Multiple types of exact solutions and conservation laws of new coupled (2 + 1)-dimensional Zakharov–Kuznetsov system with time-dependent coefficients

Multiple types of exact solutions and conservation laws

of new coupled ( 2 + 1 ) -dimensional Zakharov–Kuznetsov system with time-dependent coefficients

BIKRAMJEET KAUR¹and R K GUPTA^{2,3 ,∗}

1School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147 004, India

2Department of Mathematics and Statistics, Central University of Punjab, Mansa Road, Bathinda 151 001, India

3Department of Mathematics, School of Physical and Mathematical Sciences, Central University of Haryana, Mahendergarh123 031, India

∗Corresponding author. E-mail: rajeshateli@gmail.com

MS received 29 October 2018; revised 11 March 2019; accepted 13 March 2019

Abstract. This paper investigates the new coupled(2+1)-dimensional Zakharov–Kuznetsov (ZK) system with time-dependent coefficients for multiple types of exact solutions by using the Lie symmetry transformation method.

Similarity transformation reduces the system of equations into ordinary differential equations and further, these are solved for solutions having bright, dark and singular solitons as well as periodic waves. Also, the solutions appeared in terms of time-dependent coefficientβ(t)and analysed graphically to show the effect of this arbitrary function. It is proved that the given system is nonlinear self-adjoint, and some conservation laws are obtained by applying the new conservation theorem.

Keywords. Lie’s infinitesimals criterion; exact solutions; new coupled(2+1)-dimensional Zakharov–Kuznetsov system; conservation laws.

PACS Nos 02.20.Sv; 04.20.Jb; 02.30.Jr; 05.45.Yv

1. Introduction

The nonlinear physical phenomena can be dissemi- nated by solving nonlinear partial differential equations (PDEs). The evolution profile of solutions for such equa- tions plays a key role in mathematical, physical and engineering sciences. Thus, it is required to develop an algorithm to find a variety of solutions. The present work addresses to obtain multiple types of exact solutions for a system of equations by the symmetry transfor- mation method [1–6]. This method reduces the number of independent variables by one at each step. This enables us to reduce the nonlinear PDEs to nonlinear ordinary differential equations (ODEs) which can be solved for exact solutions [7]. Further, the Lie symme- tries are used to construct conservation laws [8–15]. The present report describes one such nonlinearly evolved (2+1)-dimensional new coupled Zakharov–Kuznetsov (ncZK) system with time-dependent coefficients as follows:

1 ≡u_t +α(t)(uv)x+γ (t)(vw)x

+β(t)(ux x+uyy)x =0,

2 ≡vt +λ(t)(uw)x+β(t)(vx x+vyy)x =0, 3 ≡wt +λ(t)(uv)x +β(t)(wx x+wyy)x =0, (1.1) whereα(t),β(t),γ (t)andλ(t)are arbitrary functions oft.

The ncZK system models dust-acoustic solitary waves evolved in magnetised dusty plasmas, nonlinear ion- acoustic waves and hot isothermal electrons [16–21].

With constant coefficients, system (1.1) is solved suc- cessfully by Elboree [22], Wei and Tang [23], Khalique [24] and Fahmy [25] for different types of exact solu- tions. System (1.1) with time-dependent coefficients has not been investigated yet by the symmetry trans- formation method for exact solutions as well as for conservation laws. Therefore, the main motive of the present paper is to investigate system (1.1) for exact 0123456789().: V,-vol

solutions and conservation laws via the symmetry trans- formation method.

The paper is structured as follows. Section2presents the stepwise procedure for determining the Lie infinites- imal symmetries, symmetry groups and optimal sys- tems. Further, in §3, similarity solutions and reduced ODEs are obtained corresponding to vector fields of optimal system, and reduced ODEs are analysed for getting multiple types of exact solutions. Then, some of the solutions are represented graphically. In §4, the nonlinear self-adjointness of the system is proved and the new conservation theorem is employed to determine the non-trivial conservation laws.

2. Lie symmetry transformations

This section presents the Lie symmetry transformation criterion to find exact solutions for system (1.1). For Lie infinitesimal symmetries, consider an infinitesimal generator of the following form [1]:

=ξ1 ∂

∂t +ξ2 ∂

∂x +ξ3 ∂

∂y +η1 ∂

∂u +η2 ∂

∂v +η3 ∂

∂w, (2.1)

where ξ1, ξ2, ξ3, η1, η2 and η3 are functions of (t,x,y,u, v, w). The third-order prolongation Pr³of for system (1.1) is given as follows:

Pr³=+η₁^t ∂

∂ut +η^x₁ ∂

∂ux +η^{x x x}₁ ∂

∂ux x x

+η^{x yy}₁ ∂

∂ux yy +η^t₂ ∂

∂ut +η₂^x ∂

∂vx

+η^{x x x}₂ ∂

∂vx x x +η^{x yy}₂ ∂

∂vx yy +η^t₃ ∂

∂ut

+η^x3

∂

∂wx +η^{x x x}3

∂

∂wx x x +η^{x yy}₃ ∂

∂wx yy, (2.2) whereη^t₁,η₂^t,η^t₃,η^x₁,η₂^x,η₃^x,η₁^{x x x},η^{x x x}₂ ,η₃^{x x x},η₁^{x yy},η^{x yy}₂ andη₃^{x yy} are the extended infinitesimals [1]. Equation (2.1) represents the Lie point symmetry of the ncZK system if the following conditions hold:

Pr³(1)|₁₌0,2=0,3=0=0, Pr³(2)|1=0,2=0,3=0 =0,

Pr³(3)|1=0,2=0,3=0=0. (2.3) The following symmetry equations are obtained from eq. (2.3):

η₁^t +α(t)ξ1(uv)x+α(t)

×(uη^x₂+η1vx+η^x₁v+uxη2) +γ(t)ξ1(vw)x+γ (t)

×(vη₃^x+η2wx +η^x₂w+vxη3) +β(t)ξ1ux x x+β(t)η1^{x x x}

+β(t)ξ1ux yy+β(t)η₁^{x yy} =0, η2^t +λ(t)ξ1(uw)x+λ(t)

×(uη^x3+η1wx+η1^xw+uxη3) +β(t)ξ1vx x x+β(t)η^{x x x}₂ +β(t)ξ1vx yy+β(t)η^{x yy}₂ =0, η₃^t +λ(t)ξ1(uv)x+λ(t)

×(uη^x₂+η1vx+η^x₁v+uxη2) +β(t)ξ1wx x x+β(t)η₃^{x x x}

+β(t)ξ1wx yy+β(t)η₃^{x yy} =0. (2.4) Using the extended infinitesimals in eq. (2.4) and by equating the coefficients of like derivatives ofu,vandw, we have obtained the over-determined system of linear PDEs and the solution of over-determining equations reads as follows:

ξ1= 1 β(t)

3a1

β(t)dt+a4

, ξ2 =a1x+a3, ξ3 =a1y+a2,

η1 =a5u, η2 =a5v, η3 =a5w, (2.5) wherea₁,a₂,a₃,a₄,a₅ are arbitrary constants and the coefficientsα(t),β(t),γ (t),λ(t)follow the subsequent conditions:

β(t)γ (t)(2a1+a5)+(−βtγ (t)+β(t)γ(t))ξ1 =0, β(t)α(t)(2a1+a5)+(−βtα(t)+β(t)α(t))ξ1=0, β(t)λ(t)(2a1+a5)+(−βtλ(t)+β(t)λ(t))ξ1 =0.

(2.6) From eq. (2.5), the one-dimensional Lie algebra [1,11]

is generated by the following vector fields:

1 = 1

β(t)∂t, 2 =∂x, 3=∂y, 4 =u∂u+v∂v+w∂w,

5 =x∂x+y∂y+3

β(t)dt

β(t) ∂t. (2.7)

These vector fields generate one-parameter groups G():

G₁:(x,t,y,u, v, w)→

x,t+

β(t),y,u, v, w

, G₂:(x,t,y,u, v, w)→(x +,t,y,u, v, w), G3:(x,t,y,u, v, w)→(x,t,y+,u, v, w), G4:(x,t,y,u, v, w)→(x,t,y,ue, ve, we), G5:(x,t,y,u, v, w)

→

xe,t+3 β(t)dt

β(t) ,ye,u, v, w

, (2.8)

where variousGi (i =1,2, . . . ,5)represent the sym- metry groups and we can state the following theorem that leads to the optimal system [1,11].

Theorem 2.1. Ifu = P(x,t,y), v = Q(x,t,y), w = R(x,t,y)represents the solution of system(1.1), so are the functions

(u⁽¹⁾, v⁽¹⁾, w⁽¹⁾)=

P

x,t− β(t),y

, Q

x,t− β(t),y

, R

x,t− β(t),y

, (u⁽²⁾, v⁽²⁾, w⁽²⁾)=(P(x −,t,y),Q(x −,t,y),

R(x−,t,y)),

(u⁽³⁾, v⁽³⁾, w⁽³⁾)=(P(x,t,y−),Q(x,t,y−), R(x,t,y−)),

(u⁽⁴⁾, v⁽⁴⁾, w⁽⁴⁾)=(eP(x,t,y),eQ(x,t,y), eR(x,t,y)),

(u⁽⁵⁾, v⁽⁵⁾, w⁽⁵⁾)=

P

xe⁻,t−3

β(t)dt β(t) ,ye⁻

, Q

xe⁻,t−3 β(t)dt β(t) ,ye⁻

, R

xe⁻,t−3

β(t)dt β(t) ,ye⁻

. (2.9) It describes that for each subgroup of symmetry groups Gi (i = 1,2, . . . ,5), there will be a family of group- invariant solutions and in the present case there are an infinite number of such subgroups. Hence, it is not feasi- ble to enlist all the possible group-invariant solutions for ncZK system. The group-invariant solution introduces the concept of an optimal system.

The optimal system is obtained using the adjoint rep- resentation and is given in terms of the Lie series as follows:

Ad(exp(i))j =j −[i, j] +²

2 [i,[i, j]] + · · · , (2.10)

Table 1. Adjoint table.

Ad 1 2 3 4 5

1 1 2 3 4 5−31

2 1 2 3 4 5−2

3 1 2 3 4 5−3

4 1 2 3 4 5

5 1e³ 2e 3e 4 5

where is the real parameter.[i, j] = ij −ji

is defined as the Lie bracket. The non-zero Lie brackets from Lie algebra (2.7) are obtained as follows:

[1, 5] = −[5, 1] =31, [2, 5] = −[5, 2] =2,

[3, 5] = −[5, 3] =3. (2.11) The Lie series (2.10) and Lie commutation relation (2.11) further help to write down the adjoint table (table1) of the ncZK system.

Then, the adjoint table is used to construct the optimal system generated by the following vector fields:

(i) 5+ρ4,

(ii) 4+μ3+θ2+ν1, (iii) 3+r2+s1,

(iv) 2+p1, (v) 1,

whereρ,μ,θ,ν,r,sandpare arbitrary constants. These vector fields are used in the next section for similarity reductions.

3. Similarity reductions and exact solutions

For similarity reductions with respect to the vector fields described in the optimal system, the following charac- teristic equations are used:

dt ξ1 = dx

ξ2 = dy ξ3 = du

η1 = dv η2 = dw

η3. (3.1)

3.1 Vector field5+ρ4

The characteristic equation (3.1) for this vector field 5+ρ4gives the following invariants:

ζ1 = x

(

β(t)dt)^1/3, ζ2 = y (

β(t)dt)^1/3, u(x,t,y)= F(ζ1, ζ2)

β(t)dt _ρ/3

,

v(x,t,y)=G(ζ1, ζ2)

β(t)dt _ρ/3

, w(x,t,y)= H(ζ1, ζ2)

β(t)dt _ρ/3

. (3.2)

Using eq. (2.6), the time-dependent coefficients can be written as follows:

α(t)= p1β(t)

β(t)dt

₋₍2+ρ)/3

, γ (t)= p₂β(t)

β(t)dt

₋₍2+ρ)/3

, λ(t)= p3β(t)

β(t)dt

₋₍2+ρ)/3

, (3.3)

where p₁, p₂and p₃are arbitrary constants. Using the invariants and coefficient functions given in eqs (3.2) and (3.3), the ncZK system is transformed into the fol- lowing reduced PDEs:

F_ζ1ζ1+F_ζ2ζ2−Fρ−3p1F G_ζ1−3p1F_ζ1G−3p2G H_ζ1

−3p2G_ζ1H−3F_ζ1ζ1ζ1−3F_ζ1ζ2ζ2 =0, G_ζ1ζ1+G_ζ2ζ2−Gρ−3p3H F_ζ1−3p3H_ζ1F

−3G_ζ1ζ1ζ1−3G_ζ1ζ2ζ2 =0,

H_ζ1ζ1+H_ζ2ζ2−Hρ−3p₃F G_ζ1−3p₃F_ζ1G−3H_ζ1ζ1ζ1

−3H_ζ1ζ2ζ2 =0. (3.4)

We are interested in non-trivial solutions for the given system, but the above reduced PDEs possess only trivial solutions. Hence, this case is not physically important.

3.2 Vector field4+μ3+θ2+ν1

In this case, the invariants and the corresponding vari- able coefficients are obtained as follows:

ζ1 = −θ ν

β(t)dt+x, ζ2 = −μ

ν

β(t)dt+y, u(x,t,y)=F(ζ1, ζ2)e^1/ν

β(t)dt, v(x,t,y)=G(ζ1, ζ2)e^1/ν

β(t)dt, w(x,t,y)= H(ζ1, ζ2)e^1/ν

β(t)dt, α(t)=q₁β(t)e^−1/ν

β(t)dt, γ (t)=q2β(t)e^{−1/νβ(t)dt}, λ(t)=q3β(t)e^−1/ν

β(t)dt, (3.5)

whereq1, q2 andq3 are arbitrary constants. Invariants (3.5) yield the following reduced PDEs for the ncZK system:

F_ζ1θ−F_ζ2μ+F+q₁νF G_ζ1+q₁νF_ζ1G+q₂νG H_ζ1 +q2νG_ζ1H+νF_ζ1ζ1ζ1+νF_ζ1ζ2ζ2 =0,

G_ζ1θ+G_ζ2μ−G−q3νHF_ζ1−q3νH_ζ1F

−νG_ζ1ζ1ζ1−νG_ζ1ζ2ζ2 =0,

−H_ζ1θ −H_ζ2μ+H+q3νF G_ζ1+q3νF_ζ1G

+νH_ζ1ζ1ζ1+νH_ζ1ζ2ζ2 =0. (3.6) By applying the Lie symmetry transformation algorithm on the reduced PDEs, the following infinitesimals are obtained:

ξ¹=q5, ξ²=q4, η¹=η²=η³ =0, (3.7) whereq4andq5are arbitrary constants. The new invari- ants are obtained using the characteristic equation

dζ1

ξ¹ = dζ2

ξ² = dF η¹ = dG

η² = dH η³ as follows:

ζ =q4ζ1−q5ζ2, F(ζ1, ζ2)= f(ζ ),

G(ζ1, ζ2)=g(ζ ), H(ζ1, ζ2)=h(ζ ). (3.8) Finally, we obtained the reduced ODEs for the ncZK system as follows:

f_ζq4θ+f_ζq5μ+f+q1νf g_ζq4+q1νf_ζq4g+q2νgh_ζq4

+q2νg_ζq4h+νf_ζζζq₄³+νf_ζζζq₅²q4=0, g_ζq4θ −g_ζq5μ−g−q3νh f_ζq4−q3νh_ζc4f

−νg_ζζζq₄³−νg_ζζζq₅²q4=0,

−h_ζq4θ+h_ζq5μ+h+q3νf g_ζq4

+q3νf_ζc4g+νh_ζζζq₄³+νh_ζζζq₅²q4 =0. (3.9) The solution of system (3.9) in the form of power series is considered as follows:

f(ζ )=

∞ n=0

Dnζⁿ, g(ζ )=

∞ n=0

Enζⁿ, h(ζ )=

∞ n=0

Knζⁿ, (3.10) whereDn,EnandKn are unknown coefficients and are to be determined later. From the substitution of (3.10) into the reduced ODEs (3.9), we obtained the following recurrence relations:

Dn+3 = −1

(νq₄³+νq₅²q4)(n+1)(n+2)(n+3)

×

(q4θ+q5μ)(n+1)Dn+1+Dn

+νq1q4

_n

l=0

(n−l+1)(DlEn−l+1+ElDn−l+1)

+νq2q4

_n

l=0

(n−l+1)(ElKn−l+1+KlEn−l+1) ,

En+3= 1

(νq₄³+νq₅²q4)(n+1)(n+2)(n+3)

×

(q4θ −q5μ)(n+1)En+1−En

−νq3q4

_n

l=0

(n−l+1)(KlDn−l+1+DlKn−l+1) ,

K_n+3= 1

(νq₄³+νq₅²q4)(n+1)(n+2)(n+3)

×

(q4θ −q5μ)(n+1)Kn+1−Kn

−νq₃q_{4 n}

l=0

(n−l+1)(D_lE_n−l+1+E_lD_n−l+1) , (3.11) whereD0,D1,D2,E0,E1,E2,K0,K1,K2are arbitrary constants, and

D3 = −1

6(νq₄³+νq₅²q4)[(q4θ+q5μ)D1

+D₀+νq₁q₄(D₀E₁+E₀D₁) +νq2q4(E0K1+K0E1)],

E₃ = 1

6(νq₄³+νq₅²q4)[(q₄θ−q₅μ)E₁

−E0+νq3q4(K0D1+D0K1)],

K3 = 1

6(νq₄³+νq₅²q₄)[(q4θ−q5μ)K1

−K0+νq3q4(D0E1+E0D1)].

In this case, the explicit solutions are obtained in the following form:

u(x,t,y)=e^1/ν

β(t)dt

×

D0+D1ζ +D2ζ²− 1 6(νq₄³+νq₅²q4)

×[(q4θ +q5μ)D1+D0+νq1q4(D0E1+E0D1) +νq₂q₄(E₀K₁+K₀E₁)]ζ³

− ∞ n=1

1

(νq₄³+νq₅²q4)(n+1)(n+2)(n+3)

×

(q4θ +q5μ)(n+1)Dn+1+Dn

+νq1q4

_n

l=0

(n−l+1)(DlEn−l+1+ElDn−l+1)

+νq2q4

_n

l=0

(n−l+1)(ElKn−l+1

+KlEn−l+1) ζⁿ⁺³ , v(x,t,y)=e^1/ν

β(t)dt

×

E0+E1ζ +E2ζ²+ 1 6(νq₄³+νq₅²q₄)

×[(q4θ −q5μ)E1−E0

+νq3q4(K0D1+D0K1)]ζ³ +

∞ n=1

1

(νq₄³+νq₅²q₄)(n+1)(n+2)(n+3)

×

(q4θ−q5μ)(n+1)En+1−En

+νq3q4

_n

l=0

(n−l+1)(KlDn−l+1

+DlKn−l+1) ζⁿ⁺³ , w(x,t,y)=e^1/ν

β(t)dt

×

K0+K1ζ +K2ζ²+ 1 6(νq₄³+νq₅²q4)

×[(q4θ −q5μ)K1−K0

+νq3q4(D0E1+E0D1)]ζ³ +

∞ n=1

1

(νq₄³+νq₅²q4)(n+1)(n+2)(n+3)

×

(q4θ−q5μ)(n+1)Kn+1−Kn

+νq₃q_{4 n}

l=0

(n−l+1)(D_lE_n−l+1

+E_lD_n−l+1) ζⁿ⁺³ , (3.12) where

ζ=q4

−θ ν

β(t)dt+x

−q5

−μ ν

β(t)dt+y

. 3.3 Vector field3+r2+s1

The invariants, coefficient functions and reduced PDEs are found to be as follows:

ζ1 = −r s

β(t)dt+x, ζ2= −1 s

β(t)dt+y, u(x,t,y)=F(ζ1, ζ2), v(x,t,y)=G(ζ1, ζ2), w(x,t,y)= H(ζ1, ζ2), α(t)=c1β(t),

γ (t)=c2β(t), λ(t)=c3β(t) (3.13) and

−F_ζ1r −F_ζ2+c1s F G_ζ1+c1s F_ζ1G+c2sGH_ζ1 +c2sG_ζ1H+s F_ζ1ζ1ζ1+s F_ζ1ζ2ζ2 =0, G_ζ1r +G_ζ2−c3sHF_ζ1−c3s H_ζ1F−sG_ζ1ζ1ζ1

−sG_ζ1ζ2ζ2 =0,

−H_ζ1r −H_ζ2+c3s F G_ζ1+c3s F_ζ1G+s H_ζ1ζ1ζ1

+s H_ζ1ζ2ζ2 =0, (3.14)

wherec1,c2 andc3are arbitrary constants. Once again the application of similarity transformation gives the infinitesimals as follows:

ξ¹ =c5, ξ² =c4, η¹ =η² =η³=0, (3.15) wherec4andc5are arbitrary constants. The correspond- ing invariants and ODEs are obtained as follows:

ζ =c4ζ1−c5ζ2, F(ζ1, ζ2)= f(ζ ),

G(ζ1, ζ2)=g(ζ ), H(ζ1, ζ2)=h(ζ ) (3.16) and

−f_ζc4r + f_ζc5+c1s f g_ζc4+c1s f_ζc4g+c2sgh_ζc4

+c2sg_ζc4h+s f_ζζζc³₄+s f_ζζζc²₅c4 =0, g_ζc4r −g_ζc5−c3sh f_ζc4−c3sh_ζc4f −sg_ζζζc³₄

−sg_ζζζc²₅c4=0,

−h_ζc4r+h_ζc5+c3s f g_ζc4+c3s f_ζc4g+sh_ζζζc₄³

+sh_ζζζc²₅c4=0. (3.17)

These ODEs are solved by the computational software Maple and finally the exact solutions for the ncZK sys- tem become

(i) c2 = 6c²₇(12c²₇c²₅c₄²+c²₄c1c8+6c²₇c₄⁴+6c₇²c⁴₅+c²₅c1c8) c²₈c3

, u(x,t,y)= 8A−c5+r c4

2c3c4s −6B c3

tanh²

c6+c7

−(r c₄−c₅)

β(t)dt+s(−c₄x+c₅y) s

, v(x,t,y)= −c8(8A−c5+r c4)

12c₄s B +c8tanh²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

, w(x,t,y)= −c8(8A−c5+r c4)

12c4s B +c₈tanh²

c₆+c₇

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

.

(3.18) (ii) c₂= − 3c²₁

16c3,

u(x,t,y)= −4A−c5+r c4

2c3c4s + 6B c3

sech²

c₆+c₇

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

, v(x,t,y)= 2(−4A−c₅+r c₄)

3c1sc4 + 8B c1

sech²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

, w(x,t,y)= 2(−4A−c5+r c4)

3c₁sc₄ + 8B c₁ sech²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

. (3.19)

(iii) c₂ = 6c²₇(12c²₇c52c²₄+c²₄c1c8+6c₇²c⁴₄+6c²₇c⁴₅+c²₅c1c8) c₈²c3

,

u(x,t,y)= −−4A−c5+r c4

2c3c4s +6B c3

csch²

c₆+c₇

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

, v(x,t,y)= −c₈(−4A−c₅+r c₄)

12c4s B +c8csch²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

, w(x,t,y)= c8(−4A−c5+r c4)

12c4s −c8csch²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

. (3.20) (iv) c₂ = 6c₇²(12c²₇c52c²₄+c²₄c1c8+6c²₇c⁴₄+6c²₇c⁴₅+c₅²c1c8)

c²₈c₃ ,

u(x,t,y)= −−8A−c5+r c4

2c4sc3 + 6B c3

cot²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

, v(x,t,y)= −c8(−8A−c5+r c4)

12c4s B +c8cot²

c6+c7

−(r c₄−c₅)

β(t)dt+s(−c₄x +c₅y) s

, w(x,t,y)= c8(−8A−c5+r c4)

12c₄s B −c8cot²

c6+c7

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

.

(3.21) (v) c₂= − 3c²₁

16c3, u(x,t,y)= − C

2c₃ +6Bc₈²

c3c₇²JacobiNS²

c7+c8

−(r c4−c5)

β(t)dt+s(−c4x +c5y) s

,c6

, v(x,t,y)= 2C

3c1 −8Bc²₈

c1c²₇ JacobiNS²

c7+c8

−(r c₄−c₅)

β(t)dt+s(−c₄x+c₅y) s

,c6

, w(x,t,y)= −2C

3c1 +8Bc²₈

c₁c²₇ JacobiNS²

c7+c8

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

,c6

. (3.22)

(vi) c2 = − 3c₁² 16c3, u(x,t,y)= − C

2c3 +6c²₈c₆²B

c3c²₇ ×JacobiSN²

c₇+c₈

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

,c₆

,

v(x,t,y)= 2C

3c₁ − 8c₈²c²₆B

c1c²₇ ×JacobiSN²

c7+c8

−(r c4−c5)

β(t)dt+s(−c4x+c5y) s

,c6

, w(x,t,y)= −2C

3c1 +8c²₈c²₆B

c1c₇² ×JacobiSN²

c7+c8

−(r c₄−c₅)

β(t)dt+s(−c₄x+c₅y) s

,c6

.

(3.23)

Here

A=c₇²c³₄s+c₇²c₄sc²₅, B =c²₇

c²₅+c²₄ ,

C = 4c²₈c₄³sc²₆+4c²₈c₄³s+4c²₈c4sc²₅c²₆+4c82c4sc²₅−c₅+r c4

c4s , (3.24)

whereci are arbitrary constants.

The dark, bright and periodic wave solutions are pre- sented graphically in figures 1–6 for solutions (3.18), (3.19) and (3.23), respectively, by considering suitable parametric values. Figures1a–1c show the dark soliton solution (3.18) foru,vandw, respectively whenβ(t)= sin(t)by three-dimensional (3D) plots. Figures 2a–2c describe the effect of coefficient function β(t) on the wave profile of the solution by two-dimensional (2D)

plots when it is linear in t, exponential function of t and trigonometric function oft. Figures3a–3c show the bright soliton solution (3.19) at particular β(t) = e^t and figures 4a–4c show the effect of various β(t) on the solution profile. Figures 5a–5c and 6a–6c show the periodic wave profiles of solution (3.23) by 3D plots for β(t) = t and 2D plots for different β(t), respectively.

Figure 1. Dark soliton solution (3.18) profile withc1 =0.25,c3 = −1,c4 =1,c5 =2,c6 = 0.5,c7 =0.75,c8 = 1.5, r=0.5,s=1,y=1,β(t)=sin(t): (a)u, (b)v, (c)w.

(a) (b) (c)

Figure 2. Effect ofβ(t)on the profile of the solution (3.18) withc1=0.25,c3= −1,c4=1,c5=2,c6=0.5,c7=0.75, c8=1.5,r =0.5,s=1,y=1,t =1: (a)u, (b)v, (c)w.

Figure 3. Bright soliton profile of solution (3.19) withc1=0.25,c3=1,c4=1,c5=2,c6=0.5,c7=0.75,c8=1.5, r=0.5,s=1,y=1,β(t)=e^t: (a)u, (b)v, (c)w.

(a) (b) (c)

Figure 4. Effect of time-dependent coefficientβ(t)on solution (3.19) withc1 =0.25,c3=1,c4=1,c5=2,c6=0.5, c7=0.75,c8=1.5,r=0.5,s=1,y=1,t =1: (a)u, (b)v, (c)w.

Figure 5. Periodic wave profile of solution (3.23) withc1=0.25,c3= −1,c4=1,c5=2,c6=0.5,c7=0.75,c8=1.5, r=0.5,s=1,y=1,β(t)=t: (a)u, (b)v, (c)w.

3.4 Vector field2+ p1

By using similar procedure as described in the previous section, the reduced ODEs in this case are obtained as follows:

−p f_ζζζr₄²+ f_ζ −r1p f g_ζ −r1p f_ζg−r2pgh_ζ

−r₂pg_ζh− p f_ζζζr₅²=0,

pg_ζζζr₄²−g_ζ +r3ph f_ζ +r3ph_ζ f + pg_ζζζr₅²=0,