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=NΛ, 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 Λρ. The initial state of the gas is assumed to be close to a 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 Λρ. 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.