Ion-acoustic shocks in multicomponent plasma with relativistic positron beam

Abstract

In this investigation, propagation properties of ion-acoustic shocks (IAShs) are studied in electron–ion plasma embedded with relativistic positron beam. Employing reductive perturbation technique, we have derived the Burgers’s equation. Further, by considering the higher order effects, inhomogeneous Burgers-type equation and its solution are derived. It is found that the inclusion of higher-order corrections, results in creating new type of shocks called humped IAShs. Behaviour of different types of IAShs is seen to be dependent on the ion temperature, beam density and kinematic viscosity of electrons.

Appendix

The coefficients of Equations (29)–(37) are expressed as follows:

$$\begin{aligned} D_i&= \frac{R^2 (3R(v^2_{p}+\sigma ) - 2B V_{p} )}{2}, \\ E_i&= R^2 V_{p} (2C + \eta _i),\\ F_i&= \frac{BR}{2} - V_{p} R^2 + V_{p} D_i,\\ G_i&= R C - V_{p} E_i,\\ H_i&= 3 R^2 + 3 D_i, \\ I_i&=3 E_i, \\ D_e&= \frac{S^2(2\beta _1 V_{p}B-3(4S-1))}{2},\\ E_e&= S^2(2\beta _1 V_{p}C + \eta _e V_{p}) ,\\ F_e&= \frac{BS-2V_{p}(S^2-D_e)}{2}, \\ G_e&= SC + V_{p} E_e, \\ H_e&= 3(S^2 + D_e),\\ I_e&= 3 E_e,\\ D_b&= c^2_1 PQ^2(2c^2_1 P(1-2v_{bo}(V_{p}-v_{bo})P) + 9Q \\&-2B\beta _2(V_{p}-v_{bo})),\\ E_b&= 2\beta _2 C 2c^2_1(V_{p}-v_{bo})P Q^2 ,\\ F_b&= c^2_1PQ (B_22c^2_1 PQ (V_{p}-v_{bo})) + D_b (V_{p}-v_{bo}) ,\\ G_b&= 2c^2_1PQC - E_b (V_{p}-v_{bo}) ,\\ H_b&= \frac{3D_b}{2c^2_1P} + P^2 Q^2 3v_{bo}2c^2_1(V_{p}-v_{bo}) - 3c^2_1PQ^2 + \frac{9Q^2}{2} \\ I_b&= \frac{3E_b}{2c^2_1P}. \\ \end{aligned}$$

Various coefficients involved in Equations (50)–(53) are expressed as follows:

$$\begin{aligned} T_i&= R(3V_pRF_i - V_pJ_i + \sigma M_i - 2H_iR\sigma - 3\sigma D_iR \\&\quad + 3R^3\sigma - 2BF_i),\\ U_i&= R(G_iRV_p - K_iV_p + \sigma N_i + I_iR\sigma - 2F_i\eta _i \\&\quad -2CF_i - BG_i) ,\\ V_i&= R(G_iRV_p - L_iV_p + \sigma O_i + 3E_iR\sigma - 2F_i\eta _i - BG_i),\\ W_i&= \frac{E_iC}{V_p} + \frac{I_iCR\sigma }{V_p} - RG_i(\eta _i+C),\\ T_e&= S(\beta _1 B(2F_e+S^2 V_p) + \beta _1V_pJ_e - \beta _1V_p S F_e\\&\quad - \beta _1V^2_p S^3 + \beta _1V^2_p S D_e + D_e - M_e),\\ U_e&= S(\beta _1 C(2F_e+S^2 V_p) + \beta _1 BG_e + \beta _1V_pK_e \\&\quad + 2\eta _e F_e - N_e),\\ V_e&= S(\beta _1 BG_e + \beta _1V_pL_e - \beta _1V_p S G_e + \beta _1V^2_p S E_e + E_e \\&\quad + 2\eta _e F_e - O_e),\\ W_e&= S(\beta _1 CG_e + \beta _1V_pE_eC + \eta _e G_e+ \frac{C}{V_p}(I_e-3E_e) ),\\ T_b&= -\beta _2 Q(J_b(V_p-v_{bo}) + 2BF_b) + 2c^2_1PQ(\alpha M_b + D_b)\\&\quad - 2\beta _2 v_{bo}2c^2_1(V_p-v_{bo})^2 P^3Q^3[B - 2c^2_1(V_p-v_{bo})PQ] \\&\quad - \beta _2 PQ^2 2c^2_1(V_p-v_{bo})[2F_b + D_b(V_p-v_{bo})\\&\quad + 6v_{bo}(V_p-v_{bo})PF_b]- B\beta _2Q(2c^2_1PQ)^2,\\ V_b&= -\beta _2Q(BG_b + (V_p-v_{bo})L_b) + 2c^2_1 PQ(\alpha O_b - E_b) \\&\quad + \beta _2 2c^2_1(V_p-v_{bo})^2PQ^2(E_b-2v_{bo}PG_b), \\ U_b&= -\beta _2Q(2CF_b + BG_b + (V_p-v_{bo})K_b) + 2c^2_1 PQ\alpha N_b \\&\quad - \beta _2 2c^2_1(V_p-v_{bo}) PQ^2G_b (1+2v_{bo}(V_p-v_{bo})P)\\&\quad - C\beta _2 Q(2c^2_1PQ)^2(2v_{bo}(V_p-v_{bo})^2P+1), \\ W_b&= \frac{CQ(\beta _2(V_p-v_{bo})G_b + \alpha I_b 2c^2_1 P)}{(V_p-v_{bo})},\\ J_i&=D_i(2B-3RV_p) - 3RF_i, \\ K_i&=2CD_i + E_i(RV_p-B) - RG_i ,\\ L_i&=E_i(RV_p-B) - RG_i ,\\ M_i&=\frac{3J_i + 5H_iRV_p + 21RF_i - 2BH_i}{V_p}, \\ N_i&=\frac{3K_i + 9RG_i - I_iRV_p - 2CH_i + I_iB}{V_p}, \\ O_i&=\frac{3L_i + 3RG_i - 3I_iRV_p + I_iB}{V_p},\\ J_e&= 2BD_e - 3S(F_e + V_p D_e), \\ K_e&= 2CD_e + BE_e - SG_e - SV_pE_e ,\\ L_e&= BE_e - SG_e - SV_pE_e ,\\ M_e&= \frac{-2BH_e + 5SV_pH_e + 21SF_e +3J_e}{V_p}, \\ N_e&= \frac{-2CH_e - BI_e + SV_pI_e + 9SG_e + 3K_e}{V_p}, \\ O_e&= \frac{-BI_e + 3SG_e + 3SV_pI_e + 3L_e}{V_p}, \\ J_b&= 2BD_b - 2c^2_1PQ3(F_b + D_b(V_p-v_{bo})) ,\\ K_b&= 2CD_b - BE_b - 2c^2_1PQ(G_b - E_b(V_p-v_{bo})), \\ L_b&= -BE_b - 2c^2_1PQ(G_b - E_b(V_p-v_{bo})),\\ M_b&= \frac{-2BH_b+3QF_b}{(V_p-v_{bo})} + \frac{3J_b+18QF_b}{ 2c^2_1P(V_p-v_{bo})} + 2 2c^2_1 PQH_b \\&\quad+ 18v_{bo}PQ(F_b + 2c^2_1 PQ^2(V_p-v_{bo})) +3QH_b, \\ N_b&= \frac{-2CH_b+BI_b}{(V_p-v_{bo})} + \frac{3(K_b+3Q)}{ 2c^2_1P(V_p-v_{bo})} + 6v_{bo}PQG_b\\&- 2c^2_1PQI_b, \\ O_b&= \frac{BI_b+3QG_b}{(V_p-v_{bo})} + \frac{3L_b}{ 2c^2_1P(V_p-v_{bo})} + 6v_{bo}PQG_b \\&- 3QI_b. \end{aligned}$$