Dynamics of Sound Waves in an Interacting Bose Gas

We consider a non-relativistic quantum gas of $N$ bosonic atoms confined to a box of volume $Λ$ in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, $ρ=\frac{N}Λ$, of the gas. We study the time evolution of coherent excitations above the ground state of the gas in a regime of large volume $Λ$ and small ratio $\fracΛρ$. 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 $Λ$. 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 $\fracΛρ$. 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.

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