Effects of group velocity and multiplasmon resonances on the modulation of Langmuir waves in a degenerate plasma

We study the nonlinear wave modulation of Langmuir waves (LWs) in a fully degenerate plasma. Using the Wigner–Moyal equation coupled to the Poisson equation and the multiple scale expansion technique, a modified nonlocal nonlinear Schrödinger (NLS) equation is derived which governs the evolution of LW envelopes in degenerate plasmas. The nonlocal nonlinearity in the NLS equation appears due to the group velocity and multiplasmon resonances, i.e., resonances induced by the simultaneous particle absorption of multiple wave quanta. We focus on the regime where the resonant velocity of electrons is larger than the Fermi velocity and thereby the linear Landau damping is forbidden. As a result, the nonlinear wave-particle resonances due to the group velocity and multiplasmon processes are the dominant mechanisms for wave–particle interaction. It is found that in contrast to classical or semiclassical plasmas, the group velocity resonance does not necessarily give rise the wave damping in the strong quantum regime where $\hslash k\sim m{v}_F$ with $\hslash$ denoting the reduced Planck's constant, $m$ the electron mass, and $v_F$ the Fermi velocity; however, the three-plasmon process plays a dominant role in the nonlinear Landau damping of wave envelopes. In this regime, the decay rate of the wave amplitude is also found to be higher compared to that in the modest quantum regime where the multiplasmon effects are forbidden.

Figure 1

The normalized resonant velocities ($\sim v_F$) are plotted against the normalized wave number $k(\sim {\lambda }_F^{-1})$ for two different values of the dimensionless quantum parameter $H$ to show different parameter regimes, namely semiclassical (e.g., $0<k\ll 0.59$ for $H=1$), modest quantum ($0<k\lesssim 0.59$ for $H=1$ and $0<k\lesssim 0.75$ for $H=0.5$) and strong quantum ($0.591\lesssim k\lesssim 0.9$ for $H=1$ and $0.75\lesssim k\lesssim 1$ for $H=0.5$) regimes. In the legends, $v_{\mathrm{res}}^n$ denotes the resonant velocity, where $n=l,2,3,g$, respectively, correspond to the velocities for the linear, two-plasmon, three-plasmon, and group velocity resonances.

Figure 2

The normalized frequency shift ${\mathrm{\Omega }}_r(\sim {\omega }_p)$ [Eq. (40), panel (a)] and the energy transfer rate $\mathrm{\Gamma }(\sim {\omega }_p)$ [Eq. (41), panel (b)] are plotted against the normalized wave number of modulation $K(\sim {\lambda }_F^{-1})$ for different values of the carrier wave number $k(\sim {\lambda }_F^{-1})$ that correspond to semiclassical and modest quantum regimes.

Figure 3

The same as in Fig. 3 but in the strong quantum regime. In the legends, $2P,3P$, and $GV$, respectively, stand for two-plasmon, three-plasmon, and group velocity resonance effects.

Figure 4

The absolute value of the decay rate $DR\equiv |{\left(1-i\tau /{\tau }_0\right)}^{-1/2}|$ is shown against the normalized time variable $\tau \left({\omega }_p^{-1}\right)$ in different parameter regimes as in the legend.