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We report the experimental studies of a parametric excitation of a secondsound (SS) by a first sound (FS) in a superfluid helium in a resonance cavity.The results on several topics in this system are presented: (i) The linearproperties of the instability, namely, the threshold, its temperature andgeometrical dependencies, and the spectra of SS just above the onset weremeasured. They were found to be in a good quantitative agreement with thetheory. (ii) It was shown that the mechanism of SS amplitude saturation is dueto the nonlinear attenuation of SS via three wave interactions between the SSwaves. Strong low frequency amplitude fluctuations of SS above the thresholdwere observed. The spectra of these fluctuations had a universal shape withexponentially decaying tails. Furthermore, the spectral width grew continuouslywith the FS amplitude. The role of three and four wave interactions arediscussed with respect to the nonlinear SS behavior. The first evidence ofGaussian statistics of the wave amplitudes for the parametrically generatedwave ensemble was obtained. (iii) The experiments on simultaneous pumping ofthe FS and independent SS waves revealed new effects. Below the instabilitythreshold, the SS phase conjugation as a result of three-wave interactionsbetween the FS and SS waves was observed. Above the threshold two new effectswere found: a giant amplification of the SS wave intensity and strong resonanceoscillations of the SS wave amplitude as a function of the FS amplitude.Qualitative explanations of these effects are suggested.Comment: 73 pages, 23 figures. to appear in Phys. Rev. B, July 1 st (2001

Parametric generation of second sound in superfluid helium: Linear stability and nonlinear dynamics