Linear and nonlinear waves in quantum plasmas with arbitrary degeneracy of electrons

Abstract

The purpose of this review is to revisit recent results in the literature where quantum plasmas with arbitrary degeneracy degree are considered. This is different from a frequent approach, where completely degeneracy is assumed in dense plasmas. The general reasoning in the reviewed works is to take a numerical coefficient in from of the Bohm potential term in quantum fluids, in order to fit the linear waves from quantum kinetic theory in the long wavelength limit. Moreover, the equation of state for the ideal Fermi gas is assumed, for arbitrary degeneracy degree. The quantum fluid equations allow the expedite derivation of weakly nonlinear equations from reductive perturbation theory. In this way, quantum Korteweg–de Vries and quantum Zakharov–Kuznetsov equations are derived, together with the conditions for bright and dark soliton propagation. Quantum ion-acoustic waves in unmagnetized and magnetized plasmas, together with magnetosonic waves, have been obtained for arbitrary degeneracy degree. The conditions for the application of the models, and the physical situations where the mixed dense–dilute systems exist, have been identified.

Appendix: Functions defined after collecting first and second order terms from perturbation theory

The functions $f_{1}$ to $f_{9}$ defined in the Eq. (116) are given as follows:

$$\begin{aligned} f_{1} =\frac{\partial n_{i1}}{\partial \tau }+\frac{\partial }{\partial \xi }\left( n_{i1}v_{ix1}\right) ,\quad f_{2}=\frac{\partial n_{e1}}{\partial \tau }+\frac{\partial }{\partial \xi }\left( n_{e1}v_{ex1}\right) , \end{aligned}$$

(a1)

$$\begin{aligned} f_{3} =\frac{\partial v_{ix1}}{\partial \tau }+v_{ix1}\frac{\partial }{\partial \xi }v_{ix1}-\Omega \text { }v_{iy1}B_{z1},\quad f_{4}=-v_{0}\frac{\partial v_{ix1}}{\partial \xi }+\Omega \text { }v_{ix1} B_{z1}, \end{aligned}$$

(a3)

$$\begin{aligned} f_{5} =\frac{\partial v_{ex1}}{\partial \tau }+v_{ex1}\frac{\partial }{\partial \xi }v_{ex1}+\delta \, \Omega \,v_{ey1}B_{z1}-\delta \, \alpha \, n_{e1}\frac{\partial n_{e1}}{\partial \xi }-\delta \, \frac{H^{2}}{4}\quad \frac{\partial ^{3}n_{e1}}{\partial \xi ^{3}}, \end{aligned}$$

(a5)

$$\begin{aligned} f_{6} =-v_{0}\frac{\partial v_{ey1}}{\partial \xi }+\Omega \,v_{ex1} B_{z1},\quad f_{7}=-\Omega \frac{\partial B_{z1}}{\partial \tau }, \end{aligned}$$

(a6)

$$\begin{aligned} f_{8} =\left( n_{e1}-n_{i1}\right) v_{ex1},\quad f_{9}=\frac{c_{\mathrm{s}}^{2} }{c^{2}}\left( n_{e1}v_{ey1}-n_{i1}v_{iy1}\right) . \end{aligned}$$

(a8)