Overtaking Collision and Phase Shifts of Dust Acoustic Multi-Solitons in a Four Component Dusty Plasma with Nonthermal Electrons

1 Introduction

The nonlinearities cannot be ignored when the amplitudes of the waves are sufficiently large. The nonlinearities come from the harmonic generation involving fluid advection, the nonlinear Lorentz force, trapping of particles in the wave potential, etc. The nonlinearities in plasmas contribute to the localisation of waves, leading to different types of interesting coherent structures (namely solitary structures, shock waves, vortices, etc.) which are important from both theoretical and experimental points of view. Research on the nonlinear propagation of dust acoustic (DA) waves has been developed rapidly in the last few decades, both in laboratory and space plasmas [1–10]. It is well known that Rao et al. [11] first theoretically worked on the DA waves in an unmagnetised dusty plasmas. Solitary waves are omnipresent in dusty plasma, and due to this reason, investigations of nonlinear solitary waves in dusty plasma have received considerable attention [12–14].

But to study DA solitons and shocks [15–17] most of the researchers have considered the negatively charged dust only. The consideration of only negatively charged dust in a plasma is valid only when dust charging process by collection of plasma particles (viz. electron and ions) is much more important than the other charging processes. There are important charging processes by which the dust grains can be positively charged [3, 18, 19]. Dust grains become positively charged by three mechanisms. These are (i) photo emission in the presence of a flux of ultraviolet photons, (ii) thermionic emission induced by radiative heating, and (iii) a secondary emission of electrons from the surface of the dust grains.

Examples of the existence of plasma with both positively and negatively charged dust particle can be found in several astrophysical plasmas such as the Earth’s mesosphere [20], cometary tails [3, 21], magnetosphere of Jupiter [3, 21, 22]. Chow et al. [23] explained the situations under which smaller dust particles become positively charged and larger particles become negatively charged. Sayed and Mamun [24] investigated solitary waves in four component plasmas where they considered both positively and negatively charged dust particles. Chatterjee and Roy [25] studied the nonthermal electron effect on the four component dusty plasma. Mandal et al. [26] also studied the double layers in four component dusty plasma. El-Taibany and Sabry [27] studied the DASWs and double layers in a magnetised dusty plasma with nonthermal ions and dust charged variation. Roy et al. [28] studied the dressed soliton in four component dusty plasma. Also Roy et al. [29] investigated double layers in a four component dusty plasma with kappa distributed electrons.

Zabusky and Kruskal [30] were first to remark that when solitary waves undergo a collision then they preserve their shape and velocities after the collision. For the collision of solitary waves, some phenomena have been observed in the laboratory [31–33] and the same can be explained in the solution of two solitary waves of Korteweg–de Vries (KdV) equations. In a one- (or quasi-one-) dimensional system, the solitons may interact between them in two different ways. One is the overtaking collision and the other is the head-on collision [34]. Because of the multi-solitons solutions of the KdV equation, where waves travel in the same direction, the overtaking collision of solitary waves can be studied by the inverse scattering transformation method [35]. The collision of solitary waves phase shifts [36] show the effect of the collisions.

The role of non-thermal electron distribution on characterization of solitary waves has been reported, and attempts have been made to explain the observation of solitary wave structures with density depression [37, 38]. The non-thermal distributions associated with the particle flows resulting from the force fields present in the space and astrophysical plasma are of relevance for super-thermal particles. It is known that electron and ion distributions play important roles in the formation of nonlinear structures. Thus, it is significant to study the coherent nonlinear wave structures of electrons or ions, which do not follow the Boltzmann distribution.

In this study, we have considered a four component dusty plasma comprising ions, electrons, and positively and negatively charged dust grains, where ions follow the Boltzmann distribution and electrons are non-thermal. Standard reductive perturbation technique (RPT) [39] has been used to derive the KdV equation, and the solutions of KdV equation are analysed in terms of different plasma parameters. We are particularly interested in the observation of the phase shifts due to the overtaking collision of solitary waves.

This article is organized as follows. The basic governing equations describing the dusty plasma model are stated and KdV equation is derived by using reductive perturbation technique in Section 2. In Section 3, results are shown and discussed in some detail. Finally, Section 4 covers our conclusions.

2 Basic Equations and Derivation of KdV Equation

We consider a four component dusty plasma consisting of Boltzmann distributed ions, non-thermal electrons, and negatively and positively charged dust grains. The basic equations are given by

(1)∂n1∂t + ∂(n1u1)∂x = 0, (1)

(2)∂u1∂t + u1∂u1∂x = ∂ϕ∂x, (2)

(3)∂n2∂t + ∂(n2u2)∂x = 0, (3)

(4)∂u2∂t + u2∂u2∂x = −αβ∂ϕ∂x, (4)

(5)∂2ϕ∂x2 = n1 − (1 − μi + μe)n2 − μie−ϕ+ μe(1 − β1σϕ + β1σ2ϕ2)eσϕ, (5)

where n₁ and n₂ are the number densities of negatively and positively charged dust particles, respectively. At equilibrium, we have n_i0 + Z₂n₂₀=n_e0 + Z₁n₁₀. Equations (1) and (2) are respectively the continuity equation and momentum equation for the negatively charged dust particles, and (3) and (4) are counterparts for the positively charged dust particles. Equation (5) is the Poission equation. Here, u₁ and u₂ are negative and positive dust field speed normalized to C1 = Z1kBTi/m1;ϕ is the electric potential normalised to k_BT_i/e. The space variable x and time variable t are normalised to λD = kBTi/4πZ1e2n10 and ωp1−1 = m1/4πZ12e2n10, respectively. Furthermore, α=Z₂/Z₁, β=m₁/m₂, μ_e=n_e0 + Z₁n₁₀, μ_i=n_i0/Z₁n₁₀, σ=T_i/T_e, and Z₁ and Z₂ are the number of electrons and protons residing on a negative and positive dust particle, respectively. β₁=4γ₁/(1 + 3γ₁), where γ₁ determines the proportion of fast electrons and m₁ and m₂ are masses of the negative and positive dust particles, respectively. T_i and T_e are ion and electron temperatures, respectively, k_B is the Boltzmann constant, and e is the charge of the electrons.

Now, we derive the KdV equation from (1) to (5) employing the RPT. The independent variables are the stretched variables given by Schamel [40, 41] ξ=ϵ^1/2 (x − v_ot), τ=ϵ^3/2t. The dependent variables are expanded as

(6)n1 = 1 + ϵn1(1)+ϵ2n1(2)+ϵ3n1(3)+⋯, (6)

(7)n2 = 1 + ϵn2(1) + ϵ2n2(2) + ϵ3n2(3) + ⋯, (7)

(8)u1 = 0 + ϵu1(1) + ϵ2u1(2) + ϵ3u1(3) + ⋯, (8)

(9)u2 = 0 + ϵu2(1) + ϵ2u2(2) + ϵ3u2(3) + ⋯, (9)

(10)ϕ = 0 + ϵϕ(1) + ϵ2ϕ(2) + ϵ3ϕ(3) + ⋯, (10)

where ϵ is a small non-zero parameter proportional to the amplitude of the perturbation. Now, substituting (6)–(10) into (1)–(5) and considering the lowest order of ϵ, we obtain the dispersion relation as

(11)V02 = 1 + αβ(1 + μe − μi)μi + μeσ − μeβ1σ. (11)

In the next higher order of ϵ, we eliminate the second order perturbed quantities from a set of equations to obtain the required KdV equation

(12)∂ϕ(1)∂τ + Aϕ(1)∂ϕ(1)∂ξ + B∂3ϕ(1)∂ξ3 = 0, (12)

where the nonlinear coefficient A and the dispersion coefficient B are given by the following relations:

(13)A = 3α2β2(1 + μe − μi) − 3 − V04(μeσ2 − μi)2V0[1 + αβ(1 + μe − μi)], (13)

(14)B = V032[1 + αβ(1 + μe − μi)]. (14)

Let us replace ξ by ξ̅B^1/3, ϕ⁽¹⁾ by −6ϕ̅⁽¹⁾A⁻¹B^1/3, and τ by τ̅, then (12) is transformed to the following standard KdV equation:

(15)∂ϕ¯(1)∂τ¯ − 6ϕ¯(1)∂ϕ¯(1)∂ξ¯ + ∂3ϕ¯(1)∂ξ¯3 = 0. (15)

3 Results and Discussion

Now our aim is to obtain the two-solitons and three-solitons solutions of (15) and to study the interaction between them. To do so, we employ Hirota’s bilinear method [42]. Though Hirota’s method is well known, for the sake of completness we give here a gist of the method for KdV equation. Using the transformation ϕ̅⁽¹⁾=−2(log f)ξ̅ξ̅ in the standard KdV (15), one can obtain the bilinearized form of (15) as

(16)ffξ¯ τ¯ − fξ¯fτ¯ + ffξ¯ ξ¯ ξ¯ ξ¯ − 4fξ¯fξ¯ ξ¯ ξ¯ + 3fξ¯ ξ¯2 = 0. (16)

By using the Hirota-D operator [43], we get

(17)Dξ¯Dτ¯{f ⋅ f} = 2(ffξ¯τ¯ − fξ¯fτ¯), (17)

(18)Dξ¯4{f ⋅ f} = 2(ffξ¯ ξ¯ ξ¯ ξ¯ − 4fξ¯fξ¯ ξ¯ ξ¯ + 3fξ¯ ξ¯2) (18)

Using (17) and (18) in (16), we get the Hirota bilinear form (for details see [42, 44])

(19)(Dξ¯Dτ¯ + Dξ¯4){f ⋅ f} = 0. (19)

To construct two-solitons solution, we use the Hirota’s perturbation technique and we insert f = 1 + ϵ(eθ¯1 + eθ¯2) + ϵ2f2 into (19), where θ̅_i=k_iξ̅ + ω_iτ̅ + α_i, i=1, 2. The coefficient of different powers of ϵ will give ωi = −ki3, f2 = a12eθ¯1 + θ¯2 with a₁₂=(k₁ − k₂)²/(k₁ + k₂)²; a₁₂ determines the phase shifts of the respective solitons after overtaking takes place.

Finally considering ϵ=1, we have the two-soliton solution of the KdV (15) as

(20)ϕ¯(1) = −2k12eθ¯1 + k22eθ¯2 + a12eθ¯1 + θ¯2(k22eθ¯1 + k12eθ¯2) + 2(k1 − k2)2eθ¯1 + θ¯2(1 + eθ¯1 + eθ¯2 + a12eθ¯1 + θ¯2)2 (20)

Hence, the two-soliton solution of the KdV equation (12) is given by

(21)ϕ(1) = 12B1/3Ak12eθ1 + k22eθ2 + a12eθ1 + θ2(k22eθ1 + k12eθ2) + 2(k1 − k2)2eθ1 + θ2(1 + eθ1 + eθ2 + a12eθ1 + θ2)2, (21)

with θi = kiB1/3ξ − ki3τ + αi.

When τ>>1, e−(θ1 + θ2), e−(2θ1 + θ2), e−(θ1+2θ2) are non-dominant terms. Neglecting the non-dominant terms and after some rearrangement, we get

(22)ϕ(1) = 12B1/3A[a12k12e−θ1(e−θ1 + a12)2 + a12k22e−θ2(e−θ2 + a12)2] (22)

Using the result e^−x/(1 + e^−x)²=sech²(x/2)/4 and writing a₁₂=e^{ln∣a₁₂∣}, we get the asymptotic solution of (12)

(23)ϕ(1) ≈ −3B1/3A[k122sech2{k12B1/3(ξ − B1/3k12τ − Δ1)}+ k222sech2{k22B1/3(ξ − B1/3k22τ − Δ2)}], (23)

where Δi = ±2B1/3kiln|a12|, (i = 1, 2).

It is to be noted that the phase shifts Δ₁ and Δ₂ are of the same sign, and both of them are proportional to B^1/3 and the amplitude of the the solitons, a result consistent with those obtained in the study of head-on collision [45, 46].

Similarly the three-soliton solution of (12) has the form

(24)ϕ(1) = −2∂2∂ξ2(ln[g(ξ, τ)]), (24)

where g(ξ, τ) = 1 + eθ1 + eθ2 + eθ3 + α122eθ1 + θ2 + α232eθ2 + θ3 + α312eθ3 + θ1 + α2eθ1 + θ2 + θ3α122 = (k1 − k2k1 + k2)2,α232 = (k2 − k3k2 + k3)2,α312 = (k3 − k1k3 + k1)2,α2 = α122α232α312,θi = kiB1/3ξ − ki3τ + αi, i = 1, 2, 3.

For τ>> 1 this solution is asymptotically transformed into a superposition of three single-soliton solutions as

(25)ϕ(1) ~ −∑i = 13Aisech2[ki2B1/3(ξ − ki2B1/3τ + Δ′i)], (25)

where Ai=3B1/3ki22A, (i = 1, 2, 3) are the amplitudes, Δ′1 = ±2B1/3k1log|αα23|,Δ′2 = ±2B1/3k2log|αα31|, and Δ′3 = ±2B1/3k3log|αα12| are the phase shifts of the solitons. It is to be noted that, unlike in the two-soliton case, the phase shifts for three-soliton collision are different from each other.

Per soliton theory, the shape and velocities of the solitons do not change after collision. It means that their shapes in the remote past are same as those in future. But in finite time they may collide and merge together and eventually form a single soliton at τ=0, say, and after τ >> 1 they regain their original shapes. This behaviour is clearly shown in this article.

In Figures 1 and 2, time evaluation of the interaction of compressive two-solitons ϕ⁽¹⁾ vs. ξ are plotted for the several values of τ. In Figure 1a, we show that for τ=−5 the larger amplitude soliton is behind the smaller amplitude soliton. Then, in Figure 1b at τ=−1, the two solitons merge and become one soliton at τ=0, shown in Figure 1c. But at τ=1, they separate from each other, which is shown in Figure 1d, and then finally they depart from each other when τ=5, shown in Figure 1e. The combined profile of the two soliton is shown in Figure 2. It can be clearly seen from the exact two-soliton solution and asymptotical solution that the amplitude of the merge soliton is greater than the amplitude of the shorter soliton but less than the amplitude of the taller soliton (e.g. see the solution given by Drazin et al. [44, p. 76]), which are obtained using the inverse scattering method).

Figure 1:

Variation of compressive two-solitons profiles for different values of τ with k₁=1, k₂=2, α=1, β=4, β₁=0.1, μ_e=0.3, μ_i=0.7, σ=0.15, α₁=1, α₂=1.

Figure 2:

Variation of the combined compressive two-solitons profile for different values of τ with same values of the other parameters are the same as Figure 1.

In Figures 3–6, the effect of β₁ on compressive two solitons ϕ⁽¹⁾ vs. ξ is shown keeping other parameters fixed. In Figure 3, we show the profile of two solitons for β₁=0.3 with τ=−5, k₁=1, k₂=2, α=1, β=4, μ_e=0.3, μ_i=0.7, σ=0.15, α₁=1, α₂=1. Then, in Figure 4, we show the profile of two solitons for β₁=0.6 with other parameters the same as Figure 1. In Figure 5, we show the profile of two soliton for β₁=0.9 with other parameters the same as Figure 1. A variation of the compressive two-solitons profiles for different values of β₁ is shown in Figure 6.

Figure 3:

Compressive two-solitons profile for β₁=0.3 and other parameters the same as Figure 1.

Figure 4:

Compressive two-solitons profile for β₁=0.6 and other parameters the same as Figure 1.

Figure 5:

Compressive two-solitons profile for β₁=0.9 and other parameters the same as Figure 1.

Figure 6:

Variation of the compressive two-solitons profiles for different values of β₁ with same values of the other parameters the same as Figure 1.

Figure 7 shows time evaluation of the interaction of compressive three-solitons solution ϕ⁽¹⁾ vs. ξ for different values of τ. At τ=−10 the larger amplitude soliton is behind the smaller amplitude solitary wave. Then, two solitons merge and become one soliton at τ=0. But at τ=10 they separate from each other and then finally each appears as a separate soliton acquiring their original speed and shape. In Figure 8, we show the variation of the combined compressive three-soliton profiles for different values of τ.

Figure 7:

Variation of the compressive three-solitons profiles for different values of τ, with k₁=1, k₂=2, k₃=3, α=1, β=4, β₁=0.1, μ_e=0.3, μ_i=0.7, σ=0.15, α₁=1, α₂=1.

Figure 8:

Variation of the combined compressive three-solitons profiles for different values of τ with same values of the other parameters the same as Figure 7.

In Figures 9–12, the effect of β₁ on compressive three solitons ϕ⁽¹⁾ vs. ξ has been plotted with fixed values of the other parameters. In Figure 9, we show the profile of three soliton for β₁=0.3 with τ=−5, k₁=1, k₂=2, k₃=3, α=1, β=4, μ_e=0.3, μ_i=0.7, σ=0.15, α₁=1, α₂=1. Then, in Figure 10, we show the profile of three soliton for β₁=0.6 with other parameters as in Figure 9. In Figure 11, we show the profile of three soliton for β₁=0.9 with other parameters as in Figure 9. A variation of the compressive three-soliton profiles for different values of β₁ is shown in Figure 12.

Figure 9:

Compressive three-solitons profile for β₁=0.3, and other parameters the same as Figure 7.

Figure 10:

Compressive three-solitons profile for β₁=0.6, and other parameters the same as Figure 7.

Figure 11:

Compressive three-solitons profile for β₁=0.9 and other parameters the same as Figure 7.

Figure 12:

Variation of the compressive three-solitons profiles for different values of β₁, with same values of the other parameters the same as Figure 7.

In Figure 13, we present the variation of phase shift for two solitons against β₁ with k₁=1, k₂=2, α=1, β=4, μ_e=0.5, μ_i=0.5, σ=0.15. The phase shift increases with the increase of β₁. We plotted the variation of phase shift with μ_i for β₁=0.3 (solid line), 0.6 (dotted line), and 0.9 (dashed line) in Figure 14. The phase shift is monotonically decreasing for each β₁ with the increase of the parameter μ_i.

Figure 13:

Graph of the phase shift against β₁. Other parameters are the same as those in Figure 1.

Figure 14:

Graphs of the phase shift of the two-solitons against μ_i for different values of β₁.

Figure 15 shows the variation of the phase shift for respective solitons against β₁ when the values of the other parameters are kept fixed. As before, the phase shift increases with increased β₁, as the value of B increases with increased β₁.

Figure 15:

Graphs of the phase shift of three-solitons against β₁. Other parameters are the same as those in Figure 7.

4 Conclusions

In this work, we presented the nature of the nonlinear propagation and interaction of dust acoustic two solitons and three solitons in a four component dusty plasma consisting of negatively and positively charged cold dust fluids, non-thermal electrons, and Boltzmann distributed ions. The KdV equation is derived by using RPT, and it is transformed to the standard KdV equation with the help of suitable transformation. Using the Hirota direct method, we obtained two-soliton and three-soliton solutions to the KdV equation. Propagations of two solitons and three solitons have been discussed in detail. It has been observed that the larger soliton moves faster, approaches the smaller one, and, after the overtaking collision, both resume their original shapes and speeds. However, it should be noted that the KdV equation describes multi-soliton solutions depending on the initial conditions. Hirota’s method is an innovative, powerful method by which we can obtain, in principle, any number of solutions for many nonlinear partial differential equations. With the two-soliton solution, it was found that k_iΔ_i, (i=1, 2) are the same values. However, in the three-soliton case, phase shift of any soliton is different from the others. Our present study may be helpful in understanding the nonlinear features of the two-soliton and three-soliton solutions in Earth’s mesosphere [20], cometary tails [1, 47], and Jupiter’s magnetosphere [3, 22], where non-thermal electrons and Boltzmann distributed ions are present.