Chapter 3 Head-on collision of ion acoustic solitary waves in electron-positron-ion plasma with superthermal electrons and positrons. 73
3.1 Introduction The study of linear and nonlinear wave propagation of electron-positron plasma attracted a remarkable interest during the past three decades or so [176]-[178],[15]. Positrons are created in the interstellar medium when the atoms become interacted by the cosmic ray nuclei [194]-[195]. It is well established that the presence of positrons in electron-ion plasma changes the linear and nonlinear wave features [20]. So electronpositron-ion plasmas have been considered extensively in different plasma models [180]-[184]. The electrons and positrons have the same mass but opposite charge, so it is expected that they will be described by a similar distribution. The distribution function that can better model such particle velocity distribution function is known as the generalized Lorengian or kappa distribution [91] with functional dependence of the form f 0 (v) [1+v 2 /(kθ 2 )] (κ+1). The spectral index κ, is a measure of the slope of the energy spectrum of the super thermal electrons and positrons forming the tail of the velocity distribution function. Several observations in astrophysical plasmas, namely, solar wind, auroral zone plasma, magnetosphere [36] [92]-[97] insists one to follow the super thermal or non Maxwellian distribution. Moreover the kappa (super thermal) distribution has also been used to analyze and interpret spacecraft data on the earth s magnetospheric plasma sheet [98], Jupiter [99] and Saturn [100]. In limit κ the distribution approaches a Maxwellian distribution [ exp( v 2 /θ 2 ])]. E.L.EL-Awady et al [22] obtained the nonlinear ion acoustic solitary waves (IASWs) which accompany collision-less e-p-i plasma with high-energy electrons and positrons (represented by a kappa distribution). Solitons are solitary waves with the remarkable property that the solitons preserve their form asymptotically even when they undergo a collision. The term was first 74
coined by Zabusky and Kruskal [61]. In a one dimensional system, the solitons may interact between them in two different ways. One is overtaking collision and it can be studied by the inverse scattering transformation method [63]. Another one is head-on collision [144] where the angle between two propagation directions of two solitons is π. This has been investigated by many researchers. For such a collision, the solution of two solitary waves via two Korteweg-de vries (KdV) equations can explain resonance phenomena and these have been observed in shallow water wave experiments [187], in plasmas experiments [188], in two core optical fiber [189] and in fluid filled elastic tubes [190]. For a head-on collision between two solitary waves travelling to positive and negative directions, one must search their evolution and so one need to employ a suitable asymptotic expansion to solve original fluid equations. The unique effects due to the collision are their phase shift and trajectories. Many authors investigated the head-on collision of two solitary waves in different plasma models using extended Poincare-Lighthill- Kuo (PLK) method [147]-[156]. For example J.N Han et al [149] studied head-on collisions of IASWs in a weakly relativistic e-p-i plasma. By using the extended PLK method they obtained the KdV equations, the analytical phase shift and the trajectories after the head-on collision of two solitary waves in e-p-i plasma. Chatterjee et al [156] investigated head-on collision of ion acoustic solitary waves in an electron-positron-ion plasma with superthermal electrons but they have taken the Maxwell distribution of positrons. As electrons and positrons have the same mass but opposite charge, it is expected that they will be described by a similar distribution. To the best of our knowledge there is no detailed investigations about the interaction of superthermal electron, superthermal positron and cold ions in collision-less unmagnetized plasma. Thus in this paper we want to study the same mentioned above by the extended PLK method. Two opposite directional solitary waves are propagated and head-on collision of these two waves occurs. The phase shift and trajectories of 75
the two solitons after the collision are deduced. The effects of superthermal distribution parameter κ on the phase shifts on the characteristics of head-on collisions are also included. 3.2 Basic Equations and Derivation of KdV Eqs and phase shifts Let us consider a normalized ion-fluid model with super thermal electrons, super thermal positrons and cold ions in order to study the head on collision of IASWs which accompanied the nonlinear dynamics of the low frequency one-dimensional IASWs in this three component plasma. The basic equations are n i t + (n iu i ) x u i t + u (u i ) i x 2 ψ 2 x = 0; (3.1) = ψ x ; (3.2) n e = 1 p p 1 p n p n i ; (3.3) where n e = (1 ψ κ ) κ+1 2, n p = (1+σ ψ κ ) κ+1 2, σ = T e /T p and p = n p0 /n e0 = 1 n i0 /n e0 Equations (3.1)-(3.3) are the continuity equation, momentum equation and Poisson equation respectively. T e,p are the electron and positron temperature respectively, κ is the spectral index. n α is the perturbed density and n α0 is the normalized density of α species (where α = e,p,i stands for electron, positrons and ions respectively). u i is the ion fluid speed normalized by c s = (K B T e /m i ) 1/2. The electric potential Ψ is normalized by K B T i /e. space x and time t are normalized by λ D = K B T i 4πe 2 n i0 76
and ωp1 1 = m i 4πe 2 n i0 respectively. K B is the Boltzmann constant and e is the electron charge and m i is mass of the ion. Now we assume that two solitons α and β in the plasma which are asymptotically far apart in the initial state and travel towards each other. After some time they interact, collide and then depart. We also assume that the solitons have small amplitudes ε (where ε is the small parameter characterizing the strength of non linearity ) and the interaction between two solitons is weak. Hence we expect that the collision will be quasi elastic and so it will only cause shifts of the post collision trajectories (phase shift). In order to analyze the effects of a collision, we employ an extended PLK method. According to this method, the dependent variables are expanded as n i = 1 + ε 2 n 1 + ε 3 n 2 + ε 4 n 3 + ; (3.4) u i = u 0 + ε 2 u 1 + ε 3 u 2 + ε 4 u 3 + ; (3.5) ψ = 0 + ε 2 ψ 1 + ε 3 ψ 2 + ε 4 ψ 3 + ; (3.6) The independent variables are given by ξ = ε(x c 1 t) + ε 2 P 0 (η,τ) + ε 3 P 1 (η,ξ,τ) + ; (3.7) η = ε(x + c 2 t) + ε 2 Q 0 (ξ,τ) + ε 3 Q 1 (η,ξ,τ) + ; (3.8) τ = ε 3 t; (3.9) where ξ and η denote the trajectories of two solitons travelling toward to each other, and c 1 and c 2 are the unknown phase velocity of IASWs. The variables of P 0 (η,τ) and Q 0 (ξ,τ) are also to be determined. Proceeding in a similar manner in chapter 2, Substituting Eqs (3.4) (3.9) into Eqs (3.1) (3.3) and equating the quantities with equal power of ε, we obtain coupled equations in different orders of ε. Finally we get λu 3 = (3.10) 77
( ψ ξ τ + Aψ ξ + where A = 1 2 [ ψ ξ ξ + ψ ξ B 3 ξ )dη + ( ψ η 3 τ Aψ ψ η η η ψ η B 3 η )dξ 3 (C P0 η Dψ η) 2 ψ ξ ξ 2 dξdη (C Q0 ξ Dψ ξ) 2 ψ η η 2 dξdη; (1 p)κ [3 (κ 1 (κ 1 2 )(1+pσ) (κ+1 2 )(1 pσ2 ) 2 )(1+pσ)]1/2 (1 p)κ (1+pσ)κ (1 p)κ (1 p)κ [ (κ 1 (κ 1 2 )(1+pσ) + (κ+1 2 )(1 pσ2 ) ] 2 )(1+pσ)]1/2 (1 p)κ (1+pσ)κ (p 1)κ (κ 1 )(1+pσ)] 2 2[ and D = 1[ (κ 1 2 )(1+pσ)]1/2 2 and λ = 2 The phase velocity c 1 = (1 p)κ (κ 1 2 )(1+pσ) + u 0 and c 2 = ],B = 1 2 [ (1 p)κ (κ 1 2 )1+pσ]3/2, C = (1 p)κ (κ 1 2 )(1+pσ) u 0 are also obtained, where u 0 is the drift fluid velocity. The first term in the right hand side of Eqn (3.10) will be proportional to η because the integrand function is independent of η. The second term in the right hand side of Eqn (3.10) will be proportional to ξ because the integrand function is independent of ξ. Thus the two terms of Eqn (3.10) are all secular terms, which must be eliminated in order to avoid spurious resonances. Hence, we have ψ ξ τ + Aψ ψ ξ ξ ξ + ψ ξ B 3 ξ 3 = 0; (3.11) ψ η τ Aψ ψ η η η ψ η B 3 η 3 = 0; (3.12) The third and fourth terms in Eqn (3.10) are not secular terms in this order, they could be secular in the next order. Hence we have C P0 η C Q0 ξ = Dψ η ; (3.13) = Dψ ξ ; (3.14) Equations (3.11) and (3.12) are the two side travelling wave KdV equations in the reference frames of ξ and η respectively. One of their special solutions are ψ ξ = ψ A sech 2 [( Aψ A 12B )1/2 (ξ 1 3 Aψ Aτ)]; (3.15) 78
ψ η = ψ B sech 2 [( Aψ B 12B )1/2 (η + 1 3 Aψ Bτ)]; (3.16) where ψ A and ψ B are the amplitudes of the two solitons α and β in their initial positions. The leading phase changes due to the collision can be calculated from Eqs (3.13) and (3.14) are given by P 0 (η,τ) = D C (12Bψ B A )1/2 [tanh( Aψ B 12B )1/2 (η + 1 3 Aψ Bτ) + 1]; (3.17) Q 0 (ξ,τ) = D C (12Bψ B A )1/2 [tanh( Aψ B 12B )1/2 (ξ 1 3 Aψ Aτ) 1]; (3.18) Therefore, up to O(ε 2 ), the trajectories of the two solitary waves for head-on interactions are ξ = ε(x c 1 t) + ε 2D C (12Bψ B A )1/2 [tanh( Aψ B 12B )1/2 (η + 1 3 Aψ Bτ) + 1] + ;(3.19) η = ε(x + c 2 t) + ε 2D C (12Bψ B A )1/2 [tanh( Aψ B 12B )1/2 (ξ 1 3 Aψ Aτ) 1] + ;(3.20) To obtain the phase shifts after a head-on collision of the two solitons, we assume that the solitons α and β are asymptotically far from each other at the initial time ( t = ) i.e. soliton α is at ξ = 0,η = and soliton β is at η = 0,ξ = +, respectively. After the collision (t = + ), soliton α is far to the right of soliton β, i.e. soliton α is at ξ = 0,η = + and soliton β is atη = 0,ξ =. Using Eqs (3.17) and (3.18) we obtain the corresponding phase shift P 0 and Q 0 as follows P 0 Q 0 = 2ε 2D C (12Bψ B A )1/2 ; (3.21) = 2ε 2D C (12Bψ A A )1/2 ; (3.22) 3.3 Discussion In this paper we have investigated the head-on collision between two IASWs in a threecomponent unmagnetized plasma comprising super thermal electrons, super thermal 79
positrons and cold ions using the extended PLK method. The effects of temperature ratio σ on the phase shift and amplitude ψ A and ψ B are studied where ψ A and ψ B are the amplitudes of the two solitons α and β in their initial positions. Fig 5. Plotted the phase shift Q 0 with p. κ=2 (solid line), κ=8 (dashed line), σ = 0.8. The magnitude of phase shift due to the collision can be calculated from Eqs (3.21) and (3.22). We remarked that each of the solitons has a negative phase shift due to collision. The magnitudes of the phase shift are related to the physical parameters 80
i.e. ε, σ, p, κ, ψ A and ψ B. This dependence is shown in Figs (5-7). Here we consider all physical quantities are dimensionless and ε = 0.1 and ψ A = ψ B = 1. Q 0 stands for Q 0 in fig 5 to fig 7. The phase shifts physically means that the solitons reduce or gain their velocity during the interaction stage. Fig 6. Plotted the phase shift Q 0 with κ. p = 0.3, σ = 0.8(solid line), σ = 1.2 (dashed line),σ = 1.8 (dotted line). 81
In Fig.5 we have plotted the phase shift Q 0 with the concentration of positron component (p) for several values of κ. From Fig 5 it is seen that the magnitude of the phase shift decreases with both p and κ. For a given value of κ, the change is weaker, especially when p 1. The phase shift monotonically decreases as p increases. Fig 7. Plotted the phase shift Q 0 with σ. κ = 2 (solid line), κ = 1000 (dashed line), p =.3. When the graph of positive phase shifts ( Q 0 ) is plotted against κ for several values of σ, it is seen from Fig 6 that initially the phase shift decreases for σ < 1.8 (for σ =.8 82
(solid line), σ = 1.2 (dashed line)) and for σ = 1.8 (dotted line)). For κ 3 they are almost steady. It has also been observed that the phase shifts does not change i.e constant for σ = 1.8. It is a interesting feature that there is no change of phase shift during collision for σ = 1.8. In Fig 7 we have plotted the phase shift Q 0 with the temperature ratio (σ) between electron and positron for several values of κ. It is found that the phase shift is decreasing with the increase of σ. But it is also seen that for higher values κ the rate of decrease is low, κ = 2 (solid line) and κ = 1000(dashed line). Here one can observed that the distribution is non-maxwellian for κ < 2. The phase shift decreases monotonically but significantly for κ (Maxwellian) i.e for higher values of κ the phase shift decreases slowly. 83
Fig 8. Plotted ψ ξ (solid line) and ψ η (dashed line) against ξ and η respectively, for several values of τ, where ε = 0.1, p =.3, κ = 8, and ψ A = ψ B = 1. In Fig.8 we have plotted special solution ψ ξ and ψ η for several values of τ for time evolution of head-on collision. It is seen that solitary wave solution ψ ξ shifted towards right as time increases where as solution ψ η shifted towards left as time increases, which agrees with our analytical results. This results obtained so far indicate that the super thermal electrons and positrons play a significant role in the soliton collision. 84
3.4 Summary In this paper we have consider a normalized ion-fluid model with superthermal electrons, superthermal positrons and cold ions. We derive the KdV equation and finally the phase shifts by using the extended version of PLK method. It has been observed that eqs (3.11) and (3.12) are the two side travelling wave KdV equations in the reference frames ξ and η, respectively. It can be assumed that the solitary waves represented by eqs (3.11) and (3.12) are initially at( t = ) i.e. soliton α is at ξ = 0,η = and soliton β is at η = 0,ξ = +, respectively. After that at time t = 0 they collide and depart from each other. An important changes of phase i.e phase shift occurs and they have been calculated from eqs (3.21) and (3.22). Positive and negative phase shift take place due to the collision. The positive phase shift is plotted against parameters including temperature ratio σ, positrons and electrons density ratio p and finally the the spectral index κ. It has been observed that the positive magnitude of phase shift depends on these parameters. It has been found that the rate of change of phase shift of the solitons becomes slow when the distribution changes its nature from non-maxwellian to Maxwellian. Finally we have investigated the head-on collision between two IASWs in e-p-i plasma by using the extended Poincare-Light hill-kuo method. We have derived the KdV equation and phase shift of head-on collision between two IASWs. 85