Magnetoacoustic Nonlinear Solitary and Freak Waves in Pair-Ion Plasma

Papihra Sethi; Kuldeep Singh; N.S. Saini

doi:10.1515/zna-2019-0031

Article Publicly Available

Magnetoacoustic Nonlinear Solitary and Freak Waves in Pair-Ion Plasma

Papihra Sethi , Kuldeep Singh and N.S. Saini

Published/Copyright: May 21, 2019

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From the journal Zeitschrift für Naturforschung A Volume 74 Issue 9

Abstract

An investigation of magnetoacoustic nonlinear solitary and freak waves in a magnetised collisionless pair-ion plasma using two-dimensional magnetohydrodynamic model is presented. The reductive perturbation method is used to obtain the Kadomtsev–Petviashvili (KP) equation. The solitary wave solution of KP equation is examined. Further, on modulating KP equation, the nonlinear Schrödinger equation is derived with the help of appropriate transformation. The influence of various plasma parameters such as magnetic field strength, number density of ions, and temperature ratio of negative to positive ions, etc. on the propagation characteristics of solitary waves and first- as well as second-order magnetoacoustic freak waves in pair-ion plasma is examined.

Keywords: Freak Waves; Magnetoacoustic Waves; Pair-Ion Plasma; Solitary Waves

1 Introduction

During the past few years, pair-ion plasma has invoked a lot of recognition among plasma physics community due to their importance in understanding various astrophysical environments such as neutron stars, active galactic nuclei, pulsar magnetosphere, quasars, etc. [1], [2], [3]. Pair-ion plasma is very different from ordinary plasma and thus generally represents a novel state of matter with unique thermodynamics properties. The general electron-ion plasma is subject to various temporal and spatial scale lengths because of significant change in their masses, whereas pair plasmas are subject to the same time scale for thermodynamic equilibrium due to equal masses. The pair-ion plasma is physically not different from an electron–positron plasma except that the pair-ion plasma generation in laboratory experiments is more stable. It is difficult to generate the stable electron–positron plasma in laboratory due to weak source of positrons obtained using accelerator-based sources [4] and from radioactive sources [5]. The production of low-density positrons and short annihilation time makes various collective modes difficult to analyze in electron–positron plasma. Thus, much attention is being paid to laboratory for the stable generation of pair-ion plasmas such as hydrogen (H±) and fullerene (C60±). The fullerene–ion plasmas were first produced experimentally by Oohara and Hatakeyama [6] by the impact ionisation of fullerenes to overcome the short time scale problem of electron–positron plasma. The fullerene pair-ion plasmas in the direction of static external magnetic field give rise to the generation of three electrostatic modes, viz. ion acoustic wave, the intermediate frequency wave, and the ion plasma wave. Hydrogen pair-ion plasma production by catalytic ionisation was investigated by Oohara and Fukumasa [7]. The pair-ion plasma is expected to be used for the synthesis of dimers directly from carbon allotropes as well as for nanotechnology. During recent years, many authors have investigated pair plasmas collective modes in pair-ion–electron plasma [8], [9], as well as in nonlinear wave structures in pure pair-ion plasma [10], [11], [12], [13], [14], [15], [16], [17] and pair-ion–dust plasma [18], [19]. The nonlinear electrostatic structures in homogeneous, unmagnetised pair-ion plasma were studied by [13]. Kaladze et al. [16] reported the study of the periodic and solitary waves in an unmagnetised plasma consisting of positively and negatively charged ions of same mass but different temperatures.

Nonlinear wave motion in a magnetised plasma has attracted a great deal of interest among plasma physicists because of their applications in fusion plasmas [20], Earth’s magnetotail [21], space plasmas [22], etc. The magnetoacoustic wave is a kind of magnetohydrodynamic wave in which the wave propagates perpendicular to magnetostatic fields and is partially transverse and partially longitudinal as a result of compression of both magnetic field and density, respectively. As the phase speed of magnetoacoustic wave is greater than the Alfvén wave, thus magnetoacoustic waves are considered as fast magnetohydrodynamic waves. The magnetoacoustic wave is imperative in plasma heating, in fusion devices, in solar wind plasma, in acceleration, and in the earth’s magnetosphere [23], [24], [25], [26], [27], [28]. Magnetoacoustic waves are widespread in the universe. They were first time observed by OGO-3 spacecraft [29]. Magnetoacoustic waves are of significant importance due to their important role in scattering of energetic electrons in the magnetosphere through Landau resonance interaction [30], [31]. In the magnetoacoustic wave, the restoring forces are the gradient of the compressional stresses between the magnetic field lines and fluid pressure gradient. The magnetoacoustic wave becomes essentially an acoustic wave when fluid pressure exceeds the magnetic pressure. On the other hand, if the magnetic field is strong enough that magnetic pressure is much larger than the fluid pressure, then magnetoacoustic phase velocity becomes equal to Alfvén wave velocity propagating in the perpendicular direction. In recent years, the investigation of nonlinear magnetoacoustic solitary wave structures has been studied by a number of authors [32], [33], [34], [35], [36], [37]. Ruan et al. [34] reported magnetoacoustic solitary waves in pair-ion–electron plasmas. They derived the Korteweg–de-Vries (KdV) equation in a pair-ion plasma containing electrons as impurity and observed the effect of various parameters like negative ion density, magnetic field, etc. Magnetosonic wave in pair-ion electron collisional plasmas was studied by Hussain and Hasnain [35]. They considered (C60±) positive/negative ions and electrons in a collisional plasma and derived a damped KdV equation. Rarefactive solitary waves are observed in their case. The propagation of magnetoacoustic solitary waves in pair-ion fullerene plasma was recently reported by Ur-Rehman et al. [37]. The KdV equation is derived, and both linear and nonlinear effects appearing through ion-inertial length are discussed.

In the first part of this investigation, we have described the two-dimensional study of magnetoacoustic waves by deriving Kadomtsev–Petviashvili (KP) equation. Magnetoacoustic KP equation has been studied by different authors [38], [39], [40], [41]. Magnetoacoustic solitary waves in a warm collisional plasma were studied in a two-dimensional plasma from KP–Burgers equation [38]. They discussed both the planar and the lump solution and observed that the solutions decay with time in the weak collisional limit. Magnetoacoustic solitary waves were investigated in a magnetised relativistic two-dimensional multi-ion plasma consisting of electrons and light and heavy ions [39]. They obtained the n-soliton solution and dromion solution of potential of a physical field. Cylindrical KP equation for magnetoacoustic solitary waves in a collisionless weak relativistic plasma was reported by Wang et al. [40]. They studied the influence of magnetic field on the phase velocity, width, and amplitude of cylindrical KP solitons. Two-dimensional study of cylindrical fast magnetoacoustic solitary waves in a dusty plasma was reported by Liu et al. [41]. They considered a three-component model consisting of ions, electrons, and dust in a nonplanar regime and derived a cylindrical KP equation.

In the second part of this investigation, we have focussed our study on the investigation of freak waves in pair-ion plasma. Freak waves that are localised both in time and space domains are often referred to as monster, violent, extreme, or giant waves and happen to arise suddenly from a relatively calmer sea often known as “WANDTs” (waves that appear from nowhere and disappear without a trace) [42], [43], [44]. Freak waves are short lived and have small probability to appear suddenly out of normal waves. They have few times higher amplitude than the solitary waves and are observed in coastal waters. Thus, they are a threat for various hydrotechnic constructions. This makes freak waves an important problem to study. Various propelling methods such as Darboux transformation method, Bäcklund transformation, Riccati method, and Hirota’s bilinear method have been employed to obtain the exact solutions of nonlinear evolution equations [45], [46], [47], [48]. The dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation (NLSE) was investigated by Wang et al. [46]. Using the modified Darboux transformation, the hierarchies of breather wave and rogue wave solutions are generated from the trivial solution. Feng and Zhang [47] reported the higher-order soliton, breather, and rogue wave solutions of the coupled NLSE by applying the Darboux transformation method. Qin et al. [48] employed the extended homoclinic test method and obtained the rational breather wave and rogue wave solutions of the (3+1)-dimensional generalised KP equation. Also, using the Riccati equation method, analytical bright soliton, dark soliton, and travelling wave solutions have been derived. Over the last many years, numerous investigations on the study of freak waves in unmagnetised and magnetised plasmas have been reported [49], [50], [51], [52], [53], [54], [55], [56], [57], [58]. The influence of various physical parameters on the characteristics of freak waves has been studied in such different plasma environments. In order to study the characteristics of freak waves, it is necessary to derive NLSE and its rational solutions. The study of freak waves in magnetoacoustic electron-ion plasma was reported by El-Awady et al. [58]. They observed the influence of various physical parameters such as plasma number density, magnetic field strength, and electron temperature on the characteristics of rogue waves. Owing to the importance of pair-ion plasma as well as freak waves, the study of magnetoacoustic nonlinear solitary and freak waves is of paramount interest. To the best of our knowledge, no such investigation of the magnetoacoustic nonlinear solitary and freak waves in two-dimensional pair-ion plasma has been reported so far. In the present investigation, we have focussed our study considering transverse perturbation leading to formation of KP solitons (which are solutions of KP equation) and then derived NLSE to describe the magnetoacoustic freak waves. The manuscript is structured as follows. In Section 2, fluid model equations are described. Section 3 presents derivation of KP equation and its solution. The numerical analysis of solution of the KP equation is discussed in Section 4. In Section 5, NLSE is derived, and its solutions are discussed. In Section 6, numerical analysis of rational solutions of NLSE is presented. Section 7 is devoted to conclusions of important results.

2 Fluid Model

We consider a two-fluid magnetised homogeneous pair-ion plasma (H^±) having ions of equal masses and opposite charges with magnetic field taken along the z axis, i.e. B0=B0z^, where z^ is unit vector in the z axis direction. The quasi-neutral condition at equilibrium is n+0=n−0=n0 where n+0 and n−0 are the equilibrium number densities of positive and negative ions. The positive and negative ions have same mass m. The set of normalised equations (continuity and momentum equations) for nonlinear magnetoacoustic waves in plasmas is given as follows:

(1)∂n+∂t+∇.(n+u+)=0,

(2)∂u+∂t+(u+.∇)u+=(E+(u+×B))−βn+∇n+,

(3)∂n−∂t+∇.(n−u−)=0,

(4)∂u−∂t+(u−.∇)u−=−(E+(u−×B))−σβn−∇n−,

Faraday’s law is given as

(5)∇×E=−∂B∂t,

and Ampere’s law is

(6)∇×B=J+α∂E∂t,

where

(7)J=n+u+−n−u−.

Here n−,n+,T− and T₊ are the number densities and temperatures of the negative and positive ions with T+≠T−. The number densities n_± are normalised by their unperturbed values n±0. The velocities are normalised by positive ion Alfvén speed uA=B04πmn+0, the electric field is normalised by muAΩe, Ω=eB0mc is the ion gyro-frequency, and the magnetic field is normalised by muAΩe. The space is normalised by r=rΩ/uA, and time is normalised by t=tΩ. The temperature ratio of the ion species is defined as σ=T−/T+. The other parameters are given as α=uA2c2 and β=Vs2uA2, Vs=T+/m. The propagation of wave is considered along the x and y axes, i.e. ∇=(∂∂x,∂∂y,0).

The normalised equations in component form can be written as

(8)∂n±∂t+∂∂x(n±u±x)+∂∂y(n±u±y)=0,

(9)∂u+x∂t+u+x∂u+x∂x+u+y∂u+x∂y=Ex+u+yBz−βn+∂n+∂x,

(10)∂u+y∂t+u+x∂u+y∂x+u+y∂u+y∂y=Ey−u+xBz−βn+∂n+∂y,

(11)∂u−x∂t+u−x∂u−x∂x+u−y∂u−x∂y=−Ex−u−yBz−βσn−∂n−∂x,

(12)∂u−y∂t+u−x∂u−y∂x+u−y∂u−y∂y=−Ey+u−xBz−βσn−∂n−∂y,

(13)−∂Ex∂y+∂Ey∂x=−∂Bz∂t,

(14)∂Bz∂y=n+u+x−n−u−x+α∂Ex∂t,

(15)−∂Bz∂x=n+u+y−n−u−y+α∂Ey∂t

3 Derivation of the KP Equation and Its Solution

To study the two-dimensional magnetoacoustic waves in pair-ion plasma, reductive perturbation technique is employed to derive the KP equation. We consider the following stretched coordinates as

(16)ξ=ε(x−λt),η=ε2y and τ=ε3t,

where λ is the wave velocity and ε is a very small parameter, which portrays the strength of nonlinearity.

The dependent variables are expanded as

(17)(nsBzusxEy)=(1B000)+∑j=1∞ε2j(nsjBzjusxjEyj)

(18)(usyEx)=∑j=1∞ε2j+1(usyjExj)

where s = ±. Using stretching co-ordinates from (16) and substituting the expansions of (17) and (18) into (8) to (15) and after collecting the coefficients of different powers of ε, we find the first-order evolution equations in the following forms:

(19)n+1=n−1=1λu+x1,

(20)u+x1=u−x1,

(21)Ey1=u+x1B0,

(22)B1=B0λu+x1,

(23)∂Ex1∂ξ=β(1−σ)(αB02+2)λ∂2u+x1∂ξ2−B0∂u+x1∂η,

where

(24)λ=B02+β(1+σ)2+αB02,

Collection of coefficients of higher order in ε yields the following second-order equations.

(25)−λ∂n+2∂ξ+∂n+1∂τ+∂u+x2∂ξ+∂u+y1∂η+∂(n+1u+x1)∂ξ=0,

(26)−λ∂n−2∂ξ+∂n−1∂τ+∂u−x2∂ξ+∂u−y1∂η+∂(n−1u−x1)∂ξ=0,

(27)−λ∂u+x2∂ξ+∂u+x1∂τ+u+x1∂u+x1∂ξ=Ex2+B0u+y2+B1u+y1−β∂n+2∂ξ+βn+1∂n+1∂ξ,

(28)−λ∂u−x2∂ξ+∂u−x1∂τ+u−x1∂u−x1∂ξ=−Ex2−B0u−y2+B1u−y1−βσ∂n−2∂ξ+βσn−1∂n−1∂ξ,

(29)−λ∂u+y1∂ξ=Ey2−B0u+x2−B1u+x1−β∂n+1∂η,

(30)−λ∂u−y1∂ξ=−Ey2+B0u+x2+B1u+x1−βσ∂n+1∂η,

(31)−∂Ex1∂η+∂Ey2∂ξ=λ∂B2∂ξ−∂B1∂τ,

(32)∂B1∂η=u+x2+n+1u+x1−n−1u−x1−u−x2−αλ∂Ex1∂ξ,

(33)∂B2∂ξ=u+y2+n+1u+y1−n−1u−y1−u−y2−αλ∂Ey2∂ξ+α∂Ey1∂τ,

By eliminating second-order quantities (n+2, n−2, u+x2, u−x2, E_x2, E_y2) from (25) to (33), one can finally obtain the KP equation as follows:

(34)∂∂ξ(∂u+x1∂τ+au+x1∂u+x1∂ξ+b∂3u+x1∂ξ3)+c∂2u+x1∂η2=0,

where nonlinear coefficient

(35)a=λc(B02+β(1+σ))(6−β(1−σ)λ2),

dispersion coefficient

(36)b=λc(B02+β(1+σ))(2λ3B02+β2(1+σ2)λB02−2λβ(1+σ)B02−β2(1−σ)2(αB02+2)λB0),

and weak dispersion coefficient,

(37)c=λ2.

In the limiting case, on neglecting the displacement current in (7) and letting c = 0, (34) reduces to KdV equation, which agrees with (41) of Ur-Rehman et al. [37].

By using new co-ordinate transformation frame χ=k(ξ+η−Uτ), where k represents dimensionless wave number and k = 1, the KP equation (34) is integrated using appropriate boundary conditions (e.g. ψ(χ),ψ′(χ)→0 as |χ|→∞), admits solitary wave solution given as [59]

(38)ψs(ξ,η,τ)=3(U−c)asech2(ξ+η−Uτw),

where w=4bU−c.

4 Numerical Analysis of Solution of the KP Equation

As ψs(ξ,η,τ) is a function of coefficients a, b, c (see (38)), which are further the functions of strength of the magnetic field (via B₀), number density of ions (n₀) (which is appearing in α via u_A), and temperature ratio of negative to positive ions (via σ); thus, it becomes imperative to study their effects on the characteristics of two-dimensional solitary wave structures. We have chosen hydrogen plasma parameters [60], [61] as n+0=1×1016m−3, T+=0.2 eV, T−=0.15 eV, B0=0.1T for the study of KP solitary, first-order, and second-order magnetoacoustic freak waves in pair-ion plasma. These data agree with β < 1, i.e. when the magnetic field compression dominates the density compression. Here, it is illustrated that both nonlinear coefficient (a) and dispersion coefficient (b) are positive, and thus, positive potential KP solitary wave structures are formed.

$Figure 1: (Color online) The variation of magnetoacoustic KP soliton profile for parameters B0=0.1T,n0=1015m−3,σ=0.5${B_{0}}=0.1T,{n_{0}}{=10^{15}}{m^{-3}},\sigma=0.5$ (black solid), B0=0.2T${B_{0}}=0.2T$ (blue dotted), n0=1016m−3${n_{0}}{=10^{16}}{m^{-3}}$ (red dot dashed), σ = 0.9 (green dashed), keeping other parameters fixed.$

Figure 1:

(Color online) The variation of magnetoacoustic KP soliton profile for parameters B0=0.1T,n0=1015m−3,σ=0.5 (black solid), B0=0.2T (blue dotted), n0=1016m−3 (red dot dashed), σ = 0.9 (green dashed), keeping other parameters fixed.

The effect of magnetic field is illustrated in Figure 1. On increasing magnetic field from B0=0.1T (black solid) to B0=0.2T (blue dotted), nonlinear effects dominate, and the amplitude of KP solitons is reduced, and dispersion effects dominate as a result width of KP solitons increases. Increase in magnetic field leads to increase in the Alfvén speed (uA=B04πmn+0) and decrease in plasma β(β=Vs2uA2) at same values of density and temperatures of ions. The variation of concentration of ions (n₀) on the profile of solitary waves is observed. It is found that with the increase in value of n0=1015m−3 (black solid) to n0=1016m−3 (red dot dashed), the amplitude of KP solitons reduces as depicted in Figure 1. Increase in n₀ leads to increase in plasma β and decrease in Alfvén speed (u_A). The influence of temperature ratio of negative to positive ion (via σ) is also illustrated in Figure 1, and it is found that with the increase in value of σ = 0.5 (black solid) to σ = 0.9 (green dashed), the amplitude of KP solitons is reduced. Thus, the effects of nonlinear, predominate dispersion and weak dispersion coefficients (via a, b, and c) balance each other under the influence of various physical parameters in such a way that leads to the increase or decrease of the amplitude of KP solitons. The weak dispersion via coefficient c plays a vital role in the formation of KP solitary waves and influences the behaviour of magnetoacoustic solitary waves in pair-ion plasmas.

5 Nonlinear Schrödinger Equation and Its Solution

The KP equation can be transformed to NLSE by using independent variables as u+x1(ξ,η,τ) = u+x1(X,τ) and X = Lξ + Mη – mτ.

Substituting new variables in (34), we obtain the following equation:

(39)∂u+x1(X,τ)∂τ+aLu+x1(X,τ)∂u+x1(X,τ)∂X+bL3∂3u+x1(X,τ)∂X3=0,

where L and M denote the direction cosines along ξ and η axes, respectively, with L2+M2=1, and m is an arbitrary parameter and satisfies the condition m=M2cL=(1−L2)cL. The solution of (39) with expansion of ψ(=u+x1) as weakly modulated sinusoidal wave is assumed as

(40)ψ(X,τ)=∑n=1∞εn∑l=−∞∞ψlneil(KX−ωτ),

where ω is the frequency of the given wave, and K is the carrier wave number. The new stretched coordinates ζ and T are

(41)ζ=ε(X−vgτ) and T=ε2τ.

Assuming slow variables (ζ, T) to enter the arguments of the lth harmonic amplitude ψln and phase (KX−ωτ) be responsible for all perturbed states to depend upon fast scales. For ψ(X,τ) to be real, the condition ψ−l(n)=ψl(n)* has to be satisfied, where * represents complex conjugate. The derivative operators can be written as

(42)∂∂X=∂∂X+ε∂∂ζ,

(43)∂∂τ=∂∂τ−λε∂∂ζ+ε2∂∂T,

For first-order approximation (l = 1, n = 1), one obtains linear dispersion relation

(44)ω=−L3K3b.

For first harmonics (l=1) of second approximation (n=2), the group velocity is determined as

(45)vg=−3L3K2b.

The zeroth harmonics (l=0,n=2) yields

(46)ψ02=(avg)|ψ1(1)|2.

The second harmonics (l=2,n=2) yields

(47)ψ22=(a6L2K2b)ψ1(1)2.

For n=3,l=1, after algebraic manipulations and eliminating higher-order quantities, we obtain the NLSE as [55]

(48)i∂ψ∂T+P∂2ψ∂ζ2+Q|ψ|2ψ=0.

We have assumed ψ1(1)=ψ. The nonlinear coefficient Q=a2L2P and dispersion coefficient P=−3L3bK.

The first-order rational solution (say ψ₁) of NLSE (48), which describes magnetoacoustic freak waves for pair-ion plasma, can be obtained as [62]

(49)|ψ1(ζ,T)|=(4(1+2iT)1+4T2+4ζ2P−1)eiTQ.

The solution (49) anticipates the concentration of energy of pair-ion magnetoacoustic waves into a small region from surrounding waves caused by nonlinear behaviour of plasma medium.

However, the actual wave dynamics consists of a nonlinear superposition of many simple solutions. In other words, we must take into account the nonlinear superposition of two rational solutions of first order those combined into a more complicated doubly localised structures with a higher amplitude. Higher-order freak waves have been observed in experiments like in water [63] and also theoretically predicted in plasmas [64], [65]. So, it is interesting and important to investigate the second-order freak wave solution (say ψ₂) under the influence of different physical parameters. The expression for second-order freak wave solution localised in space (ζ) and time scales (T) is given as [66]

(50)|ψ2(ζ,T)|=PQ(1+R1+iS1T1)e(iPT),

(51)R1=38−ζ42−32ζ2−6(PζT)4−10(PT)2,

(52)S1=−PT[−154+ζ4−3ζ2+4(ζT)2+4(PT)4+2(PT)2],

(53)T1=332+112ζ6+ζ48+ζ42(PT)2+916ζ2+ζ2(PT)4−332(PζT)2+23(PT)6+92(PT)4+338(PT)2.

6 Numerical Analysis of Rational Solutions

In this section, we have carried out numerical analysis of rational solutions to study the propagation properties of freak waves in pair-ion plasma. As various physical parameters such as magnetic field strength (B₀), number density of ions (n₀), temperature ratio of negative to positive ions (σ), and direction cosine (L) affect the nonlinear and the dispersion coefficients, thus it becomes imperative to investigate their role on the behaviour of freak waves.

$Figure 2: (Color online) The variation of first-order magnetoacoustic freak pulse profile for different values of B0=(a)0.1T,(b)0.14T${B_{0}}=(a)0.1T,(b)0.14T$ with fixed parameters n0=1014m−3,σ=0.75,K=2${n_{0}}{=10^{14}}{m^{-3}},\sigma=0.75,K=2$, and L = 0.9.$

Figure 2:

(Color online) The variation of first-order magnetoacoustic freak pulse profile for different values of B0=(a)0.1T,(b)0.14T with fixed parameters n0=1014m−3,σ=0.75,K=2, and L = 0.9.

$Figure 3: (Color online) The variation of first-order magnetoacoustic freak pulse profile ψ1 versus ζ for different values of n0=1014m−3${n_{0}}{=10^{14}}{m^{-3}}$(black solid), n0=1015m−3${n_{0}}{=10^{15}}{m^{-3}}$ (blue dotted), and n0=1016m−3${n_{0}}{=10^{16}}{m^{-3}}$ (green dashed) with fixed parameters B0=0.1T,σ=0.75,K=2${B_{0}}=0.1T,\sigma=0.75,K=2$, and L = 0.7.$

Figure 3:

(Color online) The variation of first-order magnetoacoustic freak pulse profile ψ₁ versus ζ for different values of n0=1014m−3(black solid), n0=1015m−3 (blue dotted), and n0=1016m−3 (green dashed) with fixed parameters B0=0.1T,σ=0.75,K=2, and L = 0.7.

$Figure 4: (Color online) The variation of first-order magnetoacoustic freak pulse profile ψ1 versus ζ for different values of L=0.7,σ=0.5$L=0.7,\sigma=0.5$ (black solid), L=0.8,σ=0.5$L=0.8,\sigma=0.5$ (blue dotted), and L=0.7,σ=0.9$L=0.7,\sigma=0.9$ (green dashed) with fixed parameter B0=0.1T${B_{0}}=0.1T$, K = 2, and n0=1016m−3${n_{0}}{=10^{16}}{m^{-3}}$.$

Figure 4:

(Color online) The variation of first-order magnetoacoustic freak pulse profile ψ₁ versus ζ for different values of L=0.7,σ=0.5 (black solid), L=0.8,σ=0.5 (blue dotted), and L=0.7,σ=0.9 (green dashed) with fixed parameter B0=0.1T, K = 2, and n0=1016m−3.

Figures 2–4 illustrate first-order freak waves solution. It can be judged that the wave solution ψ₁ sucks energy from other waves to grow almost vertically into a monster in a relatively small area in space, concentrating large amount of energy, before becoming unstable and collapsing shortly. The influence of the strength of magnetic field (via B₀) on magnetoacoustic freak waves is illustrated in Figure 2, and it is found that on increase of magnetic field the amplitude and width of magnetoacoustic freak waves are enhanced. An increase in the strength of magnetic field leads to decrease in nonlinear coefficient (Q) and increase in the dispersion coefficient (P), as a result the freak waves grow higher (i.e. amplitude and width becomes larger). Thus, magnetic field plays a crucial role on the propagation properties of freak waves. The number density of ions also plays a significant role on the behaviour of magnetoacoustic freak waves, and its impact on the behaviour of freak waves is analyzed. It is inferred that with the increase in concentration of ions n₀ both the amplitude and width of freak waves are found to reduce as depicted in Figure 3. The increase in number density of ions increases the nonlinear coefficient (Q) but decreases the dispersion coefficient (P); as a result, it leads to decrease in both amplitude and width of freak waves. The behaviour of freak wave structure is examined in Figure 4 for different values of temperature ratio of negative to positive ion (via σ). It is observed that with increase in temperature of negative ions in the system or the rise in the value of σ from σ = 0.1 (black solid) to σ = 0.9 (green dashed) the height and width of freak waves are reduced. The impact of direction cosine (via L) on the characteristics of freak waves is also observed in Figure 4. It is observed that with the increase in L both the amplitude and width of freak waves are enhanced. Increase in L from L = 0.7 (black solid) to L = 0.8 (blue dotted) leads to increase in the dispersion coefficient (P) and decrease in the nonlinear coefficient (Q); as a result, amplitude of first-order freak waves is enhanced.

$Figure 5: (Color online) Comparison of magnetoacoustic freak pulse profile of first-order |ψ1|$|{\psi_{1}}|$ and second-order |ψ2|$|{\psi_{2}}|$ with ζ.$

Figure 5:

(Color online) Comparison of magnetoacoustic freak pulse profile of first-order |ψ1| and second-order |ψ2| with ζ.

$Figure 6: (Color online) The variation of second-order magnetoacoustic freak pulse profile for different values of B0=(a)0.1T,(b)0.14T${B_{0}}=(a)0.1T,(b)0.14T$ with fixed parameters n0=1014m−3${n_{0}}{=10^{14}}{m^{-3}}$, σ=0.75,K=2$\sigma=0.75,K=2$, and L = 0.9.$

Figure 6:

(Color online) The variation of second-order magnetoacoustic freak pulse profile for different values of B0=(a)0.1T,(b)0.14T with fixed parameters n0=1014m−3, σ=0.75,K=2, and L = 0.9.

In the similar manner, the study of characteristics of magnetoacoustic freak waves from second-order solution is also carried out. From Figure 5, it is clear that the amplitude of second-order freak waves is higher than the first-order freak waves as a result of superposition of energies of first-order freak waves. The propagation characteristics of second-order freak waves has also been described, and it is analyzed that their structures are modified by the physical parameters as depicted in Figures 6–8. The influence of magnetic field strength is observed on second-order freak waves. The increase in the strength of magnetic field increases the amplitude and width of freak waves. It is clearly visible in Figure 6 that the second-order freak waves have large amplitude than the first order (Fig. 2) and appear more spiky than the first-order freak waves. Thus, the greater the magnetic field in the plasma, the greater is the enhancement in the amplitude of freak waves. In Figure 7, the effect of concentration of ions (n₀) is observed on second-order freak waves, which appears to be two times larger than the first order. It is observed that increase in concentration of ions suppresses the amplitude of both the first- and second-order freak waves due to dissipation of energy. Thus, number density of ions significantly modifies the behaviour of freak waves. It is observed in Figure 8 that increase in the temperature of negative ions (via σ) from σ = 0.1 (black solid) to σ = 0.9 (green dashed) suppresses the amplitude of freak waves, clearly indicating the dissipation of energy and resulting in the decrease of amplitude of freak waves. The impact of direction cosine L on the characteristics of freak waves is analyzed; it is found that the increase in values of L = 0.7 (black solid) to L = 0.8 (blue dotted) enhances more concentration of energy, thereby increasing the amplitude of freak waves as depicted in Figure 8. Thus, it is emphasised that the plasma physical parameters have a significant influence on the propagation characteristics of magnetoacoustic freak waves.

$Figure 7: (Color online) The variation of second-order magnetoacoustic freak pulse profile ψ2 versus ζ for different values of n0=1014m−3${n_{0}}{=10^{14}}{m^{-3}}$ (black solid), n0=1015m−3${n_{0}}{=10^{15}}{m^{-3}}$ (blue dotted), and n0=1016m−3${n_{0}}{=10^{16}}{m^{-3}}$ (green dashed) with fixed parameters B0=0.1T,σ=0.75,K=2${B_{0}}=0.1T,\sigma=0.75,K=2$, and L = 0.7.$

Figure 7:

(Color online) The variation of second-order magnetoacoustic freak pulse profile ψ₂ versus ζ for different values of n0=1014m−3 (black solid), n0=1015m−3 (blue dotted), and n0=1016m−3 (green dashed) with fixed parameters B0=0.1T,σ=0.75,K=2, and L = 0.7.

$Figure 8: (Color online) The variation of first-order magnetoacoustic freak pulse profile ψ2 versus ζ for different values of L=0.7,σ=0.5$L=0.7,\sigma=0.5$(black solid),L=0.8,σ=0.5$L=0.8,\sigma=0.5$ (blue dotted), and L=0.7,σ=0.9$L=0.7,\sigma=0.9$ (green dashed) with fixed parameter B0=0.1T${B_{0}}=0.1T$, K = 2, and n0=1016m−3${n_{0}}{=10^{16}}{m^{-3}}$.$

Figure 8:

(Color online) The variation of first-order magnetoacoustic freak pulse profile ψ₂ versus ζ for different values of L=0.7,σ=0.5(black solid),L=0.8,σ=0.5 (blue dotted), and L=0.7,σ=0.9 (green dashed) with fixed parameter B0=0.1T, K = 2, and n0=1016m−3.

7 Conclusions

We have studied the behaviour of magnetoacoustic nonlinear solitary and freak waves in a pair-ion plasma by deriving the KP equation using reductive perturbation theory and by deriving NLSE from KP equation. The solution of KP equation is studied in detail to analyze the characteristics of magnetoacoustic waves. The NLSE is derived by modulating KP equation, and its rational solutions are used to study the characteristics of freak waves. The effects of various physical parameters such as the strength of magnetic field, number density, temperature ratio of negative to positive ions, etc. on the characteristics of KP solitary and freak waves are investigated. It is inferred that both the amplitude and width of freak as well as solitary waves are strongly influenced by the plasma parameters. The present investigation might be useful to understand magnetoacoustic solitary as well as freak waves in the laboratory experiments and astrophysical environments where pair-ions such as H^± may possibly exist with distinct temperature. This study may also be of great importance to study freak waves in laboratory experiments with dust impurities and space environments.

Acknowledgements

P.S. is grateful to University Grants Commission, New Delhi, for the Rajiv-Gandhi Fellowship award. This work is supported by DRS-II(SAP) No. F 530/17/DRS-II/2015(SAP-I) UGC, New Delhi.

References

[1] L. Stenflo, P. K. Shukla, and M. Y. Yu, Astrophys. Space Sci. 117, 303 (1985).10.1007/BF00650156Search in Google Scholar

[2] N. Iwamoto, Phys. Rev. E 47, 604 (1993).10.1103/PhysRevE.47.604Search in Google Scholar

[3] G. P. Zank and R. G. Greaves, Phys. Rev. E. 51, 6079–6090, (1995).10.1103/PhysRevE.51.6079Search in Google Scholar

[4] C. Surko, M. Leventhal, and A. Passner, Phys. Rev. Lett. 62, 901 (1989).10.1103/PhysRevLett.62.901Search in Google Scholar

[5] T. Kurihara, A. Yagishita, A. Enomoto, H. Kobayashi, T. Shidara, et al., Nuclear Instm. Methods Phys. Rev. B. 171, 164 (2000).10.1016/S0168-583X(00)00074-4Search in Google Scholar

[6] W. Oohara and R. Hatakeyama, Phys. Rev. Lett. 91, 205005 (2003).10.1103/PhysRevLett.91.205005Search in Google Scholar PubMed

[7] W. Oohara and O. Fukumasa, J. Plasma Fusion Res. 5, S2106 (2010).10.1585/pfr.5.S2106Search in Google Scholar

[8] R. Sabry, W. M. Moslem, and P. K. Shukla, Phys. Plasmas 16, 032302 (2009).10.1063/1.3088005Search in Google Scholar

[9] S. A. Elwakil, E. K. El-Shewy, and H. G. Abdelwahed, Phys. Plasmas 17, 052301 (2010).10.1063/1.3383052Search in Google Scholar

[10] A. E. Dubinov, I. D. Dubinova, and V. A. Gordienko, Phys. Plasmas 13, 082111 (2006).10.1063/1.2335819Search in Google Scholar

[11] W. M. Moslem, I. Kourakis, and P. K. Shukla, Phys. Plasmas 14, 032107 (2007).10.1063/1.2710455Search in Google Scholar

[12] D. A. Diver and E. W. Laing, Phys. Plasmas 16, 092301, (2009).10.1063/1.3216545Search in Google Scholar

[13] S. Mahmood and H. Ur-Rehman, Phys. Plasmas 17, 072305 (2010).10.1063/1.3458903Search in Google Scholar

[14] A. Shah, S. Mahmood, and Q. Haque, Phys. Plasmas 17, 122302 (2010).10.1063/1.3512937Search in Google Scholar

[15] F. Verheest, Phys. Plasmas 17, 062302 (2010).10.1063/1.3435275Search in Google Scholar

[16] T. Kaladze, S. Mahmood, and H. Ur-Rehman, Phys. Scr. 86, 035506 (2012).10.1088/0031-8949/86/03/035506Search in Google Scholar

[17] K. Shahzad, A. Ur-Rehman, and H. Saleem, Phys. Plasmas 25, 022105 (2018).10.1063/1.5000572Search in Google Scholar

[18] U. M. Abdelsalam, Physica B 405, 3914 (2010).10.1016/j.physb.2010.06.027Search in Google Scholar

[19] A. Abdikian, CPP 405, 3914 (2018).Search in Google Scholar

[20] V. Land, W. J. Goedheer, and M. R. Akdim, Phys. Rev. E 72, 046403 (2005).10.1103/PhysRevE.72.046403Search in Google Scholar PubMed

[21] C. Grabbe, Phys. Rev. Lett. 84, 3614 (2000).10.1103/PhysRevLett.84.3614Search in Google Scholar PubMed

[22] W. H. Matthaeus, S. Dasso, J. M. Weygand, L. J. Milano, C. W. Smith, and M. G. Kivelson, Phys. Rev. Lett. 95, 231101 (2005).10.1103/PhysRevLett.95.231101Search in Google Scholar PubMed

[23] J. H. Adlam and J. E. Allen, Philos. Magazine 3, 448 (1958).10.1080/14786435808244566Search in Google Scholar

[24] L. Davis, R. Lüst, and A. Schulüter, Z. Naturforsch. 13a, 916 (1958).10.1515/zna-1958-1102Search in Google Scholar

[25] C. S. Gardner and G. K. Morikawa, Commun. Pure Appl. Math. 18, 35 (1965).10.1002/cpa.3160180107Search in Google Scholar

[26] Y. A. Berezin and V. I. Karpman, Sov. Phys. JETP 19, 1265 (1964).10.1143/JPSJ.19.1265Search in Google Scholar

[27] T. Kakutani and H. Ono, Phys. Soc. Jpn. 26, 1305 (1969).10.1143/JPSJ.26.1305Search in Google Scholar

[28] K. G. Klein, G. G. Howes, J. M. Tenbarge, S. D. Bale, C. H. K. Chen, et al., Astrophys. J. 755, 159 (2012).10.1088/0004-637X/755/2/159Search in Google Scholar

[29] C. T. Russell, R. E. Holzer, and E. J. Smith, J. Geophys. Res. 74, 755 (1969).10.1029/JA074i003p00755Search in Google Scholar

[30] R. B. Horne, R. M. Thorne, S. A. Glauert, N. P. Meredith, D. Pokhotelov, et al., Geophys. Res. Lett. 34, L17107 (2007).10.1029/2007GL030267Search in Google Scholar

[31] J. Li, B. Ni, L. Xie, Z. Pu, J. Bortnik, et al., Geophys. Res. Lett. 41, 4828 (2014).10.1002/2014GL060461Search in Google Scholar

[32] S. Hussain and S. Mahmood, Phys. Plasmas 18, 082109 (2011).10.1063/1.3626556Search in Google Scholar

[33] W. Masood, A. Mahmood, and H. Rizvi, Astrophys. Space Sci. 343, 273 (2013).10.1007/s10509-012-1247-7Search in Google Scholar

[34] S. S. Ruan, S. Wu, M. Raissan, and Z. Cheng, Phys. Scr. 87, 045503 (2013).10.1088/0031-8949/87/04/045503Search in Google Scholar

[35] S. Hussain and H. Hasnain, Phys. Plasmas 24, 032106 (2017).10.1063/1.4978492Search in Google Scholar

[36] N. Kaur, M. Singh, and N. S. Saini, Phys. Plasmas 25, 043704 (2018).10.1063/1.5011948Search in Google Scholar

[37] H. Ur-Rehman, S. Mahmood, and S. Hussain, Waves in Random and Complex Media 1–11 (2018). doi:10.1080/17455030.2018.1549762.Search in Google Scholar

[38] M. S. Janaki, B. K. Som, B. Dasgupta, and M. R. Gupta, J. Phys. Soc. Japan 60, 2977 (1991).10.1143/JPSJ.60.2977Search in Google Scholar

[39] W. Yun-Liang, Z. Zhong-Xiang, L. Yan-Zhen, Ni. Xiao-Dong, S. Jiang, et al., Commun. Theor. Phys. 51, 1121 (2009).10.1088/0253-6102/51/6/29Search in Google Scholar

[40] Y. Wang, Z. Zhongxiang, J. Xiangqian, H. Zhang, Y. Jiang, et al., Phys. Lett. A 355, 386 (2006).10.1016/j.physleta.2006.03.001Search in Google Scholar

[41] H. F. Liu, S. Q. Wang, Z. H. Wang, C. Z. Li, L. Yao, et al., Phys. Plasmas 18, 044501, (2011).10.1063/1.3672515Search in Google Scholar

[42] P. Müller, C. Garrett, and A. Osborne, Oceanography 18, 66 (2005).10.5670/oceanog.2005.30Search in Google Scholar

[43] C. Kharif and E. Pelinovsky, Eur. J. Mech. B 22, 603 (2003).10.1016/j.euromechflu.2003.09.002Search in Google Scholar

[44] A. Garrett and J. Gemmrich, Phys. Today 62, 57 (2009).10.1063/1.3156338Search in Google Scholar

[45] M. J. Dong, S. F. Tian, X. W. Yan, and L. Zou, Comput. Math. Appl. 75, 957 (2018).10.1016/j.camwa.2017.10.037Search in Google Scholar

[46] X. B. Wang, T. T. Zhang, and M. J. Dong, Appl. Math. Lett. 86, 298 (2018).10.1016/j.aml.2018.07.012Search in Google Scholar

[47] L. L. Feng and T. T. Zhang, Appl. Math. Lett. 78, 2133 (2018).10.1016/j.aml.2017.11.011Search in Google Scholar

[48] C. Y. Qin, S. F. Tian, X. B. Wanga, T. T. Zhang, and J. Li, Comput. Math. Appl. 75, 4221 (2018).10.1016/j.camwa.2018.03.024Search in Google Scholar

[49] D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (1999).10.1038/nature06402Search in Google Scholar PubMed

[50] G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, et al., Nature 374, 989 (2010).10.1016/j.physleta.2009.12.014Search in Google Scholar

[51] T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).10.1017/S002211206700045XSearch in Google Scholar

[52] L. Stenflo and M. Marklund, J. Plasma Phys. 76, 293 (2010).10.1017/S0022377809990481Search in Google Scholar

[53] A. Panwar and C. M. Ryu, Phys. Plasmas 21, 062104 (2014).10.1063/1.4882868Search in Google Scholar

[54] H. G. Abdelwahed, E. K. El-Shewy, M. A. Zahran, and S. A. Elwakil, Phys. Plasmas 23, 022102 (2016).10.1063/1.4940679Search in Google Scholar

[55] N. S. Saini, M. Singh, and A. S. Bains, Phys. Plasmas 22, 113702 (2015).10.1063/1.4935165Search in Google Scholar

[56] U. M. Abdelsalam, W. M. Moslem, A. H. Khater, and P. K. Shukla, Phys. Plasmas 18, 092305 (2011).10.1063/1.3633910Search in Google Scholar

[57] U. M. Abdelsalam, J. Plasma Phys. 79, 287 (2013).10.1017/S0022377812000992Search in Google Scholar

[58] E. I. El-Awady, H. Rizvi, W. M. Moslem, S. K. El-Labany, A. Raouf, and M. Djebli, Astrophys. Space Sci 349, 5 (2014).10.1007/s10509-013-1602-3Search in Google Scholar

[59] T. S. Gill, N. S. Saini, and H. Kaur, Chaos Soliton Fract. 28, 1106 (2006).10.1016/j.chaos.2005.08.118Search in Google Scholar

[60] W. Oohara and R. Hatakeyama, Phys. Plasmas 14, 055704 (2007).10.1063/1.2436854Search in Google Scholar

[61] H. Saleem, N. Batool, and S. Poedts, Phys. Plasmas 18, 052108 (2011).10.1063/1.3589474Search in Google Scholar

[62] D. H. Peregrine, J. Aust. Math. Soc. Ser. B, Appl. Math. 25, 16 (1983).10.1017/S0334270000003891Search in Google Scholar

[63] A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2, 011015 (2012).10.1103/PhysRevX.2.011015Search in Google Scholar

[64] S. Guo, L. Mei, and W. Shi, Phys. Lett. A 377, 2118 (2013).10.1016/j.physleta.2013.06.015Search in Google Scholar

[65] W. R. Sun, B. Tian, R. X. Liu, and D. Y. Liu, Phys. Rev. X 349, 366 (2014).10.1016/j.aop.2014.06.023Search in Google Scholar

[66] N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, Phys Rev. E 80, 026601 (2009).10.1103/PhysRevE.80.026601Search in Google Scholar PubMed

Received: 2019-01-29

Accepted: 2019-03-31

Published Online: 2019-05-21

Published in Print: 2019-09-25

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https://doi.org/10.1515/zna-2019-0031

Keywords for this article

Freak Waves; Magnetoacoustic Waves; Pair-Ion Plasma; Solitary Waves

Articles in the same Issue

Frontmatter
General
Electrical Conductivity of Magnesium Oxide/Molten Carbonate Eutectic Coexisting System
Atomic, Molecular & Chemical Physics
Quintet States 1s2s2p^{2 5}P and 1s2p^{3 5}S^o for Be-Like Systems
Graphene-Based Waist-Enlarged Optical Fibre Sensor for Measurement of Sucrose Concentration
Dynamical Systems & Nonlinear Phenomena
Three-Soliton Interaction and Soliton Turbulence in Superthermal Dusty Plasmas
Delta Shock Wave in a Perfect Fluid Model with Zero Pressure
Magnetoacoustic Nonlinear Solitary and Freak Waves in Pair-Ion Plasma
Hydrodynamics
Shock Wave Kinematics in a Relaxing Gas with Dust Particles
Quantum Theory
An Improved Ping-Pong Protocol Using Three-Qubit Nonmaximally Nonorthogonal Entangled States
Solid State Physics & Materials Science
Structural, Spectroscopic, and Magnetic Studies on Copper Tartrate Crystals
Features of Electrical and Photoelectric Properties of GaS(Yb) Monocrystals
Insight into the Structural, Electronic, Elastic, Mechanical, and Thermodynamic Properties of XReO₃ (X = Rb, Cs, Tl) Perovskite Oxides: A DFT Study