Lie point symmetries, conservation laws and exact

Pramana – J. Phys. (2018) 91:48 © Indian Academy of Scienceshttps://doi.org/10.1007/s12043-018-1614-1

Lie point symmetries, conservation laws and exact solutionsof (1+ n)-dimensional modified Zakharov–Kuznetsov equationdescribing the waves in plasma physics

MUHAMMAD NASIR ALI1, ALY R SEADAWY2,3,∗ and SYED MUHAMMAD HUSNINE1

1Department of Sciences and Humanities, National University of Computer and Emerging Sciences,Lahore 54000, Pakistan2Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia3Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt∗Corresponding author. E-mail: [email protected]

MS received 28 November 2017; revised 26 February 2018; accepted 27 February 2018;published online 10 August 2018

Abstract. In this study, we explore the modified form of (1 + n)-dimensional Zakharov–Kuznetsov equation,which is used to investigate the waves in dusty and magnetised plasma. It is proved that the equation follows theproperty of nonlinear self-adjointness. Lie point symmetries are calculated and conservation laws in the frameworkof the new general conservation theorem of Ibragimov are obtained. The (1/G ′), (G ′/G)-expansion and modifiedKudryshov methods are applied to extract exact analytical solutions. The so-called bright, dark and singular solutionsare also found using the solitary wave ansatz method. The results obtained in this study are new and may be ofsignificant importance where this model is used to study the waves in different plasmas.

Keywords. Modified Zakharov–Kuznetsov equation; formal Lagrangian; nonlinear self-adjointness; conservationlaws; modified Kudryshov method; solitary wave ansatz method; (G ′/G)-expansion method.

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

1. Introduction

Nonlinear partial differential equations (NLPDEs) playan important role while describing the complicated phe-nomena arising in different branches of science such asfluid flow, wave propagations, thermodynamics, opti-cal fibres, plasma physics, nonlinear networks and soilconsolidations. Therefore, solving such nonlinear mod-els is a rich area of research for the scientists because theresulting solutions can explain the physical behaviour ofthe concerned problems in the best way [1–4].

Zakharov–Kuznetsov (ZK) equation is modelled toanalyse the nonlinear development of ion-acousticwaves in a magnetised plasma comprising cold ions andhot isothermal electrons in the presence of a uniformmagnetic field [5]. It has the following form:

ut + auux + (∇2u)x = 0,

where ∇2 = ∂2x + ∂2

y + ∂2z is the Laplacian.

A variety of ZK equations are studied by themathematicians including modified ZK (mZK),

Zakharov–Kuznetsov-modified equal width (ZK-MEW), Zakharov–Kuznetsov-Benjamin–Bona–Mahony (ZK-BBM), modified Korteweg–de Vries-Zakharov–Kuznetsov (MKdV-ZK), quantum ZK,extended quantum ZK and generalised ZK equations[6–12]. Several methods such as the F-expansion method[13], extended Fan sub-equation method [14], modi-fied simple equation method [15], sine–cosine method[16], tanh, coth-method [17], exp-function method [18],Lie group analysis method [19], fractional direct alge-braic function method [20], first integral method [21],(G ′/G)-expansion method [22], enhanced (G ′/G)-expansion method [23], homotopy analysis method [24],generalised Kudryashov method [25], auxiliary equa-tion method [26], extended direct algebraic method [27]and modified differential transformation method [28]are derived for extracting exact solutions of differen-tial equations. In the theory of differential equations,conservation laws play an important role in findingexact solutions of the partial differential equations(PDEs) [20,29–34]. Existence of a large number ofconservation laws for a particular differential equation

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48 Page 2 of 9 Pramana – J. Phys. (2018) 91:48

indicates its integrability [35]. In the literature, differentmethods are suggested to obtain the conservation lawsof differential equations. These methods include thedirect construction method [36,37], multiplier method[38], Noether approach [39], partial Lagrangianapproach [40] and new general conservation theorem byIbragimov [41].

We study the (1 + n)-dimensional mZK equation ofthe form

G = ut + au2ux1+ (∇2u)x1

= 0, (1)

where ∇2 = ∂2x1

+ ∂2x2

+ · · · + ∂2xn

is the n-dimensionalLaplacian.

The paper is arranged as follows. In §2, formalLagrangian is defined and the adjoint equation is calcu-lated for eq. (1). Lie point symmetries and conservationlaws of eq. (1) are also calculated. In §3, exact solu-tions are obtained using the solitary wave ansatz method.Moreover, the (1/G ′), (G ′/G)-expansion and modifiedKudryshov methods are also applied to obtain someother exact analytical solutions, whereas the physicaldescription of the solutions has been presented in §4.We conclude the paper in §5.

2. Symmetries, formal Lagrangian, self-adjointnessand conservation laws

In this section, symmetries of eq. (1) are derived andthen conservation laws with the aid of a new generalconservation theorem are obtained.

2.1 Lie point symmetries

Consider the following infinitesimal generator X of theform:

X = ξ1∂t + ξ2∂x1+ · · · + ξn+1∂xn + ϕ∂u, (2)

where ξ i (1 ≤ i ≤ n + 1) are functions of the variables(t, x1, x2, . . . , xn , u).Applying the following invarianceproperty, we obtain

X3[ut + au2ux1+ (∇2u)x1

]|ut=−au2ux1−(∇2u)x1

= 0,

where X3 is the third-order prolongation of operator X[42]. After expanding and separating w.r.t. the deriva-tives of u, one can obtain the system of determiningequations in the unknown functions ξ i (1 ≤ i ≤ n + 1)

and ϕ. The solution of this determining system providesthe following set of symmetries:

X1 = ∂t ,

X2 = 3t∂t +n∑

j=1

x j∂x j − u∂u, for 1 ≤ j ≤ n,

Xi j = x j∂xi − xi∂x j , for 2 ≤ i < j ≤ n.

2.2 Formal Lagrangian and nonlinearself-adjointness

The formal Lagrangian of eq. (1) can be written as

L = v[ut + a, u2ux1+ (∂2u)x1

]. (3)

Here v is a new dependent variable whose value willbe determined later. This value of the new dependentvariable v will classify the type of self-adjointness. Nowthe adjoint equation for eq. (1) is

G∗ ≡ δL

δu= 0, (4)

where

δL

δu= Lu − Dt (Lut) − Dx1

(Lux1)

−(

n∑

k=1

Dx1D2xk(Lux1 xk xk

)

)(5)

and Dt , Dx1, Dx2

, . . . , Dxn represent the total deriva-tives. From eqs (4) and (5), we obtain the requiredadjoint equation

G∗ = −(vt + a, u2vx1

+∑

vx1 xk xk

)= 0. (6)

The related definitions from the literature are given tounderstand the self-adjointness [43].

DEFINITION 1

A nonlinear differential equation

G(x, u, u(1), . . . , u(m)) = 0 (7)

of order m with r independent variables x = (x1, x2,

. . . , xr ) and the dependent variable u is said to be strictlyself-adjoint if its adjoint equation becomes equivalent tothe original on substituting u for v.

DEFINITION 2

Differential eq. (7) is called nonlinearly self-adjoint ifit is extracted from its adjoint equation by putting v =σ , where σ is the non-zero function of the dependentvariable, independent variables and derivatives of thedependent variable.

Pramana – J. Phys. (2018) 91:48 Page 3 of 9 48

It is observed from eq. (6) that eq. (1) is strictlyself-adjoint. However, we shall try to find the conserva-tion laws with nonlinear self-adjointness under non-zerosubstitution v of the form

v = σ(t, x1, x2, . . . , xn, u) �= 0.

For the calculation of v, we use the condition

G∗ |v=σ(t,x1,x2,...,xn,u) = μ(ut + a, u2, ux1+ (∇2u)x1

),

(8)

where μ is the undetermined coefficient. Putting thevalues of derivatives of v in eq. (8) and equating thecoefficients of the derivatives of u on both sides yieldthe following system:

σu = −μ,

σu,xi= 0, σuu = 0, for 1 ≤ i ≤ n,

σt + au2σx1+

n∑

i=1

σx1 xi xi= 0. (9)

Solving the above system, we obtain

v = k1u + f (x2, x3, . . . , xn), (10)

where k1 is any arbitrary constant and f (x2, x3, . . . , xn)is an arbitrary function.

2.3 Conservation laws

Ibragimov’s theorem is used for calculating the con-servation laws of eq. (1). This theorem depends on theself-adjointness and formal Lagrangian. The statementof this theorem is as follows [41].

Theorem. Any symmetry (Lie point symmetry, LieBäcklund symmetry, or non-local symmetry) operatorof the form given in eq. (2) of the system which containsthe original equation and its adjoint equation admits aconservation law Dk(Ck) = 0, where each componentof the conserved vector Ck is given by the formulae;

Ck = ξ kL+W [Luk− Dl(Lukl

)+DlDm(Luklm) − · · · ]

+ Dl(W )[Lukl−Dm(Luklm

) + · · · ]+ DlDm(w)[Luklm

− · · · ] + · · · , (11)

where L is the formal Lagrangian and W = η − ξ kuk .

To find the conservation laws of the (1+n)-dimensionalmZK equation, the formal Lagrangian of eq. (1) can bewritten as

L = v

[ut + au2ux1

+ ux1 x1 x1

+1

3

(n∑

i=2

(ux1 xi xi+ uxi x1 xi

+ uxi xi x1)

)]. (12)

The components of these conservation laws are asfollows:

Ct = Wv,

C1 = W

[au2v + Dx1

Dx1(v) +

n∑

i=2

DxiDxi

(1

3v

)]

+ Dx1(W )(−Dx1

(v)) +n∑

i=2

Dxi(W )

(−Dxi

(1

3v

))

+ DxiDxi

(W ) +n∑

i=2

DxiDxi

(W )

(1

3v

),

for 2 ≤ j ≤ n

C j = W

[Dx1

Dx j

(1

3v

)+ Dx j

Dx1

(1

3v

)]

+Dx j(W )

(−Dx1

(1

3v

))

+ Dx1(W )

(−Dx j

(1

3v

))+ Dx j

Dx1(W )

(1

3v

)

+ Dx1Dx j

(W )

(1

3v

).

From the value of the new dependent variable v givenin eq. (10), the following two parameters are obtained:ϕ1 = f (x2, x3, . . . , xn),

ϕ2 = u.

With the help of these parameters, one can achieve atmost two conservation laws corresponding to a singlesymmetry.

3. Exact analytical solutions of mZK equation

For this purpose, we apply the following wave transfor-mation on eq. (1) to transform the original PDE into anODE:

ζ = x1 + x2 + · · · + xn − ωt. (13)

This will transform eq. (1) into an ODE as given below:

−ωUζ + aU 2Uζ + nUζ ζ ζ = 0, (14)

where U = U(ζ ) = u(t, x1, x2, . . . , xn) and Uζ =du/dζ . Integrating eq. (14) once with constant of inte-gration as zero, we obtain

−ωU + a

3U3 + nUζ ζ = 0. (15)


3.1 Bright soliton solution

We consider the following soliton ansatz [44]:

U = A

coshp(λζ ), (16)

where λ is the inverse width and A is the amplitudeof the soliton with p > 0 for solitons to exist. Now,substituting (16) into eq. (15), we obtain

− Aω

coshp(λζ )+

(a3

)(A3

cosh3p(λζ )

)

+ n

(Ap2λ2

coshp(λζ )− Ap(p + 1)λ2

coshp+2(λζ )

). (17)

p = 1 is obtained while equating the exponents 3p andp+ 2 from eq. (17). Now comparing the coefficients oflinearly independent terms in eq. (17) to zero, we obtainthe following system:

− ωA + nAp2λ2 = 0,(a3

)A3 − nAp(p + 1)λ2 = 0. (18)

The solution of the system yields

A = ±√

6ω

a, λ = ±

√ω

n.

Substituting the values of A and λ in eq. (16), we obtain

U1 = ±√

6ω

asech

(√ω

nζ

), (19)

where ζ is given by eq. (13) and ω is the speed of thewave.

3.2 Singular solution

Consider the ansatz of the form [44]

U = A cschp(λζ ), (20)

where A and λ are the amplitude and inverse width ofthe soliton, respectively, and p > 0 will be determinedfor the solitons to exist. Substituting (20) into eq. (15),we obtain

− ωA cschp(λζ ) + aA3

3csch3p(λζ )

+ n(Ap2λ2 cschp(λζ )

+ Ap(p + 1)λ2cschp+2(λζ )) = 0. (21)

Equating the exponents 3p and p + 2 from eq. (21), wehave p = 1.

Now comparing the coefficients of linearly indepen-dent terms in eq. (21) to zero, we obtain the followingsystem:

− ωA + nAp2λ2 = 0,(a3

)A3 + nAp(p + 1)λ2 = 0. (22)

Solving the above system, we obtain the values of ampli-tude A and inverse width λ as follows:

A = ±√−6ω

a, λ = ±

√ω

n.

Substituting values in eq. (20), we obtain

U2 = ±√−6ω

acsch

(±

√ω

nζ

), (23)


3.3 Dark solutions

Dark solutions are also called topological solitons. Forthis, we take the ansatz of the form [44]

U = U(ζ ) = A tanhp(λζ ), (24)

where λ is the inverse width of the soliton and A isthe amplitude of the soliton. By substituting (24) intoeq. (15), we obtain

− ωA tanhp(λζ ) + aA3

3tanh3p(λζ )

+ n(Apλ2((p − 1) tanhp−2(λζ )

− 2p tanhp(λζ ) + (p + 1) tanhp+1(λζ ))) = 0.

(25)

Equating the exponents 3p and p + 2 from eq. (25), weobtain p = 1.

Now comparing the coefficients of linearly indepen-dent terms in eq. (25) to zero, we obtain

− ωA − 2nAp2λ2 = 0,(a3

)A3 + nApλ2(p + 1) = 0. (26)

The solution of the above system yields

A = ±√

3ω

a, λ = ±i

√ω

2n.

Substituting the above values in eq. (24), we obtain thefollowing dark solution:

U3 = ±√

3ω

atanh

(± i

√ω√

2nζ

), (27)



3.4 (1/G ′)-expansion method

Taking eq. (15) and balancing the terms U ′′ and U 3, weobtain M = 1. Now, according to the (1/G ′)-expansionmethod [45], we look for the solution of the form

U (ζ ) = a0 + a1

(1

G ′

), (28)

where a0 and a1 are some constants to be determinedlater and1

G ′ = λ

−μ + c1λ[cosh(λζ ) − sinh(λζ )] . (29)

Substituting eq. (28) into eq. (15), and comparing thecoefficients of different powers of (1/G ′), we obtainthe following system:a

3a3

1 + 2nμ2a1 = 0,

aa0(a1)2 + 3a1nμλ = 0,

−a1ω + aa20a1 + a1nλ2 = 0,

−a0ω + a

3(a0)

3 = 0. (30)

Solving the above system, we obtainSet 1:

a0 = ±√

3ω

a, a1 = ±i

√6n

aμ, λ = −i

√2ω

n.

Set 2:

a0 = ±√

3ω

a, a1 = ∓i

√6n

aμ, λ = i

√2ω

n.

Set 3:

a0 = ±i

√3n

2 aλ, a1 = ±i

√6n

aμ, ω = −nλ2

2.

Set 4:

λ = ∓i

√2a

3na0, a1 = ±i

√6n

aμ, ω = aa2

0

3.

Substituting the above values in eq. (28), we obtainthe following solutions:

U4 = ±√

3ω

a±

( √12ω/a

−μ − c1i√

2ω/n(cosh(−i√

2ω/nζ ) − sinh(−i√

2ω/nζ ))

), (31)

U5 = ±√

3ω

a±

( √12ω/a

−μ + c1i√

2ω/n(cosh(i√

2ω/nζ ) − sinh(i√

2ω/nζ ))

), (32)

U6 = ±√

3n

2aλ ± i

√6n

aμ

(λ

−μ + c1λ[cosh(λζ ) − sinh(λζ )])

, (33)

U7 = a0 ±(

2a0μ

−μ ∓ ic1√

2a/3na0(cosh(∓i√

2a/3na0ζ ) − sinh(∓i√

2a/3na0ζ ))

). (34)

3.5 (G ′/G)-expansion method

According to the (G ′/G)-expansion method [22], thesolution of eq. (1) is

U (ζ ) = a0 + a1

(G ′

G

), (35)

where a0 and a1 are the constants to be determined. Sub-stituting eq. (35) into eq. (15) and equating the differentpowers of (G ′/G), we obtain the following system:

a

3(a1)

3 + 2na1 = 0,

aa0(a1)2 + 3a1nλ = 0,

− a1ω + a(a0)2a1 + na1(λ

2 + 2μ) = 0,

− a0ω + a

3(a0)

3 + na1λμ = 0. (36)

Solving the above system, we obtain

a0 = ±i

√3n

2aλ, a1 = ±i

√6n

a, ω = n(4μ − λ2)

2.

Substituting the above values in eq. (35), we obtain thefollowing solutions when λ2 − 4μ > 0:

U8 = ± i

√3n

2aλ ± i

√6n

a

(−λ

2+

√λ2 − 4μ

2

×⎛

⎝c1 sinh

(√((λ2 − 4μ)/2)ζ

)+ c2 cosh

(√((λ2 − 4μ)/2)ζ

)

c1 cosh(√

((λ2 − 4μ)/2)ζ)

+ c2 sinh(√

((λ2 − 4μ)/2)ζ)

⎞

⎠

⎞

⎠ . (37)


If c2 = 0, then solution U8 will be

U9 = ± i

√3n

2aλ ± i

√6n

a

(−λ

2+

√λ2 − 4μ

2

× tanh

(√λ2 − 4μ

2ζ

)). (38)

If c1 = 0, then the solution ofU8 will take the followingform:

U10 = ± i

√3n

2aλ ± i

√6n

a

(−λ

2

+√

λ2 − 4μ

2coth

(√λ2 − 4μ

2ζ

)). (39)

When λ2 − 4μ < 0

U11 = ±i

√3n

2aλ ± i

√6n

a

(−λ

2+

√4μ − λ2

2

×⎛

⎝−c1 sin

(√((4μ − λ2)/2)ζ

)+ c2 cos

(√((4μ − λ2)/2)ζ

)

c1 cos(√

((4μ − λ2)/2)ζ)

+ c2 sin(√

((4μ − λ2)/2)ζ)

⎞

⎠

⎞

⎠ . (40)

If c2 = 0, then the solution of U11 will be

U12 = ±i

√3n

2aλ ± i

√6n

a

×(

−λ

2−

√4μ − λ2

2tan

(√4μ − λ2

2ζ

)).

(41)

If c1 = 0, then the solution of U11 will take the form

U13 = ±i

√3n

2aλ ± i

√6n

a

×(

−λ

2+

√4μ − λ2

2cot

(√4μ − λ2

2ζ

)).

(42)

When λ2 − 4μ = 0

U14 = ±i

√3n

2aλ ± i

√6n

a

(−λ

2+ c2

c1 + c2ζ

), (43)

where ζ is defined in eq. (13).

3.6 Modified Kudryshov method

Here we apply the modified Kudryshov method [1] toobtain some other new analytical solutions. N = 1 isobtained while comparing the terms U3 and Uζ ζ . Now,the solution will take the form as

U(ζ ) = a0 + a1Q(ζ ), (44)

where

Q(ζ ) = 1

1 + d(θ)ζ. (45)

Substituting eq. (44) into eq. (15) and equating the coef-ficients of each power of Q(ζ ), we obtain the followingsystem:a

3a3

0 + a0ω = 0,

aa20a1 + na1(ln θ)2 − a1ω = 0,

aa0a21 − 3na1(ln θ)2 = 0,

a

3a3

1 + 2na1(ln θ)2 = 0. (46)

Solving eq. (46) givesSet 1:

a0 = i

√3n

2aln θ, a1 = −i

√6n

aln θ, ω = −i

n

2(ln θ)2.

Set 2:

a0 = −i

√3n

2aln θ, a1 = i

√6n

aln θ, ω = −i

n

2(ln θ)2.

Substituting the above values in eq. (44), we obtain thefollowing solutions:

U15 = i

√3n

2aln θ − i

√6n

aln θ

(1

1 + d(θ)ζ

), (47)

U16 = −i

√3n

2aln θ + i

√6n

aln θ

(1

1 + d(θ)ζ

), (48)

where ζ is given in eq. (13) and d, θ are the arbitraryconstants.

4. Physical description

The graphical representations of solitons has been givenin the following figures, for various values of parame-ters. Mathematica 10.4 is used to carry out simulationsand to visualise the behaviour of nonlinear waves. Fig-ure 1a is the (1 + 2)-dimensional solitary wave solutionproduced in eq. (1), for a = 1, ω = 1, t = 2, while


Figure 1. (1 + 2)-Dimensional ansatz solutions: (a) U1: a = 1, ω = 1, t = 2, (b) U2: a = −1, ω = 1, t = 2 and (c)U3: a = −1, ω = −1, t = 2.

Figure 2. (1 + 2)-Dimensional solutions by the (1/G ′)-expansion method: (a) U4: μ = 1, ω = 1, a = 1, c1 = 1, t = 2 and(b) U6: μ = 1, λ = 2, a = 1, c1 = 1, t = 2.

figure 1b represents the singular wave solution for a =−1, ω = 1, t = 2 and in figure 1c, the shock wavesolution is presented for a = −1, ω = −1, t = 2 by thesolitary wave ansatz method for eqs (19), (23) and (27),respectively.

Figure 2a shows the (1 + 2)-dimensional solutionproduced in eq. (1) by the (1/G ′)-expansion method,for μ = 1, ω = 1, a = 1, c1 = 1, t = 2 foreqs (31), while figure 2b shows the graphical repre-sentation of (1 + 2)-dimensional solution for μ = 1,λ = 2, a = 1, c1 = 1, t = 2 for eq. (33). Similarly,

the (1+2)-dimensional solution is obtained for eq. (1)by the (G ′/G)-expansion method, which is shown infigure 3a, for μ = 1, λ = 5, a = −5, c1 = 2,c2 = 5, t = 1, while figure 3b represents the (1 + 2)-dimensional solution for μ = 1, λ = 1, a = 5, c1 = 2,c2 = 5, t = 1 for eqs (38) and (43), respectively. Finally,figures 4a and 4b illustrate the (1 + 2)-dimensionalsolution extracted by the modified Kudryshov methodfor θ = 10, a = 3, d = 5, t = 1 and θ = 5,a = 5, d = 10, t = 1 from eqs (47) and (48),respectively.


Figure 3. (1 + 2)-Dimensional solutions by the (G ′/G)-expansion method: (a) U9: μ = 1, λ = 5, a = −5, c1 = 2, c2 = 5,t = 1 and (b) U14: μ = 1, λ = 1, a = 5, c1 = 2, c2 = 5, t = 1.

Figure 4. (1 + 2)-Dimensional solutions by the modified Kudryshov method: (a) U15: θ = 10, a = 3, d = 5, t = 1 and (b)U16: θ = 5, a = 5, d = 10, t = 1.

5. Conclusion

In this study, the (1 + n)-dimensional mZK equationis considered for Lie point symmetries, conservationlaws and exact solutions, which present novel results.It is established that the mZK equation is nonlinearlyself-adjoint. This property is useful in finding more con-servation laws corresponding to each symmetry of thedifferential equation. With the help of new general con-servation theorem of Ibragimov, conservation laws arederived. The solitary wave ansatz method is utilised forcalculating the exact solutions including bright, dark andsingular solutions, as shown in figure 1.

The (1/G ′), (G ′/G) expansion and modifiedKudryshov methods are also applied to obtain someother exact solutions (see figures 2–4). Theseapproaches are efficient and reliable for obtaining exactsolutions of the nonlinear differential equation. Thesesolutions may be of significance in plasma physicsincluding magnetised plasma and dust plasma, wherethis equation is modelled and used for some specialphysical phenomenon.

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Lie point symmetries, conservation laws and exact