Higher-order corrections to nonlinear dust-ion-acoustic shock waves in a degenerate dense space plasma

Abstract

A reductive perturbation technique is employed to investigate the contribution of higher-order nonlinearity and dissipation to nonlinear dust-ion-acoustic (DIA) shock waves in a three-component degenerate dense space plasma. The model consists of degenerate electron (being either ultrarelativistic or nonrelativistic), nonrelativistic ion fluid and stationary heavy dust grains. A nonlinear Burger equation and a linear inhomogeneous Burger-type equation are derived. The present model admits only compressive DIA shocks. Including these higher-order corrections results in creating new solitary wave structures “humped DIA shock” waves. For the case of ultrarelativistic (nonrelativistic) electrons, one (two) humped DIA shock is (are) created. The DIA shock wave amplitude and velocity is larger in case of ultrarelativistic electrons than of nonrelativistic electrons. It is shown that the effects of kinematic viscosity, heavy dust grains number density, and equilibrium ion number density have important roles in the basic features of the produced DIA shocks and the associated electric fields. The implications of our results to dense plasmas in astrophysical objects (e.g., non-rotating white dwarf stars) are briefly discussed.

Appendix: The coefficients D,E,F,G,H,I and J appeared in Eqs. (14) and (16)

The explicit expressions of the D, E and F coefficients presented in Eq. (14) are given by

$$\begin{aligned} D & =\frac{1}{ ( V_{p}^{2}-\alpha K_{1} ) ^{3}} \bigl\{ \bigl[ 3V_{p}^{2}+\alpha ( \alpha-2 ) K_{1} \bigr] \\ &\quad {} -2AV_{p} \bigl( V_{p}^{2}- \alpha K_{1} \bigr) \bigr\} , \end{aligned}$$

(19)

$$\begin{aligned} E & =\frac{1}{V_{p}^{2}-\alpha K_{1}}, \end{aligned}$$

(20)

$$\begin{aligned} F & =\frac{1}{ ( V_{p}^{2}-\alpha K_{1} ) ^{3}} \bigl\{ V_{p} \bigl[ \alpha ( \alpha-2 ) K_{1}+3V_{p}^{2} \bigr] \\ &\quad {} -2V_{p} \bigl( V_{p}^{2}-\alpha K_{1} \bigr) -A \bigl( V_{p}^{4}-\alpha ^{2}K_{1}^{2} \bigr) \bigr\} , \end{aligned}$$

(21)

and those coefficients; G, H, I and J, introduced in Eq. (16), are given by

$$\begin{aligned} & G =-\frac{1}{8E^{2}V_{p}} \bigl( 4+\eta^{2}E^{2} \bigr) , \end{aligned}$$

(22)

$$\begin{aligned} & H = \biggl\{ \frac{DE^{2}}{2\gamma K_{2}} \biggl[ \frac{1}{2E^{2}V_{p}}-3 ( \gamma-1 ) \biggr] \\ &\hphantom{H}\quad {} -\frac{1}{2 ( \gamma K_{2} ) ^{2}} ( \gamma-2 ) ( \gamma-1 ) E-3FE^{2}V_{p} \\ &\hphantom{H}\quad {} -\frac{3D}{2}E^{2} \bigl[ V_{p}^{2}+ \alpha K_{1} ( \alpha-2 ) \bigr] \\ &\hphantom{H}\quad {} -\frac{1}{2}E^{4} \alpha ( \alpha-3 ) ( \alpha-2 ) K_{1}+\ A ( DEV_{p}+EF ) \biggr\} , \end{aligned}$$

(23)

$$\begin{aligned} & I =-\frac{\eta}{4EV_{p}} \bigl( 2E^{2}V_{p}+2F-AE \bigr) , \end{aligned}$$

(24)

$$\begin{aligned} & J =-\frac{\eta}{4EV_{p}} \bigl( 2E^{2}V_{p}+F-AE-DV_{p} \bigr) . \end{aligned}$$

(25)