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 Lambda in physical space. The atoms interact with each other through a pair potential whose strength is inversely proportional to the density, rho = 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 Lambda/rho. 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 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 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. (C) 2016 Elsevier Inc. All rights reserved.