  We report quasi-periodic pulsations (QPPs) with double periods during three
solar flares (viz. SOL2011-Feb-15T01:44, SOL2011-Sep-25T04:31,
SOL2012-May-17T01:25). The flare QPPs were observed from light curves in
Ly$\alpha$, hard X-ray (HXR) and microwave emissions, with the Ly$\alpha$
emission recorded by the Geostationary Operational Environmental Satellite, the
HXR emission recorded by the Reuven Ramaty High-Energy Solar Spectroscopic
Imager and the Fermi Gamma-ray Burst Monitor, and the microwave emission
recorded by the Nobeyama Radio Polarimeters and Radioheliograph. By using the
Markov chain Monte Carlo (MCMC) method, QPPs with double periods of about two
minutes and one minute were first found in the Ly$\alpha$ emission. Then using
the same method, a QPP with nearly the same period of about two minutes was
also found in HXR and microwave emissions. Considering the possible common
origin (nonthermal electrons) between Ly$\alpha$ and HXR/microwave emission, we
suggest that the two-minute QPP results from the periodic acceleration of
nonthermal electrons during magnetic reconnections. The ratio between the
double periods in the Ly$\alpha$ emission was found to be close to two, which
is consistent with the theoretical expectation between the fundamental and
harmonic modes. However, we cannot rule out other possible driving mechanisms
for the one-minute QPPs in HXR/microwave emissions due to their relatively
large deviations.
