Non-linear Frequency Shifts, Mode Couplings, and Decay Instability of Plasma Waves

We present experiments and theory for non-linear plasma wave decay to longer wavelengths, in both the oscillatory coupling and exponential decay regimes. The experiments are conducted on non-neutral plasmas in cylindrical Penning-Malmberg traps, $\theta $-symmetric standing plasma waves have near acoustic dispersion $\omega (k_{z} )\propto k_{z} -\alpha k_{z}^{2} $, discretized by $k_{z} =m_{z} (\pi /L_{p} )$. Large amplitude waves exhibit non-linear frequency shifts $\delta f/f\propto A^{2}$ and Fourier harmonic content, both of which are increased as the plasma dispersion is reduced. Non-linear coupling rates are measured between large amplitude $m_{z} =2$ waves and small amplitude $m_{z} = 1$ waves, which have a small detuning $\Delta \omega =2\omega_{1} -\omega_{2} $. At small excitation amplitudes, this detuning causes the $m_{z} =1$ mode amplitude to ``bounce'' at rate $\Delta \omega $, with amplitude excursions $\Delta A_{1} \propto \delta n_{2} /n_{0} $ consistent with cold fluid theory and Vlasov simulations. At larger excitation amplitudes, where the non-linear coupling exceeds the dispersion, phase-locked exponential growth of the $m_{z} =1$ mode is observed, in qualitative agreement with simple 3-wave instability theory. However, significant variations are observed experimentally, and N-wave theory gives stunningly divergent predictions that depend sensitively on the dispersion-moderated harmonic content. Measurements on higher temperature Langmuir waves and the unusual ``EAW'' (KEEN) waves are being conducted to investigate the effects of wave-particle kinetics on the non-linear coupling rates.