https://doi.org/10.1007/s12043-018-1595-0
Modified KdV–Zakharov–Kuznetsov dynamical equation in a homogeneous magnetised electron–positron–ion plasma and 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 China
2Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
3Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt
∗Corresponding authors. E-mail: Aly742001@yahoo.com; wangmath2011@126.com 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 homogeneous magnetised electron–positron–ion plasma is analysed. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of the three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, solitary wave solutions (which represent electrostatic field potential), electric fields, magnetic fields and quantum statistical pressures are obtained with the aid of Mathematica. These new exact solitary wave solutions are obtained in different forms such as periodic, kink and antikink, dark soliton, bright soliton, bright and dark solitary wave etc. The results are expressed in the forms of hyperbolic, trigonometric, exponential and rational functions. The electrostatic field potential and electric and magnetic fields are shown graphically. These results demonstrate the efficiency and precision of the method that can be applied to many other mathematical and physical 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 of each species, the nonlinear three-dimensional modified Korteweg– de Vries–Zakharov–Kuznetsov (mKdV-ZK) equation governs the behaviour of weakly nonlinear ion-acoustic waves. The ion-acoustic wave, which is an ion time-scale phenomenon, has been studied in a two-component electron–ion plasma and both the asso- ciated linear [1–3] and nonlinear [4–6] dynamics have been investigated. In [7], the ion-acoustic wave was studied in an unmagnetised three-component electron–
positron–ion plasma. By assuming that both electrons and positrons are hot and they obey the Boltzmann dis- tribution, the linear dispersion relation is obtained. The electron–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 meaningfully when positrons are introduced into an electron–ion plasma. The positrons can be used to clarify particle transport in tokamaks and, as they have sufficient life- time, the two-component electron–ion plasma becomes a three-component electron–positron–ion plasma [14, 15]. Most of the astrophysical plasmas contain ions with electrons and positrons and the study of linear and non- linear wave propagation is important in such plasmas.
Therefore, electron–positron–ion plasma has attracted the attention of researchers [16–19] in the last decade.
To date, many mathematical models have been derived to describe the dynamics of plasma [20–25]. The mKdV–ZK equation governing the oblique propaga- tion of nonlinear electrostatic modes has been derived, and the soliton amplitudes were studied as a function
of plasma parameters such as temperatures and parti- cle number densities [26]. The Zakharov–Kuznetsov–
Burgers (ZKB) equation for the dust-ion-acoustic waves in dusty plasmas was formulated and the nonlinear wave solutions were obtained [27]. Recently, soli- tary wave solutions of (3+1)-dimensional nonlinear extended 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 were constructed [29]. Seadawy [30] investigated the stabil- ity of solitary travelling wave solutions of the mKdV-ZK equation to three-dimensional long-wavelength per- turbations and found electrostatic field potential and electric field in the form travelling wave solutions.
Furthermore, he studied the ion-acoustic solitary wave solutions 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 and shocks in a homogeneous magnetised electron–positron –ion plasma. Modified extended mapping method is introduced to find ion-acoustic solitary wave solutions of three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, elec- tric field potential, electric and magnetic fields and electron fluid pressures are obtained, which are also expressed graphically.
2. Description of the modified extended mapping method
Consider a general non-linear partial differential equa- tion in 3+1 independent variablesx,y,zandtas F
ϕ, ϕt, ϕx, ϕy, ϕz, ϕt x, ..., ϕx x, ...
=0, (1) where F is a polynomial function in ϕ(x,y,z,t) and its partial derivatives, containing nonlinear terms and highest-order derivatives. The following are the main steps of the method:
Step1. By using travelling wave transformation ϕ(x,y,z,t) = ϕ(θ), where θ = kx +l y + mz+ωtandk,l,mandωare constants, eq. (1) is reduced to the following ordinary differential equation:
P(ϕ, ϕ, ϕ, ϕ, ...), (2) where P is a polynomial in ϕ(θ)and its first- and higher-order derivatives.
Step2. 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) wherea0,a1, ...,an, andb1,b2, ...,bnare arbi- trary constants and the values ofG(θ)andG(θ) satisfy
G(θ)= 6
i=0
βiGi(θ), (4)
whereβi’s are constants to be determined such thatβn =0.
Step3. By balancing the highest-order derivative term and the nonlinear term appearing in eq. (2), determine the positive integernof eq. (3).
Step4. Substitute eq. (3) along with eq. (4) into eq.
(2), then collecting all the coefficients of the same power Gi(θ) where (i = 0,1, ...,n), and equating them to zero, a system of alge- braic equations is obtained. Solving this system of algebraic equations, give the values of all parameters and constants.
Step5. Substitute all the parameter values obtained in the previous step andϕ(θ)into eq. (3), to obtain the solutions of eq. (1).
3. Application of the described method to three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation
Consider the wave propagation in a three-dimensional homogeneous magnetised, electron–positron plasma, consisting of equal amount of cool and hot compo- nents of each species. The fluid governing equations that describe the dynamics of the cooler adiabatic species are given by (see [30])
∂ni
∂t + ∇ ·(niui)=0,
∂ui
∂t +(ui · ∇)ui = − 1
nim∇pi −qi
m∇ϕ+iui ∧ex,
∂pi
∂t +ui · ∇pi +γipi(∇ ·ui)=0, ∇2ϕ+
i
niqi +
j
Njqjexp −qjϕ
kTj =0, ne= Hpexp
eϕ
k He , np = Hpexp −eϕ
k He . (5)
First equation is the continuity equation, second and third equations are equation of motion and adiabatic pressure equation, respectively, and the last equation of the system is the Poisson equation, where ui and pi are the fluid velocities and pressures, qi and qj
are the charges of the cool and hot species, m is the common mass of the electrons and the positrons,γi is the adiabatic compression indices, the gyrofrequencies i = qiB0/m, B0 is the ambient magnetic field, the dynamics of the cooler adiabatic species, denoted by the subscript i, is a small parameter proportional to the amplitude of the perturbation. He and Hp are hot electrons and positrons with equal temperatures and equilibrium densities, ne and np are the densities of hot electrons and positrons,ϕis the electrostatic poten- tial. Using singular perturbation method, Seadawy [30]
derived the mKdV-ZK equation as
ϕt +α1ϕ2ϕx+α2ϕx x x+α3ϕxϕyy+α3ϕxϕzz=0, (6) whereα1, α2anddare given by
α1= 1
A1, α2 = A2
A1, α3= A3 A1, A1 =2
i
ω2pi(V −Ui0) (V −Ui0)2−ν2T i2, A2 =
i
ω2piqi2
15(V−Ui0)4+E1(V−Ui0)2νT i2 +E2ν4T i 2m2
(V−Ui0)2−νT i2 5
−
j
qj
2λ2D jk2Tj2, A3 =1+
i
ω2pi(V −Ui0)4 2i
(V −Ui0)2−ν2T i2, (7) where ωpi =
Niqi2/ m is the plasma frequency, λD j =
kTj/Njq2j is the Debye length and νT i =
√γipi/Nimis the thermal velocity of the speciesi. Now we apply the modified extended mapping method to solve the three-dimensional modified Korteweg–
de Vries–Zakharov–Kuznetsov equation. Consider the travelling wave transformation ϕ(x,y,z,t) = ϕ(θ), θ =kx+l y+mz+ωt, wherek,l,mandωare numbers and frequency to be determined later. Then mKdV–ZK equation becomes
ωϕ+α1kϕ2ϕ+α2k3ϕ+α3kl2ϕ+α3km2ϕ=0. (8) Balancing the highest-order derivative termϕand non- linear term ϕ2ϕ appearing in (8) we obtain n = 2.
Puttingn =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 extended direct algebraic mapping method, substituting eq. (9) and its derivatives into eq. (8) and collecting all the terms with the same power ofGi(θ), wherei =0,1, ...,n, we get a system of algebraic equations. By solving these equations, we get different sets of parameter values. By putting these parameter values in eq. (9), different fam- ilies of solutions of eq. (8) are obtained, as discussed below.
Family1.β0 =β1=β3=β5 =β6 =0
In this family, the following different sets of parameters are obtained:
a0 =0, a1= ±b1
β4, a2=0, b2 =0,
m= ±
−2α1b21−3α2k2−3α3l2
3α3 ,
ω= −1
3α1β2b21k, (10)
where
β4 >0 and −2α1b21−3α2k2−3α3l2 3α3 >0.
Substituting these parameter values given in eq. (10), for only positive values ofa1andm, along with two different values ofϕ, in eq. (9), the following two solutions in simplified forms are obtained:
ϕ11(θ)=
−β2b1sech( β2θ)
−
β2b1tanh(
β2θ), β2 >0, (11) ϕ12(θ)=
−β2b1(tan(
−β2θ) +sec(
−β2θ)), β2<0. (12) The first solution is not valid and so we consider only the second solution. Similarly, solutions are obtained for other sets of parameters.
The electric and magnetic fields are determined by the position and motion of the electrons and positrons as they move along their orbits in a homogeneous mag- netised, electron–positron plasma. The electric field is the 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
∂xxˆ −∂ϕ
∂yyˆ− ∂ϕ
∂zˆz. (13) The electric fields of the electric potentialϕ12is formu- lated as
E12 =b1sec2
−β2θ
β2−
β22sin
−β2θ
·
kxˆ+lyˆ+mˆz
, β2<0. (14) The relation of electric and magnetic fields is given by the Maxwell–Faraday equation as
∇ × E = −∂B
∂t . (15)
Using Maxwell–Faraday eq. (15), the magnetic field is formulated as
B12 = β2b1(k−m)√
−β2sin√
−β2θ
√−β2ω +
√−β2
sec2√
−β2θ
√−β2ω
·
−lxˆ+(k+m)yˆ−lzˆ
, β2 <0. (16) The graphical representations of the solution and its electric and magnetic fields are shown in figure1.
The electron fluid pressure is modelled asP =P(ne), where ne is the electric number density. The relation between the electron fluid pressure P and the electric number densityneis given as
P = mevF2
3n20 e3ne, (17)
where n0 is the equilibrium density for both electrons and ions,vF2represents the electrons Fermi velocity,me
is the mass of the electron. Using this formula, the quan- tum statistical pressure of the electron is obtained as
P12= mevF2 3n20 exp
3
−β2b1
tan
−β2θ +sec
−β2θ
. (18)
By the same method, electric potentials can be obtained and their electric fields, magnetic fields and electron fluid pressures can be formulated for the other sets of parameter values.
Family2.β0=8β22/27β4, β1 =β3=β5=0, β6 =β42/4β2
The following are the sets of parameter values in this family:
Figure 1. (a) The periodic solitary wave solutionϕ12, (b) its electric fieldE12and (c) the magnetic fieldB12.
Set1.
a0 =a1 =0, a2= ± β4b1
2√ β2
, b2 =0,
m= ±
−α1b12−6α2k2−6α3l2
6α3 ,
ω= −1
3α1β2b21k, (19)
where
β2 >0, −α1b12−6α2k2−6α3l2 6α3 >0. Set2.
a0 = a2β2
β4 , a1=0, b1 =b2 =0, m = ±
−a22α1β2−6α2β42k2−6α3β42l2
6α3β4 ,
ω= −a22α1β22k
3β42 , (20)
where
−a22α1β2−6α2β42k2−6α3β42l2 6α3β4 >0.
For the first set of parameter values, there are two solu- tions which are given below, for only positive values of a2 andm, andβ4<0.
ϕ21(θ)=
√3√
β2b1 tan(γ ) 2 cos(2γ )−1 −4√
β2b1cot2(γ ) 3
cot2(γ )−3, (21)
ϕ22(θ)=
√β2b1(4 tan2(γ )+√
3 (6 csc(2γ )−γ1γ22+csc(γ )sec(γ )γ12)) 3
γ12+3 , (22)
where(√
β2θ /√
3) = γ, tanh(γ ) = γ1 and sech(γ )
= γ2. Using formulas (13), (15) and (17), the electric field, magnetic field and electron fluid pressure for the solutionϕ21are formulated as
E21 =β2b1 sec2(γ )(−6 cos(2γ )+3 cos(4γ )+8√
3 sin(γ )cos3(γ ))
3(1−2 cos(2γ ))2 ·
kxˆ+lyˆ +mzˆ
, (23)
B21 =β2b1 (k−m)(4√
3 sin(2γ )+12 cos(2γ )+9 sec2(γ )−24
3ω(1−2 cos(2γ ))2 ·
−lxˆ+(k+m)yˆ−lzˆ
, (24)
P21=mevF2 3n20 exp
3√ 3√
β2b1 tan(γ ) 2 cos(2γ )−1
−4√
β2b1cot2(γ ) cot2(γ )−3
. (25)
The graphical representation of the electric potential and its electric and magnetic fields are shown in figure2.
Similarly, electric field, magnetic field and electron fluid pressure can be formulated and their graphs can be drawn for the second solutionϕ22.
There are four solutions for the second set of param- eter values (20). Two of them are mentioned now, the other two are similar to the first two solutionsϕ21 and ϕ22, respectively.
ϕ23(θ)= a2β2
β4 − 8a2β2tanh2 √
−β√2θ 3
3β4
tanh2
√
−β√2θ 3
+3 ,
β2<0, β4 >0, (26) ϕ24(θ)= a2β2
β4 + 8a2β2cot2 √
β2θ
√3
3β4
coth2
√ β2θ
√3
+3 ,
β2<0, β4 >0. (27) The electric field, magnetic field and electron fluid pres- sure are given below:
E23 = −8a2(−β2)3/2 sinh 2√
−β√2θ 3
√3β4
2 cosh
2√
−β√2θ 3
+1 2
·
kxˆ+lyˆ+mzˆ
, β2<0, β4>0, (28)
B23 = 8a2β22 (k−m)sinh 2√
−β√2 θ 3
√3√
−β2β4ω 2 cosh
2√
−β√2 θ 3
+1 2
·
lxˆ−(k+m)yˆ+lzˆ ,
β2 <0, β4>0, (29)
P23 = mev2F 3n20 exp
⎛
⎝3a2β2
β4
− 8a2β2tanh2 √
−β√2θ 3
β4
tanh2
√
−β√2θ 3
+3
⎞
⎠,
β2<0, β4>0. (30)
Figure 2. (a) The periodic solitary wave solutionϕ21, (b) its electric fieldE21and (c) the magnetic fieldB21.
and the graphical representations for the electric poten- tialϕ23are given in figure3.
By the same way, electric field, magnetic field and electron 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 pressures can be derived for the other sets of parameter values.
Figure 3. (a) The dark soliton travelling wave solutionϕ23, (b) its electric fieldE23and (c) the magnetic fieldB23. Family3.β0 =β1=β3=β5 =0
Following the procedure given in step 4, different sets of parameter values are obtained, which are given below:
Set1.
a0 =a1 =0, a2= ±b1
β6, b2 =0,
m = ±
−α1b21−6α2k2−6α3l2
6α3 ,
ω= −1
3α1β2b21k, (31)
where
β6 >0, −α1b12−6α2k2−6α3l2 6α3 >0. Set2.
a0 = a2β4
4β6 , a1=b1 =b2 =0,
m = ±
−a22α1−24α2β6k2−24α3β6l2
26α3β6 ,
ω= a22α1
8β2β6−3β42 k
48β62 , (32)
where
−a22α1−24α2β6k2−24α3β6l2 26α3β6 >0.
Following the procedure given in step 5 and using the first set of parameter values, 14 solutions are obtained as mentioned below.
ϕ31 =
√β2b1sech2(γ3)
β42(−sinh(2γ3))+β2β6
2 +1
sinh(2γ3)+2 cosh(2γ3)
−2√ β2√
β6β4
2
β42−β2β6( tanh(γ3)+1)2 (33)
ϕ32 =
√β2b1csch2(γ3)
β42(−sinh(2γ3))+β2β6
2 +1
sinh(2γ3)+2 cosh(2γ3) +2√
β2
√β6β4
2
β42−β2β6( coth(γ3)+1)2 (34) where √
β2θ = γ3 andβ2 > 0. The electric field and magnetic field and the electron fluid pressure for the electric potential ϕ31 are formulated below and their graphs are drawn in figure4.
E31 = β2γ52b1
β42−β2β6(γ4 +1)22
×(−β22β62( 2−1)(γ4 +1)2
+2β23/2β4β63/2(γ4+ )(γ4 +1)−β2β42β6
×
γ52 (sinh(2γ3)+ )+2
−2 β2β43
β6γ4+β44
·
kxˆ+lyˆ+mˆz ,
(35) B31= 2β2b1(k−m)
ω
2β42γ92−2β2β6(γ12 +γ9)22
×
2β23/2β4β63/2 γ10
2 +1
+2γ11
−2β22β62
2
−1(γ12 +γ9)2
−2β2β42β6
γ10 +γ11+ 2+1
−2 β2β43
β6γ10+2β44γ92
·
−lxˆ +(k+m)yˆ −lzˆ
, (36)
P31 = mevF2 3n20 exp
3√
β2b1γ52
β42(−γ10)+β2β6
2 +1
γ10+2 cosh(2γ3)
−2√ β2β6β4
2
β42−β2β6( 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 electron fluid pressure can be obtained and their graphs (see fig- ure5) can be drawn for the second solutionϕ32. ϕ33 = b1
√β2
√δ1 sinh 2√
β2θ
−2β2
√β6
β4−√
δ1 cosh 2√
β2θ , β2 >0, δ1 >0, (38) ϕ34 = −b1
√β2
√δ1 sin 2√
β2θ +2β2
√β6
β4−√
δ1 cos 2√
β2θ , β2 <0, δ1 >0, (39)
ϕ35 = b1
√ β2
√−δ1 cosh 2√
β2θ
−2β2√ β6
β4−√
−δ1 sinh 2√
β2θ ,
β2>0, δ1 <0, (40)
ϕ36 = b1
√
−β2
√δ1 cos 2√
−β2θ
−2β2√ β6
β4−√
δ1 sin 2√
−β2θ ,
β2<0, δ1 >0, (41)
Figure 4. (a) The soliton-like solution ϕ31, (b) its electric fieldE31and (c) the magnetic fieldB31.
ϕ37 = −
√β2b1sech2√
β2θ β4sinh 2√
β2θ +2√
β2β6 cosh 2√
β2θ +2√
β2
√β6
2
2√
β2β6 tanh√ β2θ
+β4
, β2 >0 (42) ϕ38 = b1sec2√
−β2θ √
−β2β4sin 2√
−β2θ
−2√
−β2
√−β2β6 cos 2√
−β2θ +β2
√β6
2
2√
−β2β6 tan√
−β2θ +β4
, β2 <0, (43) Figure 5. (a) The soliton-like travelling wave solutionϕ33, (b) its electric fieldE33and (c) the magnetic fieldB33.
ϕ39 = b1
−√
−β2β4γ11+β2
√β6γ122 −2√
−β2
√β2β6 γ11coth√ β2θ
+√ β2
√β2β6 csch2√ β2θ 2√
β2β6 coth√ β2θ
+β4
β2>0, (44)
where coth(√
−β2θ)=γ11, csch(√
−β2θ)=γ12.
ϕ310= −b1csc2√
−β2θ √
−β2β4sin 2√
−β2θ +2√
−β2√
−β2β6 cos 2√
−β2θ
+2β2√ β6
2
2√
−β2β6 cot√
−β2θ +β4
,
β2 <0, (45)
ϕ311 =
√β2b1 sech2 √
β2θ 2
4 tanh
√ β2θ 2
+4
−β2√ β6b1
tanh
√ β2θ 2
+1
β4 ,
β2>0, δ1 =0, (46)
ϕ312 = −β2√ β6b1
coth
√ β2θ 2
+1
β4
−
√β2b1 csch2 √
β2θ 2
4 coth
√ β2θ 2
+4 ,
β2>0, δ1 =0, (47)
ϕ313 =
√β2b1
e4√β2θ −16√ β2√
β6e2√β2θ −16β42 +64β2β6
8β4e2√β2θ −e4√β2θ −16β42+64β2β6
, β2 >0, (48)
ϕ314 = b1
16β2
√β6e2√β2θ +√ β2
1−64β2β6e2√β2θ ,
β2>0, β4=0. (49)
The electric field, magnetic field and the electron fluid pressure for the electric potential ϕ33 are obtained as below and their graphs (see figure5) are drawn.
E33 = 2β2b1
√δ1
2√ β2√
β6sinh 2√
β2θ
−β4cosh 2√
β2θ +√
δ1
β4−√
δ1 cosh 2√
β2θ2 .
kxˆ +lyˆ +mzˆ
, (50)
B33 = 2β2b1√
δ1 (k−m) 2√
β2
√β6sinh 2√
β2θ
−β4cosh 2√
β2θ +√
δ1
ω
β4−√
δ1 cosh 2√
β2θ2 .
−lxˆ +(k+m)yˆ −lzˆ ,
(51) P33 = mev2F
3n20
×exp 3b1
√β2√
δ1 sinh 2√
β2θ
−2β2√ β6
β4−√
δ1 cosh 2√
β2θ
, (52) forβ2 >0, δ1 >0,whereδ1=β42−4β2β6, = ±1.
Electric fields, magnetic fields and the electron fluid pressures can be formulated for other solutionsϕ34 to ϕ314and their graphs are drawn (see figure5). Similarly, solutions can be obtained for second set of parameter values.
Family4.β0 =β1=β5=β6 =0
In this family, the following different sets of parame- ter values are obtained by following the procedure in step 4.
Set1.
a0 = a1β3
4β4 , a2=b1 =b2 =0, m = ±
−a12α1−6α2β4k2−6α3β4l2
6α3β4 ,
ω= a12α1
8β2β4−3β32 k
48β42 , (53)
where
−a12α1−6α2β4k2−6α3β4l2 6α3β4 >0. Set2.
a0 =0, a1 = ±b1
β4, a2 =b2 =0,
m = ±
−2α1b21−3α2k2−3α3l2
3α3 ,
ω= −1
3α1β2b21k, (54)
where
β4 >0, −2α1b21−3α2k2−3α3l2
3α3 .
Using the first set of parameter values and following the procedure given in step 5, we obtained six solutions as given 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- tionϕ41 are formulated and their graphs (see figure6) are drawn.
E41 = 2a1β23/2√
δ2sinh√ β2θ β4−√
δ2cosh√
β2θ2 ·
kxˆ +lyˆ+mzˆ ,
δ2 >0, (56)
B41 = 2a1β23/2√
δ2(k−m)sinh√ β2θ ω
β4−√
δ2cosh√ β2θ2
·
−lxˆ+(k+m)yˆ−lzˆ
, δ2 >0, (57) P41= mevF2
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 solutionϕ41, (b) its electric fieldE41and (c) the magnetic fieldB41. 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 √
β2θ 2
β4 , δ2 =0, (62) ϕ46 = a1β3
4β4 − a1β2
1±coth √
β2θ 2
β4 , δ2=0. (63) The electric field, magnetic field and electron fluid pres- sure for the solutionϕ42are formulated and their graphs are drawn (see figure7).
E42 = − 2a1β23/2√
δ2sinh√ β2θ
√δ2cosh√ β2θ
+β4
2
·
kxˆ+lyˆ+mzˆ
, δ2 >0, (64) B42 =2a1β23/2√
δ2(k−m)sinh√ β2θ
ω√
δ2cosh√ β2θ
+β4
2
·
lxˆ−(k+m)yˆ+lˆz
, δ2 >0, (65) P42 =mevF2
3n20 exp
3a1β3
4β4 − 6a1β2sech√ β2θ β4sech√
β2θ +√
δ2
,
δ2>0, (66)
where β2 > 0, δ2 = β32 − 4β2β4. Electric fields, magnetic fields and the electron fluid pressures can be formulated for other solutions and their graphs can be drawn. Similarly, solutions can be obtained for second set of parameter values.
4. Results and discussion
The solutions obtained by the proposed method are dif- ferent from those obtained by other researchers because of the following reasons:
• Our supposed solution (3) has a structure different from other methods and has different kinds of param- eters.
• By choosing different values ofβi’s (i =0,1, ...,6), eq. (4) has many types of special solutions in the form of trigonometric, hyperbolic, exponential and rational functions.
But still some of our solutions have similarities with the following points:
• The solutionϕ33,ϕ35andϕ45 have similarities with the second, third and first solutions, respectively, of Case 1 in [30].
• Solutions ϕ41 and ϕ42 have similarities with the solutions u11 and u12, solutionsϕ43 and ϕ44 have
Figure 7. (a) The bright soliton solutionϕ42, ((b) its electric fieldE42and (c) the magnetic fieldB42.
similarities with the solutionsu13andu14, solutions ϕ45andϕ46have similarities with the solutionsu15 andu16, andϕ314has similarities with the solutions u24, respectively, of page 903 in [28].
The remaining of our solutions are new and has not been formulated before.
The electric fields in three dimensions are formulated in [30], the electric and magnetic fields and electron fluid pressure are formulated in [27] (not shown graphically), the electric field in two dimensions are formulated and presented graphically in [31]. In this paper, we for- mulated the electrostatic field potential, electric and magnetic fields and fluid pressure in two dimensions and showed the results graphically in three dimensions for slightly different parameter values. For example, one set 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, = 1, t = 2. By changing the sign of some of the val- ues, the direction of the electric and magnetic fields change.
This discussion shows the effectiveness and power of the new method ‘modified extended mapping method’.
This method can be applied to many more nonlinear partial differential equations.
5. Conclusion
In this paper, we analysed propagation of three-dimens- ional nonlinear ion-acoustic solitary waves and shocks in a homogeneous magnetised electron–positron–ion plasma. The electron–positron plasma has an important role in comprehending plasmas in the early Universe, in active galactic nuclei, in pulsar magnetosphere and in solar atmosphere. The modified extended mapping method was introduced to find different kinds of ion- acoustic solitary wave solutions of three-dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov equation, which is important for understanding many space and astrophysical phenomena. As a result, differ- ent kinds of exact travelling wave solutions, electric field potential, electric and magnetic fields and quantum sta- tistical pressures are obtained which show the reliability and effectiveness of the method.
The obtained solitary wave solutions are in the form of hyperbolic, trigonometric, exponential and ratio- nal functions which are also expressed graphically.
These solutions are important in different branches of physics and other areas of applied sciences and can help researchers to study and understand the physical interpretation of the system. Many higher-order nonlin- ear equations arising in plasma, mathematical physics, engineering, hydrodynamics and other areas of applied sciences can also be solved by this powerful, reliable and capable method.
Acknowledgements
This work was supported by NSF of China (Grants 11571140, 11671077, 11371090), Fellowship of Out- standing Young Scholars of Jiangsu Province (BK2016- 0063), NSF of Jiangsu Province (BK20150478).
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