This paper derives exact finite-temperature dynamics and stability boundaries for a Tonks-Girardeau gas in a periodically modulated harmonic trap, with both quantum and hydrodynamic descriptions reducing to the same Mathieu equation.
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Finite-temperature dynamics of a Tonks-Girardeau gas in a frequency-modulated harmonic trap
This paper derives exact finite-temperature dynamics and stability boundaries for a Tonks-Girardeau gas in a periodically modulated harmonic trap, with both quantum and hydrodynamic descriptions reducing to the same Mathieu equation.