Results in Physics 19 (2020) 103401
Symmetry analysis and conservation laws of a further modified 3D Zakharov-Kuznetsov equation
T. Goitsemang
b, D.M. Mothibi
a,*, B. Muatjetjeja
b,c, T.G. Motsumi
baDepartment of Mathematical Sciences, Sol Plaatje University, Private Bag X5008, Kimberley 8300, South Africa
bDepartment of Mathematics, Faculty of Science, University of Botswana, Private Bag 22, Gaborone, Botswana
cDepartment of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa
A R T I C L E I N F O Keywords:
Further modified 3D Zakharov-Kuznetsov equation
Lie point symmetries Conservation laws
A B S T R A C T
We study a further modified (3 +1)-dimensional(3D) Zakharov-Kuznetsov equation. The classical Lie symmetry method will be employed to obtain solutions of a further modified 3D Zakharov-Kuznetsov equation. We will also establish the multiplier method to derive conservation laws of the underlying equation. Moreover, a brief physical interpretation of the derived conserved quantities is conferred.
Introduction
Nonlinear evolution equations (NLEEs) are mostly used as models to signify physical phenomena in various fields of sciences, especially in biology, solid state physics, plasma physics, plasma waves and fluid mechanics [1–4]. It is therefore of paramount importance that solutions of such NLEEs be investigated. Finding solutions of NLEEs is a very difficult task hence only in certain cases one can explicitly obtain their solutions. In the last few decades some vital advancement has been made and many influential and effective methods for finding exact so- lutions of NLEEs have been proposed in the literature. For example, the Darboux transformation method [5], the exp-function expansion method [6], Jacobi elliptic function expansion method [7], the inverse scattering transform method [1], the sine–cosine method [8] and the Lie symmetry analysis [9–11].
In [12], authors investigated an isothermal multicomponent magnetized plasma model which resulted in the derivation of a further modified 3D Zakharov-Kuznetsov equation when taking into account higher order nonlinear effect.
The further modified 3D Zakharov-Kuznetsov equation [13] is given by
ut+auux+bu2ux+cu3ux+duxxx+uxyy+uxzz=0, (1) where a,b,c and d are plasma parameters, which describe the evolution
of various solitary waves in isothermal multicomponent magnetized plasma [13–15]. The soliton formation and propagation in nonlinear model (1), compressive and rarefactive solitary wave solutions were constructed in [16,17]. In [13], authors investigated a further modified 3D Zakharov-Kuznetsov Eq. (1) and obtained solitary wave solutions using the bifurcation method. Although many efforts have been devoted to find various methods to solve (integrable or non-integrable) nonlinear evolution equations, there is no unified method. See for example [18–21]. To the best of our ability this is for the first time that the classical Lie symmetry method is being employed to exploit solutions for a further modified 3D Zakharov-Kuznetsov equation.
The structure of this paper is organized into two parts. Firstly, we employ the classical symmetry method to derive some solutions of a further modified 3D Zakharov-Kuznetsov Eq. (1). Moreover, we construct conservation laws for Eq. (1) using the multiplier approach.
Symmetries and solutions of Eq. (1)
The objective of this section is to compute the Lie point symmetries [22,23] of a 3D further modified 3D Zakharov-Kuznetsov Eq. (1).
Thereafter, we perform symmetries reductions and solutions of Eq. (1).
Consider the vector field Γ=ξ1∂
∂t+ξ2∂
∂x+ξ3∂
∂y+ξ4∂
∂z+η∂
∂u, (2)
* Corresponding author.
E-mail addresses: [email protected] (D.M. Mothibi), [email protected] (B. Muatjetjeja), [email protected] (T.G. Motsumi).
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where ξiwith(i=1,2,3,4) and η are functions of t, x, y, z and u.
Invoking the third prolongation pr(3) Γ to Eq. (1) and expand and thereafter split the monomial yields
ξ4t =0, (3)
ξ2z=0, (4)
ξ4x=0, (5)
ξ4u=0, (6)
ξ3t =0, (7)
ξ2y=0, (8)
ξ3x=0, (9)
ξ3u=0, (10)
ξ2u=0, (11)
ξ1x=0, (12)
ξ1y=0, (13)
ξ1z=0, (14)
ξ1u=0, (15)
ξ2xx=0, (16)
ηxu=0, (17)
ηuu=0, (18)
ξ3z+ξ4y=0, (19)
− 2ηzu+ξ4yy+ξ4zz=0, (20)
− 2ηyu+ξ3yy+ξ3zz=0, (21)
ηt+cu3ηx+bu2ηx+auηx+ηxyy+ηxzz+dηxxx=0, (22) 2buη+aη+3cu2η− ξ2t+2cu3ξ2x+2bu2ξ2x+2au ξ2x+ηyyu+ηzzu=0, (23) 2buη+aη+3cu2η− ξ2t+2cu3ξ3y+2bu2ξ3y+2au ξ3y+ηyyu+ηzzu=0, (24) 2buη+aη+3cu2η− ξ2t+2cu3ξ4z+2bu2ξ4z+2au ξ4z+ηyyu+ηzzu=0, (25)
3cu2η+cu3ξ1t − cu3ξ2x+aη+2buη+bu2ξ1t+auξ1t− bu2ξ2x− auξ2x
− ξ2t+ηyyu+ηzzu=0.
(26) Solving the above system of partial differential equations, leads to the following two cases:
Case 1. a =b3c2.
In this case, the further modified 3D Zakharov-Kuznetsov Eq. (1) admits six symmetries, namely
Γ1=81c2t∂
∂t− (2b3t− 27xc2)∂
∂x+27c2y∂
∂y+27c2z∂
∂z− (6bc+18c2u)∂
∂u, Γ2=∂
∂t, Γ3= ∂
∂y,Γ4= ∂
∂z,Γ5=y∂
∂z− z∂
∂y,Γ6= ∂
∂x.
(27) We derive the solution of the further modified 3D Zakharov- Kuznetsov Eq. (1) by employing symmetry Γ2+αΓ3, which give rise to the following invariants
f=x, g=z, h=αt− y
α , ϕ=u. (28)
Making use of these invariants, we obtain the group-invariant solu- tion of Eq. (1) as u(t,x,y,z) =ϕ(f,g,h).Using this group-invariant so- lution together with Eq. (28) and apply the chain rule, Eq. (1) is transformed into the nonlinear partial differential equation
3cα2ϕggf+3c2α2ϕfϕ3+b2α2ϕfϕ+3bcα2ϕfϕ2− 3dcα2ϕfff+3cα2ϕh+3cϕhhf
=0,
(29) which admits three transitional symmetries, namely
K1= ∂
∂f,K2= ∂
∂h,K3= ∂
∂g. (30)
Taking a combination of these symmetry βK1+K2+K3, yields the following invariants:
m=h− g, n=f− βg, E=ϕ, (31)
and the group-invariant is ϕ(f,g,h) =E(m,n). Again invoking this group- invariant together with Eq. (31) and apply the chain rule, Eq. (29) transforms to:
b2α2EnE+3c2α2EnE3+6cα2βEmnn+3bcα2EnE2+3dcα2Ennn
+3cα2Emmn+3cEmmn+3cα2β2Ennn+3cα2Em=0, (32) which possess two symmetries
R1= ∂
∂m,R2= ∂
∂n. (33)
Combining these translational symmetries, R=R1 +R2, we get two invariants:
r=m− n, F=E, (34)
which yields the group-invariant solution E(m,n) = F(r). Using this group-invariant solution with Eq. (34) and apply the chain rule, Eq. (32) transforms to a third order ordinary differential equation
− 3cα2F′′′− 3c2α2F′F3− b2α2F′F− 3bcα2F′F2− 3cF′′′
− 3cα2β2F′′′+3cα2F′+6cα2βF′′′− 3dcα2F′′′=0. (35) Thus, Eq. (35) is a result of a series of symmetry reductions on a further modified 3D Zakharov-Kuznetsov Eq. (1). The integration of (35) with respect to r three times and taking the constants of integration to be zero leads to a first-order variable separable ordinary differential equation, which can be integrated easily with aid of Mathematica package. By reverting back into the original variables, we conclude that the solution [24,25] of a further modified 3D Zakharov-Kuznetsov Eq.
(1) is
u(t,x,y,z) =F(r),r=αt− y
α − x+ (β− 1)z, (36)
where
C1+r=
⎧⎨
⎩3
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
c(
1+α2(d+ (− 1+β)2) ) α2
√
F(r)
⎡
⎢⎢
⎣−
(5i+ ̅̅̅̅̅
√15)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅bc ic 30ib− 6 ̅̅̅̅̅
√15 b
√ ̅̅̅̅̅̅̅̅̅
√F(r) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1+ 6icF(r)
5ib− ̅̅̅̅̅
√15 b
√ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1+ 6icF(r)
5ib+ ̅̅̅̅̅
√15 b
√
{ EllipticE
( isinh−1
( ̅̅̅
√6 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− ic
(− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅
√F(r))⃒
⃒⃒
⃒⃒ 5i− ̅̅̅̅̅
√15 5i+ ̅̅̅̅̅
√15 )
− EllipticF (
isinh−1
( ̅̅̅
√6 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− ic
(− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅
F(r)
√ )⃒⃒
⃒⃒
⃒ 5i− ̅̅̅̅̅
√15 5i+ ̅̅̅̅̅
√15 ) }
− 4(
10b2+15bcF(r) +9c2F(r)2) ] } /(
2 ̅̅̅̅̅
√10 b2
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− F(r)3(
10b2+15bcF(r) +9c2F(r)2)
√ )
,
with C1 being a constant of integration.
We now consider symmetry Γ2+αΓ6and we get the following in- variants
f =y, g=z, h=αt− x
α , ϕ=u. (37)
Using these invariants, Eq. (1) reduces to
3cα2ϕggh+3c2α2ϕhϕ3+b2α2ϕhϕ+3bcα2ϕhϕ2+3dcϕhhh− 3cα3ϕh+3cα2ϕhff
=0,
(38) which admits four symmetries:
K1= ∂
∂h, K2= ∂
∂f,K3= ∂
∂g,K4= − g∂
∂f+f ∂
∂g. (39)
Employing the translational symmetries, yields the following invariants
m=h− βg, n=f− g, E=ϕ. (40)
Now using these invariants, Eq. (38) transforms to b2α2EmE+3c2α2EmE3+6cα2βEmmn+3bcα2EmE2+3dcEmmm
+3cα2β2Emmm+6cα2Emnn− 3cα3Em=0. (41) This equation admits two translational symmetries, viz.,
R1= ∂
∂m, R2= ∂
∂n. (42)
The combination of these translational symmetries, R= R1+R2, gives the two invariants:
r=m− n,F=E, (43)
which transforms Eq. (41) into to a nonlinear ordinary differential equation
3cα2β2F′′′+3c2α2F′F3+b2α2F′F+3bcα2F′F2
+6cα2F′′′− 3cα3F′− 6cα2βF′′′+3cdF′′′=0. (44) Solving Eq. (44) and relapsing back into the original variable, we get the solution of a further modified 3D Zakharov-Kuznetsov Eq. (1) as u(t,x,y,z) =F(r),r=αt− x
α − y− (β− 1)z, (45)
where
C1+r= − {
3
̅̅̅2 5
√ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
c (
2+d α2− 2β+β
2
√ )
F(r)(
10b2+15bcF(r) +9c2F(r)2
+ (
EllipticE (
isinh−1
( ̅̅̅
√6 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− ic
(− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅
√F(r))⃒
⃒⃒
⃒⃒ 5i− ̅̅̅̅̅
√15 5i+ ̅̅̅̅̅
√15 )
− EllipticF (
isinh−1
( ̅̅̅
√6 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− ic
(− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅
√F(r))⃒
⃒⃒
⃒⃒ 5i− ̅̅̅̅̅
√15 5i+ ̅̅̅̅̅
√15
)) ̅̅̅̅̅̅̅̅̅
√F(r)
×10i ̅̅̅
√6 b2
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− ic
(− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1− 6icF(r) (− 5i+ ̅̅̅̅̅
√15) b
√ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1+ 6icF(r) (5i+ ̅̅̅̅̅
√15) b
√ )}
/ (2 ̅̅̅̅̅
√10 b2
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− F(r)3(
10b2+15bcF(r) +9c2F(r)2)
√
).
It should be noted that the same procedure is used in case 2 below.
Case 2. a,b,c,d arbitrary but not in the form of case 1.
Here we obtain the following five symmetries:
Γ1=∂
∂t,Γ2= ∂
∂x,Γ3= ∂
∂y,Γ4= ∂
∂z,Γ5=y∂
∂z− z∂
∂y. (46)
In order to obtain symmetry reduction and solutions of a further modified 3D Zakharov-Kuznetsov Eq. (1), firstly, we consider the linear combination of time and spatial translational symmetries Γ1+Γ2+αΓ3+Γ4 (so as to obtain a travelling wave transformation), which yields the following invariants
f=t− x, g=t− y, h=αt− z, ϕ=u, (47) where ϕ is treated as a new dependent variable and f,gandh being the new independent variables. Employing these relations, Eq. (1) trans- forms to
ϕf+ϕg+ϕh− aϕf− bϕ2ϕf− cϕ3ϕf− dϕfff− ϕfgg− α2ϕfhh=0. (48) Now computing the symmetries of Eq. (48), leads to the following three symmetries, viz.,
K1= ∂
∂f,K2= ∂
∂g,K3= ∂
∂h. (49)
In order to obtain a travelling wave transformation, we take the linear combination of βK1+K2+K3, and we get the following new invariants
m=h− g, n=f− βg, E=ϕ, (50)
where E is treated as a new dependent variable and m,n as new inde- pendent variables. Invoking these invariants, Eq. (48) reduces to:
− dEnnn− cEnE3− Em− 2βEmnn+αEm+En− aEnE− bEnE2
− 2Emmn− β2Ennn− βEn=0. (51)
Eq. (51) adimits only two translational symmetries, namely R1= ∂
∂m,R2= ∂
∂n. (52)
The linear combination of these translational symmetries yields two invariants
r=m− n,F=E. (53)
Using these invariants, Eq. (51) reduces to a nonlinear ordinary differential equation
(α+β− 2)F′+ (β2− 2β+2+d)F′′′+ (cF3+bF2+aF)F′=0. (54)
Solving Eq. (54) with the help of Mathematica software and back substitution of the original variables, we conclude that the solution of a further modified 3D Zakharov-Kuznetsov Eq. (1) is
u(t,x,y,z) =F(r),r= (α+β− 2)t+ (1− β)y− z+x, (55) where
Conservation laws
Conservation laws are mathematical expressions of physical laws, such as momentum, conservation of energy and mass. We will
implement the multiplier approach to derive the low-order conservation laws of a further modified 3D Zakharov-Kuznetsov Eq. (1). A conser- vation law of a further modified 3D Zakharov-Kuznetsov Eq. (1) is a total space–time divergence expression that vanishes on the solution space ε of Eq. (1),
DiTi|ε=0, (56)
with Di being the total differential operator and Ti being the conserved vector. We determine all the second-order multipliers of a further modified 3D Zakharov-Kuznetsov Eq. (1)[26–28] and we obtain
−
2EllipticΠ (
1− A2
A3
; sin−1
( ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− F(r) +A3
− A2+A3
√ )⃒
⃒⃒
⃒ A2− A3
A1− A3
)
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− 60+30α+30β+10aF(r) +5bF(r)2+3cF(r)3
√
A3
×
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− F(r) +A1
A1− A3
√ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
− (F(r) − A2)(F(r) − A3) (A2− A3)2
√
( − A2+A3)
=C1± ir
̅̅̅̅̅
√30 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
2+d− 2β+β2
√ ,
A1=
[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅( 1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2
+4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1 3
/ (9 ̅̅̅
32
√ c) − ̅̅̅
32
√(
90ac− 25b2) /[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅(
1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2 +4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1
3− 5b
9c, A2= −
[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅( 1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2
+4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1 3
(1− i ̅̅̅
√3) 18 ̅̅̅
32
√ c +
((
1+i ̅̅̅
√3)(
90ac− 25b2) )/
{[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅( 1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2
+4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1
3×9×22/3c
⎫
⎬
⎭− 5b
9c, A3= −
[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅( 1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2
+4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1 3
(1+i ̅̅̅
√3) 18 ̅̅̅
32
√ c +
((
1− i ̅̅̅
√3)(
90ac− 25b2) )/
{[ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅( 1350abc− 250b3− 7290αc2− 7290βc2+14580c2)2
+4(
90ac− 25b2)3
√
+1350abc− 250b3− 7290αc2− 7290βc2+14580c2]1
3×9×22/3c
⎫
⎬
⎭− 5b
9c.
Λ(
t,x,y,z,u,ut,ux,uy,uz,utt,utx,uty,tz,uxx,uxy,uxz,uyy,uyz,uzz
)
=F(y,z) + 1 12
(12uzz+6u2a+3cu4+4bu3+12duxx+12uyy
)C1+C2u
where C1 and C2 are constants and F(y,z)is an arbitrary function of two independent variables. The multiplier Λ of Eq. (1) has the property Λ(ut+auux+bu2ux+cu3ux+duxxx+uxyy+uxzz)
=DtT1+DxT2+D3T3+D4T4, (57) for the arbitrary function u(t,x,y,z)[29–32], where the predetermined arguments of T1,T2,T3,T4 are of some order in derivatives of the field variable u.The computations for T1,T2,T3,T4 from Eq. (57) with the aid
Tt1= 1 20cu5+1
12bu4+1 6au3+1
2uuzz+1 2uuyy+1
2duuxx, Tx1= 1
32c2u8+1 6uuzzzz+1
2d2u2xx+1 18b2u6+1
8a2u4+1 3uyyuzz+1
6uuyyyy
− 1 6uzuzzz− 1
6uyuyzz+1 3uuyyzz+1
2adu2uxx+1
3bdu3uxx+1 4cdu4uxx
− 1 6uyuyyy− 1
6uzuyyz+1 6u2zz+1
6u2yy+13
36bu3uyy+17 60cu4uyy
+1 2au2uyy+2
3duzzuxx+1
12bu2zu2+2 15cu2yu3+2
3duyyuxx+1 8acu6 +1
2duxut+2
15cu2zu3+1 12bcu7+1
12bu2yu2+13 36bu3uzz+1
6duuxxyy
− 1 2duutx− 1
6duzuxxz+1 6abu5− 1
6duyuxxy+1
2au2uzz+17 60cu4uzz+1
6duuxxzz, Ty1= − 1
2uuty+1 3uyuxyy− 1
6uuxyyy+1 3uyuxzz− 1
6uuxyzz+1 3uxyuzz− 1
6uxuyyy
+1 3uxyuyy− 1
6uxuyzz− 2
15cuxuyu3+1 2uyut− 1
12buxuyu2− 1 6duuxxxy
− 1
6duxuxxy− 1
30cu4uxy− 1
36bu3uxy+1
3duyuxxx+1 3duxyuxx, Tz1= − 1
6duuxxxz− 1
12buxuzu2− 1
6duxuxxz− 1
30cu4uxz− 1
36bu3uxz+1 3uzuxyy
+1
3duzuxxx+1 3uzuxzz+1
3duxzuxx+1 3uxzuzz− 2
15cuxuzu3+1 3uxzuyy
+1 2uzut− 1
2uutz− 1 6uuxyyz− 1
6uuxzzz− 1 6uxuyyz− 1
6uxuzzz;
(58)
Tt2= 1 2u2, Tx2= − 1
6u2z+1 3uuyy− 1
6u2y− 1 2du2x+1
5cu5+1 3au3+1
4bu4+duuxx+1 3uuzz, Ty2= − 1
3uxuy+2 3uuxy, Tz2= − 1
3uxuz+2 3uuxz;
(59) TtF= uF,
TxF= 1 2au2F+1
3bu3F+1 4cu4F+1
3uFzz+1
3uFyy+duxxF− 1 3uzFz+1
3uyyF +1
3uzzF− 1 3uyFy, TyF= − 1
3uxFy+2 3uxyF, TzF= − 1
3uxFz+2 3uxzF.
(60) respectively.
Conclusions
In this paper we constructed solutions of a further modified 3D Fig. 1.Evolution of travelling wave solution (36), where b= 0.0001,β = 1,
α=1,d=1,c=5..
Fig. 2.Evolution of travelling wave solution (55), where c= 0.0001,β = 0.0001,α=0.0001,a=0.001,b=0.001,d=0.001..
laws (58), (59) do not contain any independent variables whereas Eq.
(60) does. Therefore, they are respectively associated with conservation of energy and momentum. We further observe that the conserved quantities for a further modified 3D Zakharov-Kuznetsov Eq. (1), contain an arbitrary function F(y,z)hence this constitutes infinitely many conservation laws. Furthermore, higher order conservation laws for a further modified 3D Zakharov-Kuznetsov equation can be derived by increasing the order of the multiplier. However, this remains to be thoroughly investigated elsewhere. see Figs. 1 and 2.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
D.M. Mothibi would like to thank the University of Botswana for their hospitality during her research visit which initiated this work. D.M.
Mothibi would also like to thank Sol Plaatje University (SPU) for their financial support.
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