  We study the breathing (monopole) oscillations and their damping in a
harmonically trapped one-dimensional (1D) Bose gas in the quasicondensate
regime using a finite-temperature classical field approach. By characterising
the oscillations via the dynamics of the density profile's rms width over long
time, we find that the rms width displays beating of two distinct frequencies.
This means that 1D Bose gas oscillates not at a single breathing mode
frequency, as found in previous studies, but as a superposition of two distinct
breathing modes, one oscillating at frequency close to $\simeq\!\sqrt{3}\omega$
and the other at $\simeq\!2\omega$, where $\omega$ is the trap frequency. The
breathing mode at $\sim\!\sqrt{3}\omega$ dominates the beating at lower
temperatures, deep in the quasicondensate regime, and can be attributed to the
oscillations of the bulk of the density distribution comprised of particles
populating low-energy, highly-occupied states. The breathing mode at
$\simeq\!2\omega$, on the other hand, dominates the beating at higher
temperatures, close to the nearly ideal, degenerate Bose gas regime, and is
attributed to the oscillations of the tails of the density distribution
comprised of thermal particles in higher energy states. The two breathing modes
have distinct damping rates, with the damping rate of the bulk component being
approximately four times larger than that of the tails component.
