Three-Soliton Interaction and Soliton Turbulence in Superthermal Dusty Plasmas

1 Introduction

During last few decades, the study of linear and nonlinear wave propagation in dusty plasma has attracted the attention of many researchers because of its applicability in different plasma devices, laboratory experiments, and space plasmas [1], [2], [3], [4], [5], [6], [7]. In addition to the usual electrons, ions, and neutral particles, a dusty plasma contains charged and massive dust grains. Because of the presence of the large number of charged particles, a dusty plasma supports different kinds of eigen modes such as the dust acoustic mode [8], dust drift mode [9], Shukla-Varma mode [10], dust lattice mode [11], dust cyclotron mode [12], dust ion acoustic (DIA) mode [13], and dust Berstain-Green-Kruskal mode [14]. Recently, researchers have shown great interest in the study of linear and nonlinear DIA wave propagation [15], [16], [17], [18], [19], [20], [21]. The existence of low-frequency DIA waves in a dusty plasma was theoretically predicted in [15] and later confirmed by Barkan et al. [22] experimentally. Recently, Pakzad et al. [23] showed that the DIA wave pattern is affected by the non-extensive parameter and also by the relative density of the plasma constituents. Very recently, Chatterjee et al. [24] studied the DIA wave in the framework of the damped forced Korteweg-de Vries (KdV) equation. They showed that the structure of the DIA solitary wave is heavily affected by the strength and frequency of the external periodic force.

It is known that the plasmas in space or in the laboratory may contain a large number of high-energy particles, which can be modeled by a Lorentzian distribution or a kappa distribution [25]. The presence of a substantially large number of superthermal particles can significantly change the rate of resonant energy transfer between the particles and plasma waves [26], [27], [28]. A three-dimensional generalised Lorentzian or kappa distribution function [29] can be written as follows:

where θ=((κ−3/2κ)(2kBTem))1/2 is the effective thermal speed, modified by spectral index κ, with T_e as the characteristic kinetic temperature and k_B is the Boltzmann constant, and Γ is the gamma function arising from the normalisation of fκ(v) such that ∫fκ(v)d3v=1. It is clear that for the physically realistic thermal speed, one requires κ>3/2. Low values of κ represent a hard spectrum with a strong non-Maxwellian (power law-like) tail having a power-law form at high speeds, while in the limit κ → ∞, the kappa distribution function reduces to the well-known Maxwell-Boltzmann distribution [27]. By integrating the kappa distribution over the velocity space, one can obtain the normalised form of hot electron number density as [30]

In weakly dispersive media, the soliton is an essential part of the nonlinear wave field, and their deterministic dynamics in the framework of the KdV equation is well studied [31], [32], [33]. Several authors have also studied the soliton and multi-soliton solutions of the KdV and KdV-like equations and their mutual interactions [34], [35], [36], [37], [38]. When a lagre number of interacting waves propagate in different directions with different velocities, their interactions lead to a fast change in the wave pattern and the characteristic of the wave field is described within the framework of statistical theory. Such a theory is termed ‘the theory of wave turbulence’ [39], [40]. Zakharov [41], [42] has discussed the chracterstics of wave turbulence in an integrable system. Soliton turbulence is specified by the kinetic equations describing the parameters of the associated scattering problem and is a specific part of wave turbulence. In 1971, Zakharov [41] for the first time described the fundamental role of pair-wise soliton collisions within the framework of the KdV equation, which was later confirmed [43], [44], [45]. Soliton turbulence in an integrable system is slightly degenerate because of the conservation of solitons in the interaction process. In turbulence theory, it is very important to know the wave field distribution and the moments (mean, variance, skewness, and kurtosis) of the random wave field, which are obtained from measurements [46], [47], [48], [49]. Pelinovsky et al. [50], [51] studied the two-soliton interaction as an elementary act of soliton turbulence in the framework of KdV and modified KdV equations. They showed that the nonlinear interaction of two solitons leads to a decrease of third- and fourth-order moments of the wave field, whereas the first and second moments remain constant. Recently, some works have been reported on soliton turbulence [52], [53], [54], [55], [56]. Very recently, Shurgalina [57], [58] studied different features of soliton turbulence in the framework of the Gardner equation with negative and positive cubic nonlinearity.

In this article, our goal is to study the properties of skewness and kurtosis of the random wave field due to three-soliton interaction and the effiect of the different plasma parameters on the skewness and kurtosis of the wave field at the time of soliton interaction. The rest of the article is organised as follows. The model equations are presented in Section 2. In Section 3, the derivation of the KdV equation and the three-soliton solution are given. Sections 4 and 5 present the effect of soliton interaction on the statistical characterstic of the wave field and the effect of different plasma parameters on soliton turbulence, respectively. Section 6 presents the conclusions.

2 Model Equations

In this work, an unmagnetised dusty plasma with cold inertial ions, stationary dust with negative charge, and inertia-less, κ-distibuted electrons is considered. The normalised basic equations governing the DIA waves are given by

where n is the ion number density normalised to n₀; n_e is the number density of the electrons; and u is the ion velocity normalised to the ion fluid speed Cs=kBTe/mi, with m_i as the mass of ions, T_e as the temperature of electrons, k_B as the Boltzman constant. The electrostatic wave potential ϕ is normalised to kBTe/e. The space and time variables are normalised to the electron Debye radius λD=kBTe/4πne0e2 and the inverse of the cold electron plasma frequency ωpi−1=mi/4πne0e2, respectively. Here, μ=Zdnd0n0, with n_{d ⁢ 0} being the number density of dust particles and Z_d the dust charge number.

3 Derivation of the KdV Equation and Its Three-Soliton Solution

To derive the KdV equation, we apply the reductive peturbation technique (RPT). According to RPT, the independent variables are

The dependent variables are expanded as

Putting (2) and (6)–(10) in (3)–(5) and comparing coefficients of lowest powers of ε, we obtain the linear propagation speed in the low-frequency limit as

with a=κ−1/2κ−3/2.

Taking the coefficients of the next higher order of ε, we obtain the following KdV equation:

where A=(3−2b(1−μ)v42v) and B=v32, with b=(κ−1/2)(κ+1/2)2(κ−3/2)2.

The coefficient A of the nonlinear term ϕ1∂ϕ1∂ξ of the KdV equation (12) becomes zero for κ=(4−3μ)2(2−3μ). So, assuming κ≠(4−3μ)2(2−3μ) (μ < 1) and making the transformations ξ=B1/3ξ¯, ϕ1=6A−1B1/3ϕ¯1, and τ=τ¯ to the KdV equation (12), we obtain the following standard KdV equation:

Using Hirota’s bilinear method, the three-soliton solution of (13) is obtained as [59]

where f(ξ,τ)=1+eθ¯1+eθ¯2+eθ¯3+A122eθ¯1+θ¯2+A232eθ¯2+θ¯3+A132eθ¯3+θ¯1+A2eθ¯1+θ¯2+θ¯3,

where α_i are the initial phase of the solitons.

Thus, the three-soliton solution of the KdV equation (12) is

where f(ξ,τ)=1+eθ1+eθ2+eθ3+A122eθ1+θ2+A232eθ2+θ3+A132eθ3+θ1+A2eθ1+θ2+θ3, with

From (16), we have

where

and

After a rigorous and very complicated mathematical calculation, N becomes

From (18), we have

Assuming τ ≫ 1, (19) can be asymptotically (up to exponentially small terms) approximated as

Therefore, for τ ≫ 1, the three-soliton solution (16) of the KdV equation (12) is asymptotically transformed into a superposition of three single solitons [38]:

where the amplitudes of the solitons are given by

and the phase shifts (Δi,i=1,2,3) of the solitons due to the interaction are given by

4 Effect of Soliton Interaction on the Statistical Characterstic of the Wave Field

In this section, we present the propagation and mutual interaction of the three solitons of the KdV equation (12). Beacuse of the complete integrability of the KdV equation (12), interaction of the solitons is elastic, and after interaction they regain the properties of the soliton [41], [43], [44], [45], [60] . Three solitons with different amplitudes propagate in time, and since the speed of the solitons is proportional to their amplitudes, the nonlinear interaction of the three solitons takes place in a certain time and space. Choosing

we observe that the interaction of the soliton elements occurs at the origin, i.e. at the point ξ = 0, τ = 0 [61]. For above choice of α₁, α₂, and α₃ and assuming η1>η2>η3, the amplitude of the resulting peak is obtained as

The pulse shape at the instant of soliton interaction is determined by the equation

Equation (23) indicates the concavity of wave profile at the strongest interaction point. The negative value of (23) indicates that the wave profile is concave downward. Therefore, the wave profile may maintain a single peak or a triple peak status at the strongest interaction point. The positive value of (23) implies that the wave profile is concave upward and it will maintain the two-peak status at τ = 0. Equation (23) is positive or negative according to whether (1−R1+R2)2−2(1−R1)(R1−R2) is negative or positive, where R1=η22η12<1 and R2=η32η12<1 with R1>R2. Figure 1 shows the region of the positive and negative values of ∂2ϕ1∂ξ2(0, 0).

Figure 1:

Positive (orange shaded) and negative (green shaded) region of ∂2ϕ1∂ξ2(0, 0).

Figure 2 shows the interaction process of the three-soliton solution of (12) for R₁ = 0.38 and R₂ = 0.08 with κ = 2.1, μ = 0.1, η1=0.4. It is observed that for significant difference between the soliton amplitude ratios R₁ and R₂ and maintaining ∂2ϕ1∂ξ2(0, 0)<0, the three solitons interact simultaneously and merge to a single peak at ξ = 0, τ = 0.

Figure 2:

Interaction process of three-soliton solution of (12) for κ = 2.1, μ = 0.1, R1=0.38, R2=0.08 with η1=0.4.

Figure 3 presents the interaction process of three solitons for R₁ = 0.38 and R₂ = 0.35 with κ=2.1,μ=0.1,η1=0.4. It is observed that when R₁ and R₂ do not differ significantly, and maintaining the negative sign of ∂2ϕ1∂ξ2(0, 0), then the two fastest solitons interact first and overtake the slowest soliton.

Figure 3:

Interaction process of three-soliton solution of (12) for κ = 2.1, μ = 0.1, R1=0.38, R2=0.35 with η1=0.4.

The three-soliton interaction process is presented in Figure 4 for R₁ = 0.85 and R₂ = 0.17 with κ = 2.1, μ = 0.1, η1=0.4, for which (23) is positive. Here, the two slowest solitons interact first and both are overtaken by the fastest soliton.

Figure 4:

Interaction process of three-soliton solution of (12) for κ = 2.1, μ = 0.1, R1=0.85, R2=0.17 with η1=0.4.

The interaction of a large number of propagating waves in conservative systems leads to a fast changing of the wave pattern. In such a scenario, the statistical theory is suitable for describing the wave field. Such a theory is called weak wave turbulence. Soliton turbulence is usually energetically harder than the ordinary weakly turbulent plasma description. The conservation laws are very importnat in the study of turbulence. For a scalar partial differential equation with two independent variables x, t and a single dependent variable u, the conservation law can be writtens as

where T and X are the ‘conserved density’ and the ‘flux’, respectively, and both are polynomials of the solution u and its derivatives with respect to the space variable x [62]. If both T and ∂X∂x are integrable over the interval (−∞,∞), then on the assumption that X → 0 as |X|→∞, (24) can be integrated to give

Equation (25) is invariant with time and is termed the ‘invariant of motion’ or ‘constant of motion’ [63], [64]. It is known that the KdV equation forms a completely integrable Hamiltonian system and, hence, posseses an infinite numbers of conserved quantities [31], [32], [33]. The first four conservation laws of the KdV equation (12) are as follows:

The first three integrals (26)–(28) correspond to the conservation of mass, momentum, and energy, respectively. These integrals are conserved in the process of the wave field evolution, and using the non-interacting solitons (21), the analytical calculation of I₁, I₂, I₃, and I₄ are as follows:

From (30)–(33), it is observed that values of the integrals I₁, I₂, I₃, and I₄ increase as the amplitudes of the interacting solitons increase.

It is well known that the dynamics of multiple solitons is affected significantly by the mutual interactions of the solitons [41], [42]. To understand the effect of three-soliton interaction on the statistical moments of the random wave field, the following integrals are considered:

The integrals in (34) are related to the statistical moments of the wave field. The first integral moment (μ₁) and the second integral moment (μ₂) represent the mean and variance of the random wave field, respectively. The first and second integral moments μ₁ and μ₂ are the same as Kruskal’s integral I₁ and I₂, and hence they are conserved for the three-soliton solution (16), which agrees with (30) and (31) (see Fig. 5). Thus, it can be said that the mean and variance of the wave field do not get affected by the nonlinear interactions of the three solitons. The third and fourth integral moments are defined by

Figure 5:

Time dependence of the integrals I₁ and I₂ in the three-soliton interaction with κ = 2.1, μ = 0.1, R1=0.38, R2=0.35 with η1=0.4.

respectively. The third and fourth moments M3(τ) and M4(τ) characterise the skewness and kurtosis (throughout this article, the word ‘kurtosis’ refers to the normalised fourth moment and not its difference from the Gaussian value 3 – ‘excess kurtosis’) of the random wave field. The statistical measure of the vertical asymmetry of the wave field is given by the skewness M3(τ), and kurtosis provides information on the probability of occurance of extreme waves [65]. The structures of μ₃ and μ₄ are different from those of Kruskal’s integrals I₃ and I₄. Therefore, the skewness M3(τ) and kurtosis M4(τ) will not be conserved in the dominant interaction region. Figure 6 presents the numerical evolution of M3(τ) and M4(τ) for the three-soliton solution (16) with κ = 2.1, μ = 0.1, R1=0.38, R2=0.35 and η1=0.4. From Figure 6, it is clear that the skewness and kurtosis decrease in the dominant interaction region of the random wave field. If the solitons do not interact with each other, then the skewness (M30) and kurtosis (M40) can be calculated using the non-interacting solitons (21)

Figure 6:

Time dependence of the skewness M3(τ) and kurtosis M4(τ) in the three-soliton interaction for κ = 2.1, μ = 0.1, R1=0.38, R2=0.35 with η1=0.4.

Also, from Figure 6 it is observed that the skewness M3(τ) calculated using (16) varies by about 7.25 % from M30 calculated using the non-interacting solitons (21) and the kurtosis M4(τ) deviates by about 8.03 % from M40 due to the three-soliton interaction. Here, the skewness and kurtosis are always positive, as all the three solitons are positive.

Figure 7:

Plot of the maximum deviation of the third and fourth moments due to interaction as a function of κ with μ = 0.3, R1=0.38, R2=0.35, and η1=0.4.

Figure 8:

Plot of the maximum deviation of the third and fourth moments due to interaction as a function of μ with κ = 2.5, R1=0.38, R2=0.35, and η1=0.4.

Figure 9:

Relative deviation in third and fourth moments due to interaction as a function of κ with μ = 0.3, R1=0.38, R2=0.35, and η1=0.4.

Figure 10:

Relative deviation in third and fourth moments due to interaction as a function of μ with κ = 2.5, R1=0.38, R2=0.35, and η1=0.4.

5 Effect of the Plasma Parameters on Soliton Turbulence

It has been observed that the soliton amplitude increases as the parameters κ and μ increase. To show the effect of the plasma parameters κ and μ on soliton turbulence, the maximum deviation Mi∗=Mimax−Mimin,(i=3,4) and the relative deviation Mi∗∗=Mimax−MiminMimax,(i=3,4) in third and fourth moments as a function of κ and μ are calculated (Figs. 7–10). From Figure 7, it is observed that an increase in the value of κ decreases the maximum deviation in third and fourth moments. Also, the maximum deviation in third and fourth moments due to the interaction of solitons decreases as the parameter μ increases (Figure 8). Therefore, the spectral index κ and the parameter μ (unperturbed dust-to-ion ratio) have strong influence on the soliton turbulence for DIA waves in a dusty plasma system.

6 Conclusions

In this work, we studied the propagation and interaction of three solitons within the framework of the KdV equation in an unmagnetised, collision-less dusty plasma consisting of κ distributed electrons. The KdV equation was derived using RPT, and by applying Hirota’s bilinear method the three-soliton solution of the KdV equation was obtained. The cancavity of the resulting peak was studied and, depending on amplitude ratios of the solitons, three different types of soliton interaction were shown. The skewness and kurtosis of the random wave field, which are crucial in turbulence theory, were calculated. It was observed that the skewness and kurtosis decrease in the soliton interaction region. Also, it was observed that the plasma parameters κ and μ have great influence on the skewness and kurtosis of the random wave field for DIA waves. The results may be helpful to understand the nonlinear features of the three-soliton solutions in Earth’s mesosphere, cometary tails, and Jupiter’s magnetosphere.