Fundamentals of Plasma Physics 2025, 44 - Wave-Wave Nonlinearities
A linear wave propagating in a plasma through inserting an antenna the plasma connected to a sine-wave generator turned to an appropriate frequency can be detected by a distant receiving probe, which is connected to a spectrum analyzer. If the generator is tuned below a critical amplitude threshold, there is a single frequency peak on the analyzer. Above the threshold, there are two additional "daughter peaks" at very different frequency. At higher amplitudes, the daughter wave amplitudes increase, while the pump wave amplitude remains the same. The lower-frequency daughter wave is at zero frequency, the pump wave beats with itself, modulating the equilibrium via a ponderomotive non-linear pressure, acting to expel plasma. It can dig itself a channel in the plasma, guiding the pump wave.
For three non-linearly coupled harmonic oscillators,
The sum of the modes of the energies is constant, though energy can be transferred between them. As ω1ω2 > 0, ω1 + ω2 = ω3, the former as both positive, and the latter is then always larger than any of the other two individually. If A2 grows, so must A1 equally, and A3 decreases, until A3 >> A1 fails. The decay of the mode 3 photon into mode 1, 2 photons can be characterized as A2 ~ eγτ.