  We study the out-of-equilibrium dynamics of a finite-temperature harmonically
trapped Tonks-Girardeau gas induced by periodic modulation of the trap
frequency. We give explicit exact solutions for the real-space density and
momentum distributions of this interacting many-body system and characterize
the stability diagram of the dynamics by mapping the many-body solution to the
solution and stability diagram of Mathieu's differential equation. The mapping
allows one to deduce the exact structure of parametric resonances in the
parameter space characterized by the driving amplitude and frequency of the
modulation. Furthermore, we analyze the same problem within the
finite-temperature hydrodynamic approach and show that the respective solutions
to the hydrodynamic equations can be mapped to the same Mathieu equation.
Accordingly, the stability diagram and the structure of resonances following
from the hydrodynamic approach is exactly the same as those obtained from the
exact many-body solution.

Quantum Gases;Quantum Physics