitation andestablish an explicit error bound tracking the accuracy of the effectivenon-linear dynamics in terms of the ratio $\frac{\Lambda}{\rho}$. We concludewith a discussion of the dispersion law of low-energy excitations, recoveringBogolyubov's well-known formula for the speed of sound in the gas, and adynamical instability for attractive two-body potentials.

Dynamics of Sound Waves in an Interacting Bose Gas