Nonlinear wave damping due to multi-plasmon resonances

For short wavelengths, it is well known that the linearized Wigner-Moyal equation predicts wave damping due to wave-particle interaction, where the resonant velocity shifted from the phase velocity by a velocity $v_q = \hbar k/2m$. Here $\hbar$ is the reduced Planck constant, $k$ is the wavenumber and $m$ is the electron mass. Going beyond linear theory, we find additional resonances with velocity shifts $n v_q$, $n = 2, 3, \ldots$, giving rise to a new wave-damping mechanism that we term \emph{multi-plasmon damping}, as it can be seen as the simultaneous absorption (or emission) of multiple plasmon quanta. Naturally this wave damping is not present in classical plasmas. For a temperature well below the Fermi temperature, if the linear ($n = 1$) resonant velocity is outside the Fermi sphere, the number of linearly resonant particles is exponentially small, while the multi-plasmon resonances can be located in the bulk of the distribution. We derive sets of evolution equations for the case of two-plasmon and three-plasmon resonances for Langmuir waves in the simplest case of a fully degenerate plasma. By solving these equations numerically for a range of wave-numbers we find the corresponding damping rates, and we compare them to results from linear theory to estimate the applicability. Finally, we discuss the effects due to a finite temperature.