Nonlinear dust-acoustic waves due to the interaction of streaming protons and electrons with dusty plasma

Chaos, Solitons and Fractals 118 (2019) 320–327

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Nonlinear dust-acoustic waves due to the interaction of streaming protons and electrons with dusty plasma Abeer A. Mahmoud a,∗, R.E. Tolba b a b

Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Centre for Theoretical Physics, British University in Egypt (BUE), El-Shorouk City, Cairo, Egypt

a r t i c l e

i n f o

Article history: Received 1 August 2018 Revised 19 November 2018 Accepted 6 December 2018 Available online 11 December 2018 Keywords: Five component dusty plasma Reductive perturbation method Homotopy perturbation transform method Shock wave Periodical wave Soliton wave

a b s t r a c t The dust-acoustic nonlinear shock like, periodical and solitary waves are inspected in dusty plasma with positive charge that interacts with streaming protons and electrons from solar wind. The evolution equation is solved using Homotopy perturbation transform method and different nonlinear solutions are obtained. The dependence of the shock like, periodical, and solitary wave profiles on the kappa distribution parameters (ki for ions and ke for electrons), ratio between the dust temperatures and effective temperature σ d are studied. The present model is of interest to realize various nonlinear waves those may propagate in the magnetosphere of Jupiter. © 2018 Published by Elsevier Ltd.

1. Introduction Few decades ago, dusty plasma becomes an interesting new branch of plasma physics, but in these days the studying of dusty plasma waves have been attracted researchers’ attention to explain many basic phenomena in plasma physics. In general, the plasma systems were considered as nonlinear system, where the nonlinearity of the plasma was considered as a result of several reasons as sudden circumstances in the particles dynamical motion, which putting the plasma states far from the equilibrium state. The plasma nonlinearity phenomena created in space on account of free energy sources and externally launched in laboratory plasmas under controlled conditions. In laboratory, the scientists succeeded in controlling the conditions to create some famous nonlinear waves as solitary wave, shock wave, surface wave and so on. However, in natural these waves were appeared and grown and no one can be expected the type appearing wave [1–21]. There are many different methods have been used for solving many different kinds of physical and mathematical problems, as Hirota’s method [22], the Backlund and Darboux transformation [23], homogenous balance method [24], Jacobi elliptic function [25], extended tanh-function methods [26], extended F-expansion methods [27], Adomian decomposition method [28], Exp-function methods [29], and the homotopy perturbation method.

∗

Corresponding author. E-mail address: [email protected] (A.A. Mahmoud).

https://doi.org/10.1016/j.chaos.2018.12.004 0960-0779/© 2018 Published by Elsevier Ltd.

In this paper, the Homotopy perturbation transform method (HPTM) is used for solving nonlinear wave-like equations of variable and constant coefficients. This method is a coupling of homotopy perturbation method and Laplace transformation with using He’s polynomials to able to transact with the nonlinear term. He’s polynomials in the nonlinear term was first introduced by He [30,31]. HPTM presents an accurate methodology to solve many types of linear and nonlinear differential equations [32]. In the present work, we use HPTM to solve the final evolution equation that describes the plasma model and we suppose that the shock, periodic and soliton waves could be produced in the space plasma due to the nonlinear interaction between the streaming electrons and ions coming from the solar wind and Jupiter magnetosphere space plasma [4,9]. This paper is organized as follows: The basic equations, which used to describe the dusty plasma system containing mobile charge positive dust, kappa-distributed ions and electrons, impinge with streaming protons and electrons are presented in Section 2. The evolution equation is derived in Section 3. The HPTM is used to solve the evolution equation in Section 4. Finally, a brief summary is given in Section 5. 2. The fluid model Let us consider homogeneous collisionless five components dusty plasma with positively charged dusty grains, kappa distributed electrons and ions, and streaming protons and electrons

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coming from solar wind. This system is described theoretically by using the fluid model. The dimensionless fluid equations, which governed the motion of dusty fluid are given by the following form [4,9]

∂ Nd ∂ ∂ + Nd ud + ud Nd = 0, ∂t ∂x ∂x

(1)

∂ ud ∂u ∂ Nd ∂φ + ud d + 3σd Nd + = 0, ∂t ∂x ∂x ∂x

(2)

The dimensionless electron streaming fluid equations are given as

∂ Nc ∂ ∂ + Nc uc + uc Nc = 0, ∂t ∂x ∂x

(3)

∂ uc ∂ uc ∂ Nc ∂φ + uc + Hc Nc − μc = 0, ∂t ∂x ∂x ∂x

(4)

and the dimensionless proton streaming fluid equations are given as

∂ Nb ∂ ∂ + Nb ub + ub Nb = 0, ∂t ∂x ∂x

(5)

∂ ub ∂u ∂ Nb ∂φ + ub b + Hb Nb + μb = 0. ∂t ∂x ∂x ∂x

(6)

Here Nd , Nc , Nb , ud , uc , and ub , are the number densities of the positive dust grains, streaming electrons and streaming protons, fluid velocities of the positive dust grains, streaming electrons and streaming protons, respectively, and φ is an electrostatic potential. The densities Nd , Nc , Nb , ne , and ni are normalized by nd0 Zd0 . The length x and the time t are normal2 e2 )1/2 and the ized by the Debye length λD = (KB Te f f /4π nd0 Zd0 inverse of dusty plasma frequency

ω−1 pd

2 e2 )1/2 , = (md /4π nd0 Zd0

while the velocities and electrostatic potential φ are normalized by the dust-acoustic speed Cd = (Zd0 KB Te f f /md )1/2 and KB Teff /e, respectively. Furthermore, σd = Td /Te f f , σc = Tc /Te f f , σb = Tb /Te f f , μb = md /(Zd0 mb ), μc = md /(Zd0 me ), γb = (nb0 Zd0 )/nd0 , δb = γb−1 , Hb = 3μb γb2 σb , Hc = 3μc γc2 σc , γc = nd0 Zd0 )/nc0 , δc = γc−1 , δi = ni0 /(nd0 Zd0 ), δe = ne0 /(nd0 Zd0 ), σi = Te f f /Ti , and σi = Te /Te f f σe = Te /Te f f , where Te(c) is the electrons (electron beam) temperature, n n Ti(b) is the ions (proton beam) temperature, and Te f f = [ Ti0 + Tee0 ]−1 i

is the effective temperature, Zd0 denotes to the unperturbed dust grain charges number. All temperatures in the above system are in the units of Kelvin. The equations from (1) to (6) are considered as six coupled nonlinear partial differential equations in seven unknown functions, and to make this system self consistence the following Poisson equation must be produced

∂ 2φ + Nb − Nc + ni − ne + Nd = 0. ∂ x2



φ ki − 3/2

−(ki −3/2)

,

and

 ne = δe 1 − σe

φ

−(ki −3/2)

ke − 3/2

,.

(9)

It is noted that the above expressions of kappa-distribution are only valid for ke(i) > 3/2, and for the limit ke(i) → ∞, it reduces to the usual Boltzmann form.

3. Derivation an evolution equation According to the reductive perturbation method the following transformations in space and time introduced to investigate the properties of the nonlinear electrostatic dust-acoustic waves (DAW)

ξ = ε 1/2 (x − Vpt )andτ = ε 3/2t ,

(10a)

where ɛ is a small parameter that measures the strength of the nonlinearity and Vp represents the dust-acoustic phase speed which normalized by Cd . We can expand the variables appearing in Eqs. (1)–(9) as

= 0 + ε 1 + ε 2 2 + ε33 + ...,

(10b)

where = [Nd , Nb , Nc , ud , ub , uc , φ ]T , with 0 = [1, δb , δc , 0, ub0 , uc0 , 0]T . Using the above stretching and expansions (10) into Eqs. (1)–(9), then collect the lowest-order inɛ, we obtain

nd1 =



1

Vp2 − 3σd

 φ1 + s1 , ud1 = 

Vp

Vp2 − 3σd

 φ1 + s 2 ,

(11)

(7)

Where ni , and ne are the ions and electrons number densities, which are assumed to follow distribution forms. The Maxwellian distribution is considered as the most used, however, observations and theorem found that the super-thermal (kappa) distribution is more suitable to describe the distributions of the high energy space plasma species. The kappa distribution is used to describe the density of hot electrons to study the electrostatic wave, and weak double layers for un-magnetized and magnetized multicomponents plasma systems, which found in the lunar wake and Aurora [33–40]. So, we use the kappa distribution to describe the density of electrons and ions as:

ni = δi 1 + σi

Fig. 1. The 3D electrostatic potential which yields from the shock like solution atσd = 9.65 ∗ 10−4 ,ke = 2, andki = 3.

(8)

δb μb φ1 + s 3 , (Vp − ub0 )2 − 3δb2 μb σb (Vp − ub0 )μb ub1 = φ1 + s 4 , (Vp − ub0 )2 − 3δb2 μb σb nb1 =

−δc μc

φ1 + s 5 , (Vp − uc0 )2 − 3δc2 μc σc (Vp − uc0 )μc uc1 = φ1 + s 6 , (Vp − uc0 )2 − 3δc2 μc σc nc1 =

(12)

(13)

and from Poisson Eq. (9) and with using Eqs. (11)–(13), we obtained the compatibility condition

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Fig. 2. The shock 2D electrostatic potential at different values of f (a) ions kappa distribution parameter ki and ke = 2, σd = 9.65 ∗ 10−4 , (b)electrons kappa distribution parameter ke and ki = 2, σd = 9.65 ∗ 10−4 .

Fig. 3. The shock 2D electrostatic potential at different values of the ratio between the dust temperature and effective temperatureσ d with ke = ki = 2.

1 δb μb − σe δe − σi δi + Vp2 − 3σd (Vp − ub0 )2 − δb2 Hb +

δc μc = 0, (Vp − uc0 )2 − δc2 Hc

(14)

where si (i = 1, ..., 6 ) are the integration constants, which are depending mainly on the boundary conditions [4]. From the next-

order of ɛ, we obtain a system of equations, which including the second-order perturbed quantities. By solving these equations and eliminating the second perturbed quantities, finally we obtain an evolution equation in the following form

∂ φ1 ∂ φ1 ∂ 3 φ1 ∂ φ1 + B 1 φ1 + B2 + B3 = 0, ∂τ ∂ξ ∂ξ ∂ξ 3 where B1 , B2 , and B3 are given by the following form.

(15)

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Eq. (15) in the following form

Dφ (ξ ,

τ ) + R1 φ (ξ , τ ) + N1 φ (ξ , τ ) = 0,

(16a)

with initial condition as

φ ( ξ , 0 ) = k ( ξ ).

(16b)

∂ is a linear differential operator with respect to τ , Where D = ∂τ 3 ∂ ) is a linear differential operator with reand R1 = (B2 ∂∂ξ 3 + B3 ∂ξ

spect to ξ and N is a nonlinear differential operator, taking Laplace transformation for Eq. (16a) and using the properties of it, yields

£ (φ (ξ ,

τ )) =

1 k (ξ ) − £ ( R1 φ ( ξ , s s

τ ) + N1 φ (ξ , τ )) ,

(17)

taking the Laplace inverse transformation for this equation, we Fig. 4. The 3D electrostatic potential which yields from the periodical solution at atσd = 9.65 ∗ 10−4 ,ke = 2, andki = 3.



     2 3(Vp − uc0 ) − δc2 Hc δc μ2c 3 Vp2 + σd δb μ2b (1−2ki )σi2 δi (1−2ke )σe2 δe B1 = B2 , 3 + 2ke −3 − 2ki −3  3 −  3 +  2 V p − 3 σd (Vp − ub0 )2 − δb2 Hb (Vp − uc0 )2 − δc2 Hc  −1 2Vp 2(Vp − ub0 )δb μb 2(Vp − uc0 )μc B2 =  , 2 +  2 +  2 Vp2 − 3σd (Vp − ub0 )2 − δb2 Hb (Vp − uc0 )2 − δc2 Hc         2 2 2(Vp − ub0 )δb μb h4 + (Vp − ub0 ) + δb2 Hb h3 μb 2h2 + h1 Vp2 + 3σd 2(Vp − uc0 )δc μc h6 + (Vp − uc0 ) + δc2 Hc h5 μc B3 = B2 + + .  2 2  2  2 V p − σd (Vp − ub0 )2 − δb2 Hb (Vp − uc0 )2 − δc2 Hc 3(Vp − ub0 ) − δb2 Hb 2



4. The solution of the evolution equation The HPTM [32] is used to solve the evolution Eq. (15). Firstly we can describe the main steps of this method, we put the evolution

Fig. 5. The periodical 2D electrostatic potential at different values of (a) ions kappa distribution parameter ki and ke = 2, σd = 9.65 ∗ 10−4 , (b)electrons kappa distribution parameter ke and ki = 2, σd = 9.65 ∗ 10−4 .

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Fig. 6. The periodical 2D electrostatic potential at different values of the ratio between the dust temperature and effective temperature withke = ki = 2.

 −£

−1





∞

∂ 3 φ j (ξ , 1 £  j B2 s ∂ξ 3 j=0

+

∞



B1  j F j [φ ]

τ)

∂ φ j (ξ , τ ) + B3 ∂ξ



,

j=0

Comparing the coefficient of like powers of  , the following approximations are obtained

 0 : φ0 (ξ , τ ) = £−1

φ0 ( ξ , 0 ) s 

 1 : φ1 (ξ , τ ) = −£−1 Fig. 7. The 3D electrostatic potential which yields from the soliton solution atσd = 9.65 ∗ 10−4 ,ke = 2, andki = 3.

1



φ (ξ , τ ) = W (ξ , τ ) − £ £ (R1 φ (ξ , τ ) + N1 φ (ξ , τ )) s     ∂ ∂3 ∂φ −1 φ (ξ , 0 ) −1 1 = £ −£ £ B3 + B2 3 φ + B1 φ , s s ∂ξ ∂ξ ∂ξ −1

(18) where W (ξ , τ ) = £ −1 ( k(sξ ) ) represents the term arising from initial conditions. Now, we apply the homotopy perturbation method [32]

φ (ξ , τ ) =

∞

 j φ j ( ξ , τ ),

(19a)

j=0

where  ∈ [0, 1] represent the auxiliary homotopy parameter, and the nonlinear term can be decomposed as

N1 φ (ξ ,

τ) =

∞

 j F j ( ξ , τ ),

(19b)

j=0

with Fj [41] are the He’s polynomials, substituting from (19) into Eq. (18), yield the following equation

j=0

 : φ2 (ξ , τ ) = −£

−1



get

∞

 2

 j φ j (ξ , τ ) = £−1

φ (ξ , 0 ) s

 3 : φ3 (ξ , τ ) = −£−1

,



1 ∂ 3 φ0 ∂ φ0 £ B2 + B3 + B1 F0 ( φ ) s ∂ξ ∂ξ 3

,

1 ∂ 3 φ1 ∂ φ1 £ B2 + B3 − B1 F1 ( φ ) s ∂ξ ∂ξ 3

,







 1 ∂ 3 φ2 ∂ φ2 £ B2 + B3 − B1 F2 ( φ ) , s ∂ξ ∂ξ 3

·······

(20)

where

∂ φ0 ( ξ , τ ) , ∂x ∂ φ1 ( ξ , τ ) ∂ φ0 ( ξ , τ ) F1 (φ ) = φ0 (ξ , τ ) + φ1 ( ξ , τ ) , ∂x ∂x ∂ φ2 ( ξ , τ ) ∂ φ1 ( ξ , τ ) F2 (φ ) = φ0 (ξ , τ ) + φ1 ( ξ , τ ) ∂x ∂x ∂ φ0 ( ξ , τ ) + φ2 ( ξ , τ ) , · · · · · · ·· ∂x F0 (φ ) =

φ0 ( ξ , τ )

(21)

Therefore, the approximate solution of the evolution equation is given by:

φ (ξ , τ ) = nlim ( φ0 ( ξ , τ ) + φ1 ( ξ , τ ) + φ2 ( ξ , τ ) + φ3 ( ξ , τ ) + . . . →∞ + φn ( ξ ,

τ )) .

(22)

The zero-order of the solution can be taken as the initial value of the state variable, which is taken as the solution of the ordinary equation at time equal to zero. For Eq. (15) we are using three different initial solutions as.

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Fig. 8. The soliton 2D electrostatic potential at different values of f (a) ions kappa distribution parameter ki and ke = 2, σd = 9.65 ∗ 10−4 , (b) electrons kappa distribution parameter ke and ki = 2, σd = 9.65 ∗ 10−4 .

Fig. 9. The 2D soliton electrostatic potential at different values of the ratio between the dust temperature and effective temperatureσ d with ke = ki = 2.



+ 4B2C12 sech2 (C1 ξ )−6B2C12 sech4 (C1 ξ ) + B3 sech2 (C1 ξ ) t,

4.1. Dust-acoustic nonlinear shock-like wave In this case the initial shock-like wave will be used as the zero order of solution, which takes the following form

φ0 (ξ , τ ) = φ (ξ , 0 ) = A1 (B1 + tanh (C1 ξ ) ), where A1 =

2B2C12 −B3 B21

(23)

.

By substituting from Eq. (23) into Eqs. (20) and (21), the first order of solution φ 1 (ξ , τ ) yields as

φ1 (ξ , τ ) = −A1C1 A1 B21 sech2 (C1 ξ ) + A1 B31 sech(C1 ξ ) sinh (C1 x )

(24)

By the same manner with using the packages of Mathematica and Maple the rest components obtained. As shown that, the analytical first order solution (24) depending mainly on the dusty plasma parameters of the system under consideration. By drawing the analytical solution, which taking to the fifth order in 3dimension, a shock wave appeared as shown in Fig. 1. The dependence of the width and steepness of the envelope shock wave on the Kappa distribution parameters (ki and ke ) were showing in Fig. 2, whereas the ions kappa parameter ki increasing the width

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and steepness of the envelope shock wave increase, but the width and steepness decreasing with increasing electrons kappa parameter ke . Fig. 3 shows the relation between the electrostatic potential and space at different values of ratio between the dust temperature and effective temperature σ d , where as shown in this figure as σ d increase the steepness will decrease. 4.2. Dust-acoustic nonlinear periodical wave Substitute the following zero order solution (initial periodical wave) for Eq. (15)

  φ0 (ξ , τ ) = φ (ξ , 0 ) = A2 B1 + cos2 (C2 ξ ) , where A2 =

2(4B2C22 −B3 ) B1 +2B21

(25)

.

Substituting by (25) into (20) and (21), yields the first order of solution as

φ1 (ξ , τ ) = 2A2C2 cos (C2 ξ ) sin (C2 ξ )

2  A2 B1 + A2 B1 cos2 (C2 ξ ) − 4B2C22 + B3 t,

(26)

By the same way the rest orders of solution obtained. By drawing the relation between electrostatic potential (which taken to the fifth order), space, and time in 3-dimention, periodical wave enveloped with constant width and amplitude as shown in Fig. 4. Figs. 5 and 6 show 2-dimension periodical waves with constant amplitude and width, and the dependence of the envelope periodical waves on the kappa distribution parameters (ki and ke ) and ratio between the dust temperature and effective temperature σ d . As shown in Fig. 5a), the increasing of both width and amplitude are depending mainly on the value of the ions kappa distribution parameter ki . The effect of both electron kappa distribution parameter ke and σ d are appearing in Figs. 5b and 6, where the width and amplitude of the periodical wave decreased by increasing both ke and σ d . 4.3. Dust-acoustic nonlinear solitary wave The solitary non-linear wave obtained by using the following zero order solution as

φ0 (ξ , τ ) = φ (ξ , 0 ) = A3 (B1 + sech(C3 ξ )),

(27)

Substituting from (27) into the orders of solution (20), yield the first order of solution as

φ1 (ξ , τ ) = A3C3 sinh (C3 ξ ) A3 B21 sech2 (C3 ξ )

+ A3 B1 sech3 (C3 ξ ) sinh (C1 x ) + B2C32 sech2 (C3 ξ )



−6B2C32 sech4 (C3 ξ ) + B3 sech2 (C3 ξ ) t, where A3 =

5B2C32 −B3 B21

(28)

.

Similarly the rest orders obtained. Studying the dependence of the envelope soliton wave on the variation for values of kappa distribution parameters (ki and ke ) and ratio between the dust temperature and effective temperature σ d are shown in Figs. 7–9. 5. Conclusion In this paper the reductive perturbation method was used to derive a non-linear evolution Eq. (15) for system of five component dusty plasma consisting of positive dust, kappa- distributed ions and electrons, impinge with streaming ions and electrons. We obtained a class of solutions represented different kinds of nonlinear waves using a little bit complex mathematical method called Homotopy perturbation transform method (HPTM). By using the Maple and Mathematica, we derived to the fifth order of solution, where we find that by drawing the solution for fifth order

doesn’t change about both the solution for forth order nor the solution for third order. These solutions will be much interesting when we use them to investigate physical phenomena in our case, which represents the interaction of dusty plasma fluid with solar wind particles. We use an available typical plasma parameters for the magnetosphere of Jupiter [1,42–44] to predict the behavior of nonlinear pulses that propagate at Jupiter magnetosphere with Zd0 = 103 , uc0 = 1.2Vc , ub0 = 45Vb , where Vc = (KB Tc /mc )1/2 is electron thermal velocity and Vb = (KB Tb /mb )1/2 is ion thermal velocity, ne0 = 0.08, ni0 = 0.1, nb0 = 5, nc0 = 6, and nd0 = 10−2 , while the integration constants si (i = 1, ..., 6 ) for simplicity are taken as arbitrary constants as s1 = 0.17, s2 = 0.2, s3 = 0.47, s4 = 0.3, s5 = 0.13, ands6 = 0.14. We interested in this studying by showing the effect of both the kappa distribution parameters (ki for ions and ke for electrons) and the ratio between the dust temperature and effective temperature σ d . Where as shown from Figs. (1–9) that both kappa distribution parameter of electron ke and the ratio between the dust temperature and effective temperature σ d are affected positively on width and amplitude of the envelope waves, vice-versa the kappa distribution parameter of ions ki affected negatively on width and amplitude of the envelops waves. References [1] Shukla P K, Mendis D A, Desai T. Advances in dusty plasmas. Singapore: World Scientific; 1997. [2] Verheest F. Waves in dusty space plasmas. Kluwer Academic Publishers; 20 0 0. [3] Shukla P K, Mamun A A. Introduction to dusty plasma physics. Bristol: Institute of Physics; 2002. [4] Tolba R E, Moslem W M, Elsadany A A, El-Bedwehy N A, El-Labany S K. IEEE Trans Plasma Sci 2017;45:9. [5] Djebli M. Phys Plasmas 2003;12. [6] Ahmed S M, Metwa M S, El-Hafeez S A, Moslem W M. Appl Math Inf Sci 2016;10(1):317–23. [7] Abdelghany A M, Abd El-Razek H N, Moslem W M, El-Labany S K. Phys Plasmas 2016;23:062121. [8] Rao N N, Shukla P K, Yu M Y. Planet Space Sci 1990;38:543. [9] Tolba R E, Moslem W M, El-Bedwehy N A, El-Labany S K. Phys Plasmas 2015;22:043707. [10] Roychoudhury R, Maitra S. Phys Plasmas 2002;9:4160. [11] El-Labany S K, Moslem W M, Safi F M. Phys Plasmas 2006;13:082903. [12] Wang Y, Zhou Z, Jiang X, Zhang H, Jiang Y, Ch Hou, Sun X, Qin R. Phys Lett A 2006;355:386. [13] Rahman O. Adv Astrophys 2017;2. [14] Hossen M R, Ema S A, Mamun A A. Commun Theor Phys 2014;62:888–94. [15] Ghai Y, Kaur N, Singh K, Saini N S. Plasma Sci Technol 2018;7:20. [16] Wang Y, Zhou Z, Jiang X, Ni X, Yu Zhang, Shen J, Qian P. Phys Lett A 2009;373:2944. [17] El-Taibany W F, Mushtaq A, Moslem W M, Wadati M. Phys Plasmas 2010;17:034501. [18] Yuan Z, Wang J, Chu M, Xia G, Zheng Z. Phys Rev E 2013;88:042901. [19] El-Bedwehy N A, El-Attafi M A, El-Labany S K. Astrophys Space Sci 2016;299:361. [20] El-Labany S K, Moslem W M, El-Shewy E K, Mowafy A E. Chaos, Solitons Fractals 2005;23:581. [21] Moslem W M. Chaos, Solitons Fractals 2006;28:994. [22] Hirota R. Direct methods in Soliton theory. Berlin: Springer; 1980. [23] Gu CH. Darboux transformation in Soliton theory and its geometric applications. Shanghai: Shanghai Scientific and Technical Publishers; 1999. [24] Engui F, Hongqing Z. Phys Lett A 1998:246. [25] Zuntao F, Shikuo L, Qiang Z. Phys Lett A 2001;290:72–6. [26] Engui F. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 20 0 0;277:212–18. [27] He J H, Xu W H. Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 20 06;30:70 0–8. [28] Wazwaz A M. Partial differential equations: methods and applications. Taylor & Francis; 2002. [29] Ghorbani A, Saberi-Nadjfi J. Int J Nonlinear Sci Numer Simul 2007;8:229–32. [30] Ghorbani A. Chaos, Solitons Fractals 2009;39:1486–96. [31] He J H. Application of homotopy perturbation method to nonlinear wave equations. Chaos, Soliton Fractals 2005:26. [32] Madani M, Fathizadeh M. Homotopy perturbation algorithm using Laplace transformation. Nonlinear Sci Lett A 2010;1:263–7. [33] Lakhina GS, Singh SV, Rubia R, Sreeraj T. A review of nonlinear fluid models for ion-and electron-acoustic solitons and double layers: application to weak double layers and electrostatic solitary waves in the solar wind and the lunar wake. Phys Plasmas 2018;25(8):080501. [34] Sreeraj T, Singh SV, Lakhina GS. Electrostatic waves driven by electron beam in lunar wake plasma. Phys Plasmas 2018;25(5):052902.

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