Modified KdV–Zakharov–Kuznetsov dynamical equation in a

Pramana – J. Phys. (2018) 91:26 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-018-1595-0

Modified KdV–Zakharov–Kuznetsov dynamical equationin a homogeneous magnetised electron–positron–ion plasmaand its dispersive solitary wave solutions

ABDULLAH1, ALY R SEADAWY2,3,∗ and JUN WANG1,∗

1Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu,People’s Republic of China2Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia3Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt∗Corresponding authors. E-mail: [email protected]; [email protected]

MS received 30 October 2017; revised 28 December 2017; accepted 16 January 2018;published online 12 July 2018

Abstract. Propagation of three-dimensional nonlinear ion-acoustic solitary waves and shocks in a homogeneousmagnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to findion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsovequation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magneticfields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wavesolutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, brightand dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential andrational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. Theseresults demonstrate the efficiency and precision of the method that can be applied to many other mathematical andphysical problems.

Keywords. Modified extended mapping method; three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation; homogeneous magnetised electron–positron–ion plasma; ion-acoustic solitary waves;electrostatic field potential; electric and magnetic fields; quantum statistical pressure; graphical representation.

PACS Nos 02.30.Jr; 47.10.A−; 52.25.Xz; 52.35.Fp

1. Introduction

In a magnetised electron–positron plasma, consist-ing of equal amount of cool and hot components ofeach species, the nonlinear three-dimensional modifiedKorteweg– de Vries–Zakharov–Kuznetsov (mKdV-ZK)equation governs the behaviour of weakly nonlinearion-acoustic waves. The ion-acoustic wave, which isan ion time-scale phenomenon, has been studied in atwo-component electron–ion plasma and both the asso-ciated linear [1–3] and nonlinear [4–6] dynamics havebeen investigated. In [7], the ion-acoustic wave wasstudied in an unmagnetised three-component electron–positron–ion plasma. By assuming that both electronsand positrons are hot and they obey the Boltzmann dis-tribution, the linear dispersion relation is obtained. Theelectron–positron plasma has a significant role in com-prehending plasmas in the early Universe [8,9], in active

galactic nuclei [10], in pulsar magnetosphere [11,12]and in the solar atmosphere [13].

The reaction of the plasma changes meaningfullywhen positrons are introduced into an electron–ionplasma. The positrons can be used to clarify particletransport in tokamaks and, as they have sufficient life-time, the two-component electron–ion plasma becomesa three-component electron–positron–ion plasma [14,15]. Most of the astrophysical plasmas contain ions withelectrons and positrons and the study of linear and non-linear wave propagation is important in such plasmas.Therefore, electron–positron–ion plasma has attractedthe attention of researchers [16–19] in the last decade.

To date, many mathematical models have been derivedto describe the dynamics of plasma [20–25]. ThemKdV–ZK equation governing the oblique propaga-tion of nonlinear electrostatic modes has been derived,and the soliton amplitudes were studied as a function

http://crossmark.crossref.org/dialog/?doi=10.1007/s12043-018-1595-0&domain=pdf

26 Page 2 of 13 Pramana – J. Phys. (2018) 91:26

of plasma parameters such as temperatures and parti-cle number densities [26]. The Zakharov–Kuznetsov–Burgers (ZKB) equation for the dust-ion-acoustic wavesin dusty plasmas was formulated and the nonlinearwave solutions were obtained [27]. Recently, soli-tary wave solutions of (3 + 1)-dimensional nonlinearextended Zakharov–Kuznetsov and mKdV-ZK equa-tions were obtained and their applications studied[28]. Different forms of new exact solutions of gen-eralised coupled Zakharov–Kuznetsov equations wereconstructed [29]. Seadawy [30] investigated the stabil-ity of solitary travelling wave solutions of the mKdV-ZKequation to three-dimensional long-wavelength per-turbations and found electrostatic field potential andelectric field in the form travelling wave solutions.Furthermore, he studied the ion-acoustic solitary wavesolutions of two-dimensional nonlinear Kadomtsev–Petviashvili–Burgers equation [31–39].

In this paper, we analyse the propagation of three-dimensional nonlinear ion-acoustic solitary waves andshocks in a homogeneous magnetised electron–positron–ion plasma. Modified extended mapping method isintroduced to find ion-acoustic solitary wave solutionsof three-dimensional modified Korteweg–deVries–Zakharov–Kuznetsov equation. As a result, elec-tric field potential, electric and magnetic fields andelectron fluid pressures are obtained, which are alsoexpressed graphically.

2. Description of the modified extended mappingmethod

Consider a general non-linear partial differential equa-tion in 3 + 1 independent variables x, y, z and t as

F(ϕ, ϕt , ϕx , ϕy, ϕz, ϕt x , ..., ϕxx , ...

) = 0, (1)

where F is a polynomial function in ϕ(x, y, z, t) andits partial derivatives, containing nonlinear terms andhighest-order derivatives. The following are the mainsteps of the method:

Step 1. By using travelling wave transformationϕ(x, y, z, t) = ϕ(θ), where θ = kx + ly +mz+ωt and k, l,m and ω are constants, eq. (1)is reduced to the following ordinary differentialequation:

P(ϕ, ϕ′, ϕ′′, ϕ′′′, ...), (2)

where P is a polynomial in ϕ(θ) and its first-and higher-order derivatives.

Step 2. Assume that the following is the solution of eq.(2).

ϕ(θ) =n∑

i=0

aiGi (θ) +

−n∑

i=−1

b−iGi (θ)G ′(θ),

(3)

where a0, a1, ..., an, and b1, b2, ..., bn are arbi-trary constants and the values ofG(θ) andG ′(θ)

satisfy

G ′(θ) =√√√√

6∑

i=0

βiGi (θ), (4)

where βi ’s are constants to be determined suchthat βn �= 0.

Step 3. By balancing the highest-order derivative termand the nonlinear term appearing in eq. (2),determine the positive integer n of eq. (3).

Step 4. Substitute eq. (3) along with eq. (4) into eq.(2), then collecting all the coefficients of thesame power Gi (θ) where (i = 0, 1, ..., n),and equating them to zero, a system of alge-braic equations is obtained. Solving this systemof algebraic equations, give the values of allparameters and constants.

Step 5. Substitute all the parameter values obtained inthe previous step and ϕ(θ) into eq. (3), to obtainthe solutions of eq. (1).

3. Application of the described method tothree-dimensional modified Korteweg–deVries–Zakharov–Kuznetsov equation

Consider the wave propagation in a three-dimensionalhomogeneous magnetised, electron–positron plasma,consisting of equal amount of cool and hot compo-nents of each species. The fluid governing equations thatdescribe the dynamics of the cooler adiabatic species aregiven by (see [30])

∂ni∂t

+ ∇ · (niui ) = 0,

∂ui∂t

+ (ui · ∇)ui = − 1

nim∇pi − qi

m∇ϕ + �i ui ∧ ex ,

∂pi∂t

+ ui · ∇pi + γi pi (∇ · ui ) = 0,

ε∇2ϕ +∑

i

niqi +∑

j

N jq j exp

(−q jϕ

kTj

)= 0,

ne = Hp exp

(eϕ

kHe

), np = Hp exp

(−eϕ

kHe

). (5)

Pramana – J. Phys. (2018) 91:26 Page 3 of 13 26

First equation is the continuity equation, second andthird equations are equation of motion and adiabaticpressure equation, respectively, and the last equationof the system is the Poisson equation, where ui andpi are the fluid velocities and pressures, qi and q jare the charges of the cool and hot species, m is thecommon mass of the electrons and the positrons, γi isthe adiabatic compression indices, the gyrofrequencies�i = qi B0/m, B0 is the ambient magnetic field, thedynamics of the cooler adiabatic species, denoted bythe subscript i , ε is a small parameter proportional tothe amplitude of the perturbation. He and Hp are hotelectrons and positrons with equal temperatures andequilibrium densities, ne and np are the densities ofhot electrons and positrons, ϕ is the electrostatic poten-tial. Using singular perturbation method, Seadawy [30]derived the mKdV-ZK equation as

ϕt + α1ϕ2ϕx + α2ϕxxx + α3ϕxϕyy + α3ϕxϕzz = 0, (6)

where α1, α2 and d are given by

α1 = 1

A1, α2 = A2

A1, α3 = A3

A1,

A1 = 2∑

i

ω2pi (V −Ui0)

((V −Ui0)2 − ν2

T i

)2 ,

A2 =∑

i

ω2piq

2i

(15(V−Ui0)

4+E1(V−Ui0)2ν2

T i+E2ν4T i

)

2m2((V−Ui0)2−ν2

T i

)5

−∑

j

q j

2λ2Djk

2T 2j

,

A3 = 1 +∑

i

ω2pi (V −Ui0)

4

�2i

((V −Ui0)2 − ν2

T i

)2 , (7)

where ωpi =√Niq2

i /εm is the plasma frequency,

λDj =√

εkTj/N jq2j is the Debye length and νT i =√

γi pi/Nim is the thermal velocity of the species i .Now we apply the modified extended mapping method

to solve the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. Consider thetravelling wave transformation ϕ(x, y, z, t) = ϕ(θ),θ = kx+ly+mz+ωt , where k, l,m and ω are numbersand frequency to be determined later. Then mKdV–ZKequation becomes

ωϕ′ +α1kϕ2ϕ′ +α2k

3ϕ′′′ +α3kl2ϕ′′′ +α3km

2ϕ′′′ = 0.

(8)

Balancing the highest-order derivative term ϕ′′′ and non-linear term ϕ2ϕ′ appearing in (8) we obtain n = 2.

Putting n = 2 in eq. (3), we get the solution of eq. (8)in the form

ϕ(θ) = a0 + a1G(θ) + a2G2(θ)

+(

b1

G(θ)+ b2

G2(θ)

)G ′(θ). (9)

Now following the procedure of the modified extendeddirect algebraic mapping method, substituting eq. (9)and its derivatives into eq. (8) and collecting all the termswith the same power of Gi (θ), where i = 0, 1, ..., n, weget a system of algebraic equations. By solving theseequations, we get different sets of parameter values. Byputting these parameter values in eq. (9), different fam-ilies of solutions of eq. (8) are obtained, as discussedbelow.

Family 1. β0 = β1 = β3 = β5 = β6 = 0

In this family, the following different sets of parametersare obtained:

a0 = 0, a1 = ±b1√

β4, a2 = 0, b2 = 0,

m = ±√

−2α1b21 − 3α2k2 − 3α3l2

3α3,

ω = −1

3α1β2b

21k, (10)

where

β4 > 0 and−2α1b2

1 − 3α2k2 − 3α3l2

3α3> 0.

Substituting these parameter values given in eq. (10), foronly positive values ofa1 andm, along with two differentvalues of ϕ, in eq. (9), the following two solutions insimplified forms are obtained:

ϕ11(θ) = √−β2b1sech(√

β2θ)

−√β2b1 tanh(

√β2θ), β2 > 0, (11)

ϕ12(θ) = √−β2b1(tan(√−β2θ)

+ sec(√−β2θ)), β2 < 0. (12)

The first solution is not valid and so we consider onlythe second solution. Similarly, solutions are obtained forother sets of parameters.

The electric and magnetic fields are determined bythe position and motion of the electrons and positronsas they move along their orbits in a homogeneous mag-netised, electron–positron plasma. The electric field isthe gradient of the scalar function ‘ϕ’, called the electro-static potential or voltage. The electric field ‘ �E’ points


from high to low electric potential regions. Mathemati-cally, the electric field is expressed as

�E = −∇ϕ = −∂ϕ

∂xx − ∂ϕ

∂yy − ∂ϕ

∂zz. (13)

The electric fields of the electric potential ϕ12 is formu-lated as

�E12 = b1 sec2 (√−β2θ)(

β2 −√

β22 sin

(√−β2θ))

· (kx + l y + mz), β2 < 0. (14)

The relation of electric and magnetic fields is given bythe Maxwell–Faraday equation as

∇ × �E = −∂ �B∂t

. (15)

Using Maxwell–Faraday eq. (15), the magnetic field isformulated as

�B12 = β2b1(k − m)(√−β2 sin

(√−β2θ)

√−β2ω

+√−β2

)sec2

(√−β2θ)

√−β2ω

· (−l x + (k + m)y − l z), β2 < 0. (16)

The graphical representations of the solution and itselectric and magnetic fields are shown in figure 1.

The electron fluid pressure is modelled as P = P(ne),where ne is the electric number density. The relationbetween the electron fluid pressure P and the electricnumber density ne is given as

P = mev2F

3n20

e3ne , (17)

where n0 is the equilibrium density for both electronsand ions, v2

F represents the electrons Fermi velocity, meis the mass of the electron. Using this formula, the quan-tum statistical pressure of the electron is obtained as

P12 = mev2F

3n20

exp(3√−β2b1

(tan

(√−β2θ)

+ sec(√−β2θ

))). (18)

By the same method, electric potentials can be obtainedand their electric fields, magnetic fields and electronfluid pressures can be formulated for the other sets ofparameter values.

Family 2. β0 = 8β22/27β4, β1 = β3 = β5 = 0,

β6 = β24/4β2

The following are the sets of parameter values in thisfamily:

Figure 1. (a) The periodic solitary wave solution ϕ12, (b) itselectric field �E12 and (c) the magnetic field �B12.

Set 1.

a0 = a1 = 0, a2 = ± β4b1

2√

β2, b2 = 0,

m = ±√

−α1b21 − 6α2k2 − 6α3l2

6α3,

ω = −1

3α1β2b

21k, (19)

where


β2 > 0,−α1b2

1 − 6α2k2 − 6α3l2

6α3> 0.

Set 2.

a0 = a2β2

β4, a1 = 0, b1 = b2 = 0,

m = ±√

−a22α1β2 − 6α2β

24k

2 − 6α3β24 l

2

6α3β4,

ω = −a22α1β

22k

3β24

, (20)

where

−a22α1β2 − 6α2β

24k

2 − 6α3β24 l

2

6α3β4> 0.

For the first set of parameter values, there are two solu-tions which are given below, for only positive values ofa2 and m, and β4 < 0.

ϕ21(θ) =√

3√

β2b1ε tan(γ )

2 cos(2γ ) − 1− 4

√β2b1 cot2(γ )

3(cot2(γ ) − 3

) , (21)

ϕ22(θ) =√

β2b1(4 tan2(γ ) + √3ε(6 csc(2γ ) − γ1γ

22 + csc(γ ) sec(γ )γ 2

1 ))

3(γ 2

1 + 3) , (22)

where (√

β2θε/√

3) = γ, tanh(γ ) = γ1 and sech(γ )

= γ2. Using formulas (13), (15) and (17), the electricfield, magnetic field and electron fluid pressure for thesolution ϕ21 are formulated as

�E21 =β2b1ε sec2(γ )(−6ε cos(2γ ) + 3ε cos(4γ ) + 8√

3 sin(γ ) cos3(γ ))

3 (1 − 2 cos(2γ ))2 · (kx + l y + mz

), (23)

�B21 =β2b1ε(k − m)(4√

3 sin(2γ ) + 12ε cos(2γ ) + 9ε sec2(γ )−24ε

3ω(1 − 2 cos(2γ ))2 · (−l x+(k + m)y − l z), (24)

P21 =mev2F

3n20

exp

(3√

3√

β2b1ε tan(γ )

2 cos(2γ ) − 1

−4√

β2b1 cot2(γ )(cot2(γ ) − 3

)

)

. (25)

The graphical representation of the electric potentialand its electric and magnetic fields are shown in figure 2.

Similarly, electric field, magnetic field and electronfluid pressure can be formulated and their graphs can bedrawn for the second solution ϕ22.

There are four solutions for the second set of param-eter values (20). Two of them are mentioned now, theother two are similar to the first two solutions ϕ21 andϕ22, respectively.

ϕ23(θ) = a2β2

β4−

8a2β2 tanh2(√−β2θε√

3

)

3β4

(tanh2

(√−β2θε√3

)+ 3

) ,

β2 < 0, β4 > 0, (26)

ϕ24(θ) = a2β2

β4+

8a2β2 cot2(√

β2θε√3

)

3β4

(coth2

(√β2θε√

3

)+ 3

) ,

β2 < 0, β4 > 0. (27)

The electric field, magnetic field and electron fluid pres-sure are given below:

�E23 = −8a2 (−β2)

3/2ε sinh(

2√−β2θε√

3

)

√3β4

(2 cosh

(2√−β2θε√

3

)+ 1

)2

· (kx + l y + mz), β2 < 0, β4 > 0, (28)

�B23 =8a2β

22ε(k − m) sinh

(2√−β2εθ√

3

)

√3√−β2β4ω

(2 cosh

(2√−β2εθ√

3

)+ 1

)2

· (l x − (k + m)y + l z),

β2 < 0, β4 > 0, (29)

P23 = mev2F

3n20

exp

⎛

⎝3a2β2

β4

−8a2β2 tanh2

(√−β2θε√3

)

β4

(tanh2

(√−β2θε√3

)+ 3

)

⎞

⎠,

β2 <0, β4 >0. (30)


Figure 2. (a) The periodic solitary wave solution ϕ21, (b) itselectric field �E21 and (c) the magnetic field �B21.

and the graphical representations for the electric poten-tial ϕ23 are given in figure 3.

By the same way, electric field, magnetic field andelectron fluid pressure can be obtained for the elec-tric potential ϕ24, and their graphs can be drawn.Electric potentials can be obtained and their elec-tric and magnetic fields and electron fluid pressurescan be derived for the other sets of parametervalues.

Figure 3. (a) The dark soliton travelling wave solution ϕ23,(b) its electric field �E23 and (c) the magnetic field �B23.

Family 3. β0 = β1 = β3 = β5 = 0

Following the procedure given in step 4, different sets ofparameter values are obtained, which are given below:

Set 1.

a0 = a1 = 0, a2 = ±b1√

β6, b2 = 0,


m = ±√

−α1b21 − 6α2k2 − 6α3l2

6α3,

ω = −1

3α1β2b

21k, (31)

where

β6 > 0,−α1b2

1 − 6α2k2 − 6α3l2

6α3> 0.

Set 2.

a0 = a2β4

4β6, a1 = b1 = b2 = 0,

m = ±√

−a22α1 − 24α2β6k2 − 24α3β6l2

26α3β6,

ω = a22α1

(8β2β6 − 3β2

4

)k

48β26

, (32)

where

−a22α1 − 24α2β6k2 − 24α3β6l2

26α3β6> 0.

Following the procedure given in step 5 and using thefirst set of parameter values, 14 solutions are obtainedas mentioned below.

ϕ31 =√

β2b1sech2(γ3)(β2

4 (− sinh(2γ3)) + β2β6((

ε2 + 1)

sinh(2γ3) + 2ε cosh(2γ3)) − 2

√β2

√β6β4

)

2(β2

4 − β2β6(ε tanh(γ3) + 1)2) (33)

ϕ32 =√

β2b1csch2(γ3)(β2

4 (− sinh(2γ3)) + β2β6((

ε2 + 1)

sinh(2γ3) + 2ε cosh(2γ3)) + 2

√β2

√β6β4

)

2(β2

4 − β2β6 (ε coth(γ3) + 1)2) (34)

where√

β2θ = γ3 and β2 > 0. The electric field andmagnetic field and the electron fluid pressure for theelectric potential ϕ31 are formulated below and theirgraphs are drawn in figure 4.

�E31 = β2γ25 b1

(β2

4 − β2β6 (γ4ε + 1)2)2

×(−β22β2

6 (ε2 − 1)(γ4ε + 1)2

+2β3/22 β4β

3/26 (γ4 + ε)(γ4ε + 1) − β2β

24β6

× (γ 2

5 ε (sinh(2γ3) + ε) + 2)

−2√

β2β34

√β6γ4 + β4

4

) · (kx + l y + mz

),

(35)

�B31 = 2β2b1(k − m)

ω(2β2

4γ 29 − 2β2β6 (γ12ε + γ9)

2)2

×(

2β3/22 β4β

3/26

(γ10

(ε2 + 1

) + 2γ11ε)

−2β22β2

6

(ε2 − 1 (γ12ε + γ9)

2

−2β2β24β6

(γ10ε + γ11 + ε2 + 1

)

−2√

β2β34

√β6γ10 + 2β4

4γ 29

)

· (−l x + (k + m)y − l z), (36)

P31 = mev2F

3n20

exp

(3√

β2b1γ25

(β2

4 (−γ10) + β2β6((

ε2 + 1)γ10 + 2ε cosh(2γ3)

) − 2√

β2β6β4)

2(β2

4 − β2β6 (ε tanh (γ3) + 1)2)

)

, (37)

where tanh(γ3) = γ4, sech(γ3) = γ5, cosh(γ3/2) =γ6, sinh(γ3) = γ7, cosh(γ3) = γ8, sinh(γ3/2) =γ9, sinh(2γ3) = γ10.

Similarly, electric field, magnetic field and electronfluid pressure can be obtained and their graphs (see fig-ure 5) can be drawn for the second solution ϕ32.

ϕ33 = b1(√

β2√

δ1ε sinh(2√

β2θ) − 2β2

√β6

)

β4 − √δ1ε cosh

(2√

β2θ) ,

β2 > 0, δ1 > 0, (38)

ϕ34 = −b1(√

β2√

δ1ε sin(2√

β2θ) + 2β2

√β6

)

β4 − √δ1ε cos

(2√

β2θ) ,

β2 < 0, δ1 > 0, (39)

ϕ35 = b1(√

β2√−δ1ε cosh

(2√

β2θ) − 2β2

√β6

)

β4 − √−δ1ε sinh(2√

β2θ) ,

β2 > 0, δ1 < 0, (40)

ϕ36 = b1(√−β2

√δ1ε cos

(2√−β2θ

) − 2β2√

β6)

β4 − √δ1ε sin

(2√−β2θ

) ,

β2 < 0, δ1 > 0, (41)


Figure 4. (a) The soliton-like solution ϕ31, (b) its electricfield �E31 and (c) the magnetic field �B31.

ϕ37 = −√

β2b1sech2(√

β2θ) (

β4 sinh(2√

β2θ) + 2

√β2β6ε cosh

(2√

β2θ) + 2

√β2

√β6

)

2(2√

β2β6ε tanh(√

β2θ) + β4

) , β2 > 0 (42)

ϕ38 = b1 sec2(√−β2θ

) (√−β2β4 sin(2√−β2θ

) − 2(√−β2

√−β2β6ε cos(2√−β2θ

) + β2√

β6))

2(2√−β2β6ε tan

(√−β2θ) + β4

) , β2 < 0, (43)

Figure 5. (a) The soliton-like travelling wave solution ϕ33,(b) its electric field �E33 and (c) the magnetic field �B33.


ϕ39 = b1(−√−β2β4γ11 + β2

√β6γ

212 − 2

√−β2√

β2β6εγ11 coth(√

β2θ) + √

β2√

β2β6εcsch2(√

β2θ))

2√

β2β6ε coth(√

β2θ) + β4

β2 > 0, (44)

where coth(√−β2θ) = γ11, csch(

√−β2θ) = γ12.

ϕ310 = −b1 csc2(√−β2θ

) (√−β2β4 sin(2√−β2θ

) + 2√−β2

√−β2β6ε cos(2√−β2θ

) + 2β2√

β6)

2(2√−β2β6ε cot

(√−β2θ) + β4

) ,

β2 < 0, (45)

ϕ311 =√

β2b1εsech2(√

β2θ2

)

4ε tanh(√

β2θ2

)+ 4

−β2

√β6b1

(ε tanh

(√β2θ2

)+ 1

)

β4,

β2 > 0, δ1 = 0, (46)

ϕ312 = −β2

√β6b1

(ε coth

(√β2θ2

)+ 1

)

β4

−√

β2b1εcsch2(√

β2θ2

)

4ε coth(√

β2θ2

)+ 4

,

β2 > 0, δ1 = 0, (47)

ϕ313 =√

β2b1

(εe4

√β2θε − 16

√β2

√β6e2

√β2θε − 16β2

4ε + 64β2β6ε)

8β4e2√

β2θε − e4√

β2θε − 16β24 + 64β2β6

, β2 > 0, (48)

ϕ314 =b1

(16β2

√β6e2

√β2θε + √

β2ε)

1 − 64β2β6e2√

β2θε,

β2 > 0, β4 = 0. (49)

The electric field, magnetic field and the electron fluidpressure for the electric potential ϕ33 are obtained asbelow and their graphs (see figure 5) are drawn.

�E33 = 2β2b1√

δ1ε(2√

β2√

β6 sinh(2√

β2θ) − β4 cosh

(2√

β2θ) + √

δ1ε)

(β4 − √

δ1ε cosh(2√

β2θ))2 .

(kx + l y + mz

), (50)

�B33 = 2β2b1√

δ1ε(k − m)(2√

β2√

β6 sinh(2√

β2θ) − β4 cosh

(2√

β2θ) + √

δ1ε)

ω(β4 − √

δ1ε cosh(2√

β2θ))2 .

(−l x + (k + m)y − l z),

(51)

P33 = mev2F

3n20

×exp

(3b1

(√β2

√δ1ε sinh

(2√

β2θ) − 2β2

√β6

)

β4 − √δ1ε cosh

(2√

β2θ)

)

,

(52)

for β2 > 0, δ1 > 0, where δ1 = β24 − 4β2β6, ε = ±1.

Electric fields, magnetic fields and the electron fluidpressures can be formulated for other solutions ϕ34 toϕ314 and their graphs are drawn (see figure 5). Similarly,solutions can be obtained for second set of parametervalues.

Family 4. β0 = β1 = β5 = β6 = 0

In this family, the following different sets of parame-ter values are obtained by following the procedure instep 4.


Set 1.

a0 = a1β3

4β4, a2 = b1 = b2 = 0,

m = ±√

−a21α1 − 6α2β4k2 − 6α3β4l2

6α3β4,

ω = a21α1

(8β2β4 − 3β2

3

)k

48β24

, (53)

where

−a21α1 − 6α2β4k2 − 6α3β4l2

6α3β4> 0.

Set 2.

a0 = 0, a1 = ±b1√

β4, a2 = b2 = 0,

m = ±√

−2α1b21 − 3α2k2 − 3α3l2

3α3,

ω = −1

3α1β2b

21k, (54)

where

β4 > 0,−2α1b2

1 − 3α2k2 − 3α3l2

3α3.

Using the first set of parameter values and following theprocedure given in step 5, we obtained six solutions asgiven below.

ϕ41 = a1β3

4β4+ 2a1β2sech

(√β2θ

)

√δ2 − β4sech

(√β2θ

) , δ2 > 0. (55)

Using formulas (13), (15) and (17), the electric field,magnetic field and electron fluid pressure for the solu-tion ϕ41 are formulated and their graphs (see figure 6)are drawn.

�E41 = 2a1β3/22

√δ2 sinh

(√β2θ

)

(β4 − √

δ2 cosh(√

β2θ))2 · (

kx + l y + mz),

δ2 > 0, (56)

�B41 = 2a1β3/22

√δ2(k − m) sinh

(√β2θ

)

ω(β4 − √

δ2 cosh(√

β2θ))2

· (−l x + (k + m)y − l z), δ2 > 0, (57)

P41 = mev2F

3n20

exp

(3a1β3

4β4+ 6a1β2sech

(√β2θ

)

√δ2 − β4sech

(√β2θ

)

)

,

δ2 > 0, (58)

ϕ42 = a1β3

4β4− 2a1β2sech

(√β2θ

)

β4sech(√

β2θ) + √

δ2, δ2 > 0, (59)

Figure 6. (a) The bright and dark solitary wave solution ϕ41,(b) its electric field �E41 and (c) the magnetic field �B41.

ϕ43 = a1β3

4β4+ 2a1β2csch

(√β2θ

)

√−δ2 − β4csch(√

β2θ) , δ2 < 0,

(60)

ϕ44 = a1β3

4β4− 2a1β2csch

(√β2θ

)

β4csch(√

β2θ) + √−δ2

, δ2 < 0,

(61)


ϕ45 = a1β3

4β4−

a1β2

(1 ± tanh

(√β2θ2

))

β4, δ2 = 0,

(62)

ϕ46 = a1β3

4β4−

a1β2

(1 ± coth

(√β2θ2

))

β4, δ2 = 0.

(63)

The electric field, magnetic field and electron fluid pres-sure for the solution ϕ42 are formulated and their graphsare drawn (see figure 7).

�E42 = − 2a1β3/22

√δ2 sinh

(√β2θ

)

(√δ2 cosh

(√β2θ

) + β4)2

· (kx + l y + mz

), δ2 > 0, (64)

�B42 =2a1β3/22

√δ2(k − m) sinh

(√β2θ

)

ω(√

δ2 cosh(√

β2θ) + β4

)2

· (l x − (k + m)y + l z

), δ2 > 0, (65)

P42 =mev2F

3n20

exp

(3a1β3

4β4− 6a1β2sech

(√β2θ

)

β4sech(√

β2θ) + √

δ2

)

,

δ2 > 0, (66)

where β2 > 0, δ2 = β23 − 4β2β4. Electric fields,

magnetic fields and the electron fluid pressures can beformulated for other solutions and their graphs can bedrawn. Similarly, solutions can be obtained for secondset of parameter values.

4. Results and discussion

The solutions obtained by the proposed method are dif-ferent from those obtained by other researchers becauseof the following reasons:

• Our supposed solution (3) has a structure differentfrom other methods and has different kinds of param-eters.

• By choosing different values of βi ’s (i = 0, 1, ..., 6),eq. (4) has many types of special solutions in theform of trigonometric, hyperbolic, exponential andrational functions.

But still some of our solutions have similarities with thefollowing points:

• The solution ϕ33, ϕ35 and ϕ45 have similarities withthe second, third and first solutions, respectively, ofCase 1 in [30].

• Solutions ϕ41 and ϕ42 have similarities with thesolutions u11 and u12, solutions ϕ43 and ϕ44 have

Figure 7. (a) The bright soliton solution ϕ42, ((b) its electricfield �E42 and (c) the magnetic field �B42.

similarities with the solutions u13 and u14, solutionsϕ45 and ϕ46 have similarities with the solutions u15and u16, and ϕ314 has similarities with the solutionsu24, respectively, of page 903 in [28].


The remaining of our solutions are new and has not beenformulated before.

The electric fields in three dimensions are formulatedin [30], the electric and magnetic fields and electron fluidpressure are formulated in [27] (not shown graphically),the electric field in two dimensions are formulated andpresented graphically in [31]. In this paper, we for-mulated the electrostatic field potential, electric andmagnetic fields and fluid pressure in two dimensionsand showed the results graphically in three dimensionsfor slightly different parameter values. For example, oneset of parameter values is: α1 = 1, α2 = −1.5, α3 =2, a2 = 2, β2 = −0.5, β4 = 1, β6 = 1, d1 =2, k = 1.7, l = 1.2, y = −2, z = −3, ε =1, t = 2. By changing the sign of some of the val-ues, the direction of the electric and magnetic fieldschange.

This discussion shows the effectiveness and power ofthe new method ‘modified extended mapping method’.This method can be applied to many more nonlinearpartial differential equations.

5. Conclusion

In this paper, we analysed propagation of three-dimens-ional nonlinear ion-acoustic solitary waves and shocksin a homogeneous magnetised electron–positron–ionplasma. The electron–positron plasma has an importantrole in comprehending plasmas in the early Universe,in active galactic nuclei, in pulsar magnetosphere andin solar atmosphere. The modified extended mappingmethod was introduced to find different kinds of ion-acoustic solitary wave solutions of three-dimensionalmodified Korteweg–de Vries–Zakharov–Kuznetsovequation, which is important for understanding manyspace and astrophysical phenomena. As a result, differ-ent kinds of exact travelling wave solutions, electric fieldpotential, electric and magnetic fields and quantum sta-tistical pressures are obtained which show the reliabilityand effectiveness of the method.

The obtained solitary wave solutions are in the formof hyperbolic, trigonometric, exponential and ratio-nal functions which are also expressed graphically.These solutions are important in different branches ofphysics and other areas of applied sciences and canhelp researchers to study and understand the physicalinterpretation of the system. Many higher-order nonlin-ear equations arising in plasma, mathematical physics,engineering, hydrodynamics and other areas of appliedsciences can also be solved by this powerful, reliableand capable method.

Acknowledgements

This work was supported by NSF of China (Grants11571140, 11671077, 11371090), Fellowship of Out-standing Young Scholars of Jiangsu Province (BK2016-0063), NSF of Jiangsu Province (BK20150478).

References

[1] N A Krall and A W Trivelpiece, Principle of plasmaphysics (McGraw-Hill, New York, 1973) p. 106

[2] S Ichimaru, Basic principles of plasma physics, A sta-tistical approach (Benjamin, New York, 1973) p. 77

[3] R J Goldston and P H Rutherford, Introduction to plasmaphysics (Institute of Physics Publishing, Bristol, 1995)p. 363

[4] A Hasegawa, Plasma instabilities and nonlineareffects (Springer, New York, 1975) p. 34

[5] R C Davidson, Methods in nonlinear plasma the-ory (Academic, New York, 1972) p. 15

[6] R A Truemann and W Baumjohann, Advanced spaceplasma physics (Imperial College, London, 1997) p.243

[7] S I Popel, S V Vladimirov and P K Shukla, Phys. Plas-mas 2, 716 (1995)

[8] M J Rees, The very early Universe edited by G W Gib-bons, S W Hawking and S Siklas (Cambridge UniversityPress, Cambridge, 1983)

[9] W Misner, K S Throne and J A Wheeler, Gravita-tion (Freeman, San Francisco, 1973) p. 763

[10] H R Miller and P J Witter, Active galacticnuclei (Springer, Berlin, 1987) p. 202

[11] P Goldreich and W H Julian, Astrophys. J. 157, 869(1969)

[12] F C Michel, Rev. Mod. Phys. 54, 1 (1982)[13] E Tandberg-Hansen and A G Emshie, The physics of

solar flares (Cambridge University Press, Cambridge,1988) p. 124

[14] C M Surko, M Leventhal, W S Crane, A Passner and FWysocki, Rev. Sci. Instrum. 57, 1862 (1986)

[15] C M Surko and T Murphy,Phys. Fluids B 2, 1372 (1990)[16] Y N Nejoh, Phys. Plasmas 3, 1447 (1996)[17] H Kakati and K S Goswami, Phys. Plasmas 5, 4229

(1998)[18] H Kakati and K S Goswami, Phys. Plasmas 7, 808

(2000)[19] Q Haque, H Saleem and J Vranjes, Phys. Plasmas 9, 474

(2002)[20] A R Seadawy, Comput. Math. Appl. 67, 172180 (2014)[21] A R Seadawy, Phys. Plasmas 21, 052107 (2014)[22] A H Khater, D K Callebaut, W Malfliet and A R

Seadawy, Phys. Scr. 64, 533 (2001)[23] A H Khater, D K Callebaut and A R Seadawy, Phys.

Scr. 67, 340 (2003)[24] S K El-Labany, W M Moslem, E I El-Awady and P K

Shukla, Phys. Lett. A 375, 159 (2010)


[25] S Mahmood, A Mushtaq and H Saleem, New J. Phys. 5,28 (2003)

[26] I J Lazarus, R Bharuthram and M A Hellberg, J. PlasmaPhys. 74, 519 (2008)

[27] A R Seadawy, Physica A 439, 124 (2015)[28] D Lu, A R Seadawy, M Arshad and J Wang, Results

Phys. 7, 899 (2017)[29] M Arshad, A R Seadawy, D Lu and J Wang, Results

Phys. 6, 1136 (2016)[30] A R Seadawy, Physica A 455, 44 (2016)[31] A R Seadawy, Math. Meth. Appl. Sci. 40, 1598 (2017)[32] Abdullah, A Seadawy and J Wang, Result Phys. 7, 4269

(2017)

[33] A R Seadawy and K El-Rashidy, Pramana – J. Phys. 87,20 (2016)

[34] A R Seadawy, Pramana – J. Phys. 89: 49 (2017)[35] M Khater, A R Seadawy and D Lu, Pramana – J. Phys.

(2018) (in press)[36] M A Helal and A R Seadawy, Comput. Math. Appl. 64,

3557 (2012)[37] M A Helal, A R Seadawy and R S Ibrahim, Appl. Math.

Comput. 219, 5635 (2013)[38] M A Helal and A R Seadawy, Z. Angew. Math. Phys.

(ZAMP) 62, 839 (2011)[39] M A Helal and A R Seadawy, Phys. Scr. 80, 350

(2009)

Documents

Modified KdV–Zakharov–Kuznetsov dynamical equation in a