Face-to-face interaction of multisolitons in spin-1/2 quantum plasma

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1 Pramana J. Phys. (2017) 88: 18 DOI /s c Indian Academy of Sciences Face-to-face interaction of multisolitons in spin-1/2 quantum plasma KAUSHIK ROY 1,, SOURAV CHOUDHURY 2, PRASANTA CHATTERJEE 2 and C S WONG 3 1 Beluti M.K.M. High School, P.O.-Beluti, Dist-Birbhum, , India 2 Department of Mathematics, Siksha Bhavana, Visva Bharati, Santiniketan , India 3 Plasma Technology Research Centre, Department of Physics, University of Malaya, 50603, Kuala Lumpur, Malaysia Corresponding author. kaushikbolpur@rediffmail.com MS received 7 May 2015; revised 26 March 2016; accepted 6 May 2016; published online 13 December 2016 Abstract. We investigate the face-to-face collision between multisolitons in spin-1/2 quantum plasma. It is studied in the framework of the model proposed by Marklund et al in Phys. Rev. E76, (2007). This study is done with the help of the extended Poincare Lighthill Kno (PLK) method. The extended PLK method is also used to obtain two Korteweg de Vries (KdV) equations and the phase shifts and trajectories during the head-on collision of multisolitons. The collision-induced phase shifts (trajectory changes) are also obtained. The effects of the Zeeman energy, total mass density of the charged plasma particles, speed of the wave and the ratio of the sound speed to Alfvén speed on the phase shifts are studied. It is observed that the phase shifts are significantly affected by all these parameters. Keywords. PACS Nos Quantum plasma; multisolitons; phase shift; Hirota bilinear method; spin-1/ h; g; Sb; Yv 1. Introduction There is currently a great deal of interest in investigating plasma modes in quantum plasmas [1 11], as such plasmas could be relevant in nanoscale electromechanical systems [12], in microplasmas and dense laser plasmas [13], and in laser interactions with atomic systems [14]. When de Broglie wavelength of charge carriers becomes comparable to the system scales (such as interparticle distances), the quantum effects should be taken into account. In quantum plasma, Fermi Dirac distribution is used to describe the system rather than Maxwell Boltzmann distribution. Now it is well known that in dense plasmas, degenerate electrons follow the Fermi Dirac pressure law and the quantum force is connected with the Bohm de Broglie potential. In classical plasmas we neglect spin effect due to random thermal orientations of the spin vector. The very important next step of including spin in a quantum plasma was taken recently. Spin plasma is developed from the Pauli Hamiltonian and generalizing the Madelung decomposition [15,16] for the two-component spinor wave function. This nonrelativistic spin theory produces new effects in, for example, propagation modes [17], magnetosonic solitons [18], instabilities and metamaterials [19]. All these quantum effects are expected to be important in low temperature and very high dense plasmas. However, it was recently shown that even for high-temperature plasmas, under specific conditions, quantum features such as spin cannot be neglected, and they can be significant [20]. Spin magnetohydrodynamics was proposed by Brodin and Marklund [21] from the nonrelativistic viewpoint. Brodin and Marklund showed that spin effects are important for low temperature, high densities and/or strongly magnetized plasmas. The latter can be found in astrophysical systems such as pulsars or magnetostars [22]. They also studied the spin dynamics through the spin evolution equation, kinetic effects associated with the spin. Marklund and Brodin [15] showed the existence of magnetosolitons in a fermionic quantum plasma. Marklund et al [18] derived the multifluid equation for spin-1/2 quantum plasmas starting from Pauli equation. Li and Han [23] investigated the head-on collisions of two magnetosonic solitary waves in a magnetized degenerate quantum plasma 1

2 18 Page 2 of 8 Pramana J. Phys. (2017) 88: 18 with effects of quantum diffraction, quantum statistics and electron spin magnetization. Asenjo et al [24] developed a new fluid formalism for relativistic quantum spin-1/2 plasmas. Chatterjee et al [25] studied the head-on collision phenomenon of the two magnetoacoustic solitons in a fermionic quantum plasma where spin effect had been studied using the extended variations of the PLK method. Fine et al [26] found that stationary nonintegrable spin-1/2 systems do not exhibit exponential sensitivity to small perturbation of Loschmidt echoes while chaotic systems of classical spins do. Sahuet al [27] studied the nonlinear propagation of arbitrary-amplitude magnetosonic solitary and shock structures in spin-1/2 quantum plasma. A parallel development in solitonic studies was the head-on collision effects. It is known that one of the interesting properties of solitons is their asymptotic preservation of form when they undergo a head-on collision (i.e. θ = π, θ being the angle between two propagation directions of two solitons). This fact was first observed by Zabusky and Kruskal [28]. Recently, extensive investigations have been made by several researchers [29 34] to study the face-to-face interaction between two ion-acoustic solitary waves (IASWs) with the help of the extended Poincare Lighthill Kuo (PLK) method in different plasma models. Head-on collision of dust-acoustic solitary waves in a fourcomponent plasma had been studied by Ghosh et al [29]. In all these investigations, the researchers observed that bidirectional solitary waves are propagated and hence head-on collision occurs. They also found the phase shifts and the trajectories of the two solitary waves after collision, which are the characteristics of the collision. It is noteworthy that a few investigations had already been done about the headon collision phenomena in quantum plasma ([35 37] and [33]). For the first time, El-Labany et al [35] found out the head-on collision between two quantum IASWs in a dense e p i plasma and have discussed the effects of both quantum diffraction parameter and the Fermi temperature ratio of positrons-to-electrons on the phase shifts. The head-on collision between two IASWs with arbitrary colliding angle in an unmagnetized ultracold quantum three-component e p i plasma had been investigated by Xu et al [37]. The propagation and interaction of IASWs in quantum e p i plasma had been investigated by Ning et al [36]. Head-on collision of dust ion-acoustic soliton in quantum pair ion plasma has also been studied by Chatterjee and Ghosh [33]. However, all these researchers investigated only head-on collison between two solitons arising out of two different KdV equations. But two-soliton solution has not been investigated much in plasma till recently. Roy et al [38] studied the overtaking collision of two ion-acoustic solitons in a plasma. Roy et al [39] also studied the head-on collision of multisolitons in e p i plasma. But the head-on collision of multisolitons in quantum plasma by taking into account spin effects, has not been studied so far. The present study has been undertaken on the head-on collision of multisolitons in the framework of the model proposed by Marklund et al [18]. We expect that the results obtained in this study can be useful in investigating strongly magnetized astrophysical plasmas like pulsars, magnetostars etc. 2. Basic equation and derivation of KdV equations Let us consider the governing equations for quantum plasma in which the electron spin-1/2 effects are included. The total mass density, the centreof-mass fluid flow velocity and the current density are defined respectively as ρ = (m e n e + m i n i ), V = (m e n e v e + m i n i v i )/ρ, j = ( en e v e + en i v i ). Here m e, n e and v e are the mass, number density and fluid velocity of the electron, and m i, n i and v i are respectively the mass, number density and fluid velocity of the ion, and e is the magnitude of the electron charge. Assuming the quasineutrality condition (n e = n i ), and taking the magnetic field along the z-axis so that B = B(x,t) z and taking the velocity V = V(x,t) x, the density as ρ(x,t), weget the following system of normalized basic equations [1,40]: b t + (bv) = 0, (1) x v t + v v x = b x (ln b) c2 s x ( + 2ω2 pe 1 2 ) b ω c ω ce x b x 2 + vb 2 x [ln(cosh(z eb)) + z e b(tanh(z e b))], (2) where ρ = ρ 0 b with b = B/B 0, c s = C s /C Al, v = V/C Al, C Al = B0 2/(μ 0ρ 0 ) is the Alfvén speed, C s = K B (T e + T i )/m i is the sound speed, ω pj = (n j0 e 2 )/(ɛ 0 m i ); j = e, i, are respectively the plasma frequencies for electron and ion, ω c (=2m e c 2 /h) is the Compton frequency, z e (=μb 0 /K B T e ) is the

3 Pramana J. Phys. (2017) 88: 18 Page 3 of 8 18 temperature-normalized Zeeman energy, where μ 0 is the permeability of the vacuum, B 0 is the strength of the magnetic field, ρ 0 is the total mass density of the charged plasma particles, T i and T e are ion and electron temperatures, K B is the Boltzmann constant and μ is the magnitude of Bohr magneton. Moreover, it is to be noted that the normalized variables are taken as t ω ci t, x ω ci x/c Al. In deriving eqs (1) and (2) the magnetic resistivity is neglected. Now we assume that the two solitons α and β in the plasma, which were asymptotically far apart in the initial state, travel towards each other. After some time they interact, collide, and then depart. We also assume that the solitons have small amplitudes proportional to ɛ (the strength of nonlinearity) and the interaction between two solitons is weak. Hence, we expect that the collision will be quasielastic. So it will only cause shifts of the post-collision trajectories (phase shift). In order to analyse the effects of collision, we employ an extended PLK method. According to this method, the dependent variables are expanded as b = 1 + ɛb 1 + ɛ 3/2 b 2 + ɛ 2 b 3 +, (3) v = ɛv 1 + ɛ 3/2 v 2 + ɛ 2 v 3 +. (4) The independent variables are given by ξ = ɛ 1/2 (x v p t) + ɛ 3/2 P 0 (η, τ) + ɛ 5/2 P 1 (ξ,η,τ)+, (5) η = ɛ 1/2 (x + v p t) + ɛ 3/2 Q 0 (ξ, τ) + ɛ 5/2 Q 1 (ξ,η,τ)+, (6) τ = ɛ 3 t, (7) where ξ and η denote the trajectories of two solitons travelling towards each other respectively and v p is the unknown phase speed normalized to C Al. The reason for taking the forms of stretched coordinates taken in eqs (5) (7) is as follows: If we neglect the higher-order terms in these equations, then the stretched coordinates are consistent with the dispersion relation for ion-acoustic waves and when reduction perturbation is used, one gets KdV equation. Moreover, the main idea behind the extended PLK perturbation method is that it will lead to uniformly valid asymptotic expansion by eliminating secular terms. One can then easily obtain the phase shifts of the solitary waves after collision. Using extended PLK method and after some long but standard calculation [25], we get the KdV equation b 11 τ + Ab b ξ + B 3 b 11 ξ 3 = 0, (8) b 12 τ Ab b η B 3 b 12 η 3 = 0, (9) where A = 3v2 p c2 s z2 e v2 sech 2 z e (3 2 z e tanh z e ), 2v p ω 2 pe B =, 2 ω ce v p ω c v p = 1 + c 2 s z ev 2 (2tanhz e + z e sech 2 z e ). Equations (8) and (9) are two side-travelling wave KdV equations in the reference frames of ξ and η, respectively. Their one-soliton solutions are, respectively, [ (AbA ) 1/2 ( b 11 = b A sech 2 ξ 1 Aτ) ] 12B 3 Ab (10) b 12 = b B sech 2 [ (AbB 12B ) 1/2 ( η Ab Bτ) ] (11) where b A and b B are the amplitudes of the two solitons in their initial positions. To obtain the phase shifts after a head-on collision of the two solitons we assume that soliton α is at ξ = 0,η = and soliton β is at η = 0,ξ =+. After the collision (t =+ ), soliton α is at ξ = 0,η =+ and soliton β is at η = 0,ξ =. We obtain the corresponding phase shift P 0 = 2ɛ 2 D ( ) 1/2 12BbB, (12) C A Q 0 = 2ɛ 2 D C where C = 2v p, ( ) 1/2 12BbA, (13) A D = v2 p + c2 s + z2 e v2 sech 2 z e (3 2z e tanh z e ). 2v p The two solitons given by eqs (10) and (11) move in opposite directions. But each of the KdV equation given by (8) and (9) has a number of soliton solutions. 2.1 Two-soliton solution We consider here two-soliton solutions of each of the KdV equation. The two-solitons for a fixed KdV equation move in the same directions, the fast moving soliton eventually overtakes the slower one. The

4 18 Page 4 of 8 Pramana J. Phys. (2017) 88: 18 two-soliton solutions of (8) and (9) are obtained using Hirota s method [41] and are given by b 11 = 12B A b 12 = 12B A k 2 1 eθ 1 + k 2 2 eθ 2 + a 12 e θ 1+θ 2 (k 2 2 eθ 1 + k 2 1 eθ 2) + 2(k 1 k 2 ) 2 e θ 1+θ 2 (1 + e θ 1 + e θ 2 + a 12 e θ 1+θ 2 ) 2 (14) k 2 1 eφ 1 + k 2 2 eφ 2 + a 12 e φ 1+φ 2 (k 2 2 eφ 1 + k 2 1 eφ 2) + 2(k 1 k 2 ) 2 e φ 1+φ 2 (1 + e φ 1 + e φ 2 + a 12 e φ 1+φ 2 ) 2 (15) with θ i = k i B 1/3 ξ k3 i τ + α i, φ i = k i B 1/3 η k3 i τ + α i, i = 1, 2, a 12 = (k 1 k 2 ) 2 /(k 1 + k 2 ) 2. When τ 1 the solutions of eqs (8) and (9) respectively transform into a superposition of two single soliton solutions [ b 11 6B k 2 { } 1 k1 A 2 sech2 2B 1/3 (ξ B1/3 k1 2 τ 1 ) { } ] + k2 2 k2 2 sech2 2B 1/3 (ξ B1/3 k2 2 τ 2 ) (16) [ b 12 6B k 2 { } 1 k1 A 2 sech2 2B 1/3( η B1/3 k1 2 τ 1) { } ] + k2 2 k2 2 sech2 2B 1/3( η B1/3 k2 2 τ 2). (17) We see from eqs (16) and (17) that now we are dealing with four solitons, two in each pair moving in the same direction, where i, i =± 2B1/3 k i ln a 12,i= 1, Three-soliton solution The three soliton of (8) and (9) have the form b 11 = 2 2 (ln[g(ξ, τ)]), (18) ξ2 b 12 = 2 2 ξ 2 (ln[g (η, τ)]), (19) where g(ξ, τ) = 1 + e θ 1 + e θ 2 + e θ 3 + α 2 12 eθ 1+θ 2 + α 2 23 eθ 2+θ 3 + α 2 31 eθ 3+θ 1 + α eθ 1+θ 2 +θ 3, g (η, τ) = 1 + e φ 1 + e φ 2 + e φ 3 + α 2 12 eφ 1+φ 2 + α 2 23 eφ 2+φ 3 + α 2 31 eφ 3+φ 1 + α eφ 1+φ 2 +φ 3, ( α12 2 = k1 k 2 k 1 + k 2 ( α31 2 = k3 k 1 k 3 + k 1 θ i = k i B 1/3 ξ k3 i τ + α i, ) 2 (, α23 2 = k2 k 3 k 2 + k 3 ) 2, ) 2, α = α2 12 α2 23 α2 31, φ i = k i B 1/3 η k3 i τ + α i, i = 1, 2, 3. For τ 1 the solution of eqs (8) and (9) respectively transform into a superposition of three single-soliton as 3 [ ] b 11 A i sech 2 ki 2B 1/3 (ξ k2 i B1/3 τ + δ i ), (20) i=1 3 [ ] b 12 A i sech 2 ki 2B 1/3( η k2 i B1/3 τ + δ i ), (21) i=1 where A i = 3Bki 2 /A, i = 1, 2, 3 are the amplitudes, δ 1 = δ 1 =± 2B1/3 log α 123 k 1 α, 23 δ 2 = δ 2 =± 2B1/3 log α 123 k 2 α, 31 δ 3 = δ 3 =± 2B1/3 log α 123 k 1 α 12 are the phase shifts of the solitons. The scattering of four solitons will be clear from figures 1a 1i. In figure 1a we see the positions of the solitons. When τ = 10, the two solitons seen on the left-hand side move to the right side and the two solitons on the right-hand side move towards the left side and as τ approaches 0, the fast solitons on each side overtake their slower partners. The overtaking can be seen from figures 1c and 1d. Figure 1e shows the merger of four solitons. It is to be noted that figures 1f and 1i are mirror images of figures 1d and 1a respectively as to be expected. Here, each soliton acquires to phase shifts, one due to head-on collision

5 Pramana J. Phys. (2017) 88: 18 Page 5 of 8 18 (a) (f) (b) (g) (c) (h) (d) (i) (e) Figure 1. Variation of the two-soliton profiles b 11 and b 12 for different values of τ with k 1 = 1, k 2 = 2.

6 18 Page 6 of 8 Pramana J. Phys. (2017) 88: 18 (a) (f) (b) (g) (c) (h) (d) (i) (e) Figure 2. Variation of the three-soliton profiles b 11 and b 12 for different values of τ with k 1 = 1, k 2 = 2,k 3 = 3.

7 Pramana J. Phys. (2017) 88: 18 Page 7 of 8 18 essentially the collisions of multisolitons (four, six) using a two-step method where we first derived two different KdV equations using PLK method and then extracted multisolution for each KdV equation using Hirota s approach. Each soliton acquires two phase shifts, one due to head-on collision, other due to overtaking collision. The effect of the parameters involved in the nonlinear and dispersion coefficient of the KdV equations on the amplitude of multisolitons had been investigated, using PLK perturbation method. Figure 3. Graph of total phase shift ( = 1 +Q 0 )against C s. C Acknowledgements The authors are grateful to the referee and former Prof. Rajkumar Roychoudhury, ISI, Kolkata, India for helpful suggestions. References Figure 4. Graph of total phase shift ( = 1 +Q 0 )against z e. and the other due to overtaking collision, as predicted by Hirota s and experience both head-on collision and overtaking collision. Similarly, the scattering of six solitons will be clear from figures 2a 2i. In figure 2a, we see the positions of the solitons. When τ = 10, the three solitons seen on the left-hand side move to the right side and the three solitons on the right-hand side move towards the left side and as τ approaches 0, the fast solitons on each side overtake their slower partners. The overtaking can be seen from figures 2c and 2d. Figure 2e shows the merger of six solitons. It is to be noted that figures 2f and 2i are mirror images of figure 2d and 2a respectively as to be expected. Figure 3 plots the phase shifts = 1 + Q 0 = against C s for different values of z e. Figure 4 shows the behaviour of 1 + Q 0 = against z e. 3. Conclusion We investigated face-to-face interaction of multisolitons in spin-1/2 quantum plasma. We have considered [1] K Roy, A P Mishra and P Chatterjee, Phys. Plasmas 15, (2008) [2] P Chatterjee, K Roy, S V Muniandy and C S Wong, Phys. Plasmas 16, (2009) [3] P Chatterjee, K Roy, Ganesh Mondal, S V Muniandy, S L Yap and C S Wong, Phys. Plasmas 16, (2009) [4] K Roy and P Chatterjee, Indian J. Phys. 85, 1653 (2011) [5] F Haas, G Manfredi and M R Feix, Phys. Rev. E62, 2763 (2000) [6] F Haas, L G Garcia, J Goedert and G Manfredi, Phys. Plasmas 10, 3858 (2003) [7] P Chatterjee, K Roy, S V Muniandy, S L Yap and C S Wong, Phys. Plasmas 16, (2009) [8] L G Garcia, F Haas, L P L de Oliviera and J Goedert, Phys. Plasmas 12, (2005) [9] M Marklund, Phys. Plasmas 12, (2005) [10] M Marklund and P K Shukla, Rev. Mod. Phys. 78, 591 (2006) [11] P K Shukla and B Eliasson, Phys. Rev. Lett. 96, (2006) [12] P A Markowich, C A Ringhofer and C Schmeiser, Semiconductor equations (Springer, Vienna, 1990) [13] K H Becker, K H Schoenbach and J G Eden, J. Phys. D 39, R55 (2006) [14] Y I Salamin, S X Hu, K Z Hatsagortsyan and C H Keitel, Phys. Rep. 427, 41 (2006) [15] M Marklund and G Brodin, Phys. Rev. Lett. 98, (2007) [16] G Brodin and M Marklun, New J. Phys. 9, 277 (2007) [17] A P Misra, Phys. Plasmas 14, (2007) [18] M Marklund, B Eliason and P K Shukla, Phys. Rev. E76, (2007) [19] G Brodin and M Marklund, New J. Phys. 10, (2008) [20] G Brodin, M Marklund and G Manfredi, Phys. Rev. Lett. 100, (2008) [21] G Brodin and M Marklund, Phys. Rev. Lett. 98, (2007) [22] M G Baring and A K Harding, Astrophys. J. 547, 929 (2001) [23] S Li and J Han, Phys. Plasmas 21, (2014)

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