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where " is a small parameter and is the wave propagation speed, to be determined latter. The dependent variables are expanded as n e;p = + "n (1) e;p + " 2 n (2) e;p + " 3 n (3) e;p + ::::::; (8) u e;p = "u (1) e;p + " 2 u (2) e;p + " 3 u (3) e;p + ::::::::::; (9) v e;p = " 3=2 v (1) e;p + " 5=2 v (2) e;p + " 7=2 v (3) e;p + :::::; (10) ' = "' (1) + " 2 ' (2) + " 3 ' (3) + :::::::::; (11) where = 1 for the electrons and = for the positrons. The perturbation method to be adopted here consists in substituting Eqs. (7)-(11) into the model Eqs. (1)-(5), and isolating, and then solving, distinct orders in ". The lowest order in " gives n (1) e;p = q e;p S e;p ' (1) ; u (1) e;p = q e;p S e;p ' (1) ; @v (1) e;p @ = q e;p S e;p @' (1) @ : (12) where S e;p = 2 2 e;p , i.e., S e = 2 2 and S p = 2 2 p . The sign of the latter may be positive or negative, depending on the values of the plasma parameters ; e;p and : Note that the (leading order) perturbation of the electric potential determines the corresponding variation of the electron and positron density and velocity. For instance, physically speaking, an electron density compression may usually be expected where a positron rarefaction is found, and vice versa; note, however, that this is not always the case, since n (1) e = 1 S e ' (1) and n (1) p = S p ' (1) bear the same sign between p 2 and p 2 p (opposite signs elsewhere). The Poisson equation provides the compatibility condition The next-order in the perturbation yields 1 2 2 + 2 2 p = i ; (13) @n(1) e;p @ 2 @n(2) e;p @ + @u(2) e;p @ + @ @ n(1) e;pu (1) e;p + u(1) e;p + @v (1) e;p @ = 0; (14) and @u (1) e;p @ @ 2 ' (1) @ 2 @u(2) e;p @ + X e;p + u (1) @u (1) e;p e;p @ + 2 e;p @n (2) e;p @ + q e;p @' (2) @ = 0; (15) q e;p n (2) e;p i ' (2) + 1 2 2 i ' (1)2 = 0 : (16) 6
Eliminating the second-order perturbed quantities and making use of the …rst-order results, we obtain a nonlinear partial-derivative equation (PDE) in the form of the cylindrical Kadomtsev-Petviashvili (CKP) equation @ @' (1) (1) @'(1) + A' + B @3 ' (1) + 1 @ @ @ @ 3 2 '(1) + 1 @ 2 ' (1) = 0 : (17) 2 2 @ 2 The nonlinearity and di¤usion coe¢ cients, A and B respectively, read A = 3 1 1 + 2 2 2 2 i 2 ; (18) 2 p B = (2 2) 2 2 + (2 2 p ) 2 2 : (19) We note that B is usually greater than zero, while the sign of A depends on speci…c parameter values, as we shall see below. Note that, combining with (13), the dependence of A and B on p is eliminated. So, A and B depend on ; i and only. To obtain an exact solitary wave solution of Eq. (17), we shall use the following substitution [21] = 2 2 ; and ' (1) = ' (1) (; ); (20) into Eq. (17). We thus obtain the standard Korteweg-de Vries (KdV) equation @' (1) @ (1) @'(1) + A' + B @3 ' (1) = 0: (21) @ @ 3 In order to obtain a steady-state travelling-wave solution of Eq. transformation (21), we introduce the = U 0 ; (22) where U 0 is a real constant representing the speed of the moving wavefront. Assuming the boundary conditions ' ! 0 and d'=d ! 0 at jj ! 1, we obtain a solitary wave solution (KdV soliton) ' (1) = ' 0 sech 2 U0 W ; (23) where ' 0 = 3U 0 =A is the maximum (potential perturbation) amplitude and W = p 4B=U 0 measures the spatial extension (width) of the localized pulse. We note that ' 0 W 2 = 12B=A = constant (for given ); in agreement with the standard KdV result [22]. However, the dependence of ' 0 and W on is more perplex, as we shall see below. Combining Eq. 7
Page 1 and 2: Nonlinear excitations in electron-p
Page 3 and 4: I. INTRODUCTION In contrast to ordi
Page 5: We shall assume that the ions are B
Page 9 and 10: propagation speed and the positron
Page 11 and 12: A. Solitary wave solution The solit
Page 13 and 14: The DL solution (42) corresponds to
Page 15 and 16: of the velocity is provided by the
Page 17 and 18: (2006); I. Kourakis, A. Esfandyari-
Page 19 and 20: Figure 7. The potential ' (1) is de
Page 21 and 22: Figure 2
Page 29: Figure 10

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