Generalised exponential rational function method for obtaining numerous exact soliton solutions to a (3 + 1)-dimensional Jimbo–Miwa equation

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Generalised exponential rational function method for obtaining numerous exact soliton solutions to a (3 + 1)-dimensional

Jimbo–Miwa equation

SACHIN KUMAR¹and DHARMENDRA KUMAR² ^,∗

1Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi 110 007, India

2Department of Mathematics, SGTB Khalsa College, University of Delhi, Delhi 110 007, India

∗Corresponding author. E-mail: dharmendrakumar@sgtbkhalsa.du.ac.in MS received 3 February 2021; revised 18 April 2021; accepted 20 April 2021

Abstract. In this work, we apply the generalised exponential rational function (GERF) method on an extended (3+1)-dimensional Jimbo–Miwa (JM) equation which describes the modelling of water waves of long wavelength with weakly nonlinear restoring forces and frequency dispersion. This JM equation is also used to construct modelling waves in ferromagnetic media and two-dimensional matter-wave pulses in Bose–Einstein condensates.

The main purpose is to construct analytical wave solutions for the (3+1)-dimensional JM equation by utilising the GERF method with the help of symbolic computations. We have also presented three-dimensional plots to observe the dynamics of obtained results. To understand physical phenomenon through different shapes of solitary waves, we discussed solitons, the interaction of multiwave solitons, lump-type solitons and kink-type solutions.

Keywords. Exact solutions; solitons; closed form solutions; generalised exponential rational function method.

PACS Nos 87.10.Ed; 04.20.Jb; 02.90.+p

1. Introduction

Differential equations are important to model real physical systems. In view of physical interpretation, a partial differential equation represents a model for the motion of long water waves in channels of shallow depth. This multiwave interaction develops a wave travelling at continuous speed without losing its sharpness, usually called ‘soliton’. In the last few decades, many methods have been used to understand the movement and interaction of solitary waves. A few of them are: the auxiliary equation method [1], the direct algebraic method [2], the F-expansion method [3], the modified Kudryashov method [4], the homogeneous balance method [5], the Hirota method [6], the sub-ODE method [7], the solitary wave ansatz method [8], the (G/G)-expansion method [9] and the Lie symmetry analysis [10–18].

These are some important methods to solve nonlinear partial differential equations. The (3+1)-dimensional Jimbo–Miwa (JM) equation is of the form [19]

ux x x y+3uxux y+3uyux x+2uyt −3ux z=0 (1)

a second member in the Kadomtsev–Petviashvili (KP) hierarchy which is used to describe fascinating three- dimensional waves in mathematical physics and other interdisciplinary areas of mathematics. Luo [20] investigated eq. (1) using the generalised bilinear method in closed form and obtained theMth-order periodic wave solution for eq. (1). Furthermore, interactions between (i) rogue wave and breather wave, (ii) the kink-like soliton and breather have been presented. Lü et al [21], with the aid of symbolic computation, obtained the bi- soliton-like solutions which represent various waves such as classical crossline bi-solitons, curved bi-solitons and bi-soliton-like breathers for eq. (1). Cai [22] applied the modified direct method for getting the Lie point symmetry groups (finite invariant transformations) and the discrete symmetry groups of non-integrable nonlinear systems. Su and Dai [23] constructed one-periodic and two-periodic wave solutions by employing multi- dimensional elliptic -function. Based on symbolic computation, Singh [24] introduced a new test function for constructing exact solitary wave solutions of eq. (1).

Qi et al [25] obtained two solitary wave solutions for eq. (1) by employing a simplified Hirota’s method and 0123456789().: V,-vol

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other dynamical aspects through some plots. Sunet al constructed not only the lump solutions of eq. (1), but also studied the interaction between lump solutions and stripe soliton solutions in his work [26]. The extended version of eq. (1) is given by

ux x x y+3uxux y+3uyux x+2uyt

−3(ux z+uyz+uzz)=0. (2) Liu et al [27] used Hirota method on eq. (2) to find various classes of different rogue wave-type solutions usingMaple. They established three types of solitons, viz., ‘algebraic solitary waves (lump)’, ‘interaction solution between algebraically decayed soliton and exponentially decayed soliton (lumpoff)’ and ‘interaction solution between the lump and exponentially localised twin soliton wave (instant or rogue wave)’. Wanget al [28] found lump wave solutions that are directly derived by taking the solution in the bilinear equation. They also discussed the interaction of multiwave solutions between the lump wave and the stripe wave.

Xuet al[29] investigated multiexponential wave solution for the extended version of (3+1)-dimensional JM equations. Liu et al [30–36] applied many new tech- niques to obtain new analytical closed form solutions of various non-linear evolution equations in the field of nonlinear dynamics, optical physics, plasma physics and so on.

In this work, an attempt has been made to gen- erate group-invariant solutions of extended (3 +1)- dimensional Jimbo–Miwa equation [37]

u_{x x x y}+k(u_xu_y)x+hu_yt−k(u_{x z}+u_yz +u_zz)=0, (3) where u is the amplitude of the nonlinear wave, h and k are constants. Recently, Khalique and Moleleki [37] explored eq. (3) using Lie point symmetries. They obtained solutions in terms of incomplete elliptic inte- gral function and constructed conservation laws using Ibragimov’s new theorem.

To fill the gap of the previous findings, we are moti- vated to find more analytical solutions via a recently developed method known as the generalised exponential rational function method (GERF method). Firstly, the solutions are classified into the number of families and divided into the corresponding cases. To the best of our knowledge, such kinds of abundance of solutions has never been established in earlier literature. In this work, we study the problem of finding group-invariant solutions of the (3+1)-dimensional Jimbo–Miwa equation using the ‘new generalised exponential rational function method’ [38,39].

This paper is organised as follows: In §2, we introduce the generalised exponential function method. In §3, we obtain exact solitary wave solutions and some particular

three-dimensional shapes. In §4, we give results and discussions of our new results and in §5, we present the conclusion of our work.

2. Generalised exponential rational function method for eq. (3)

This method is invented by Ghanbari and Inc in [39].

The GERF method has the following steps:

Step 1: We set the given PDE for F(x,y,z,t) in the form(F,Fx,Fy,Fz,Fx x, . . .) =0. Using the trans- formationF =F(w)andw=αx+βy+γz−ct, we convert the given nonlinear partial differential equation to the following differential equation:

(, , , . . .)=0, (4)

where α, β, γ and c are arbitrary constants, and they need to be determined later in the manuscript.

Step2: Let us suppose that JM eq. (3) has the following assumed solution:

(w)= p1e^q¹^w+ p2e^q²^w

p3e^q³^w+ p4e^q⁴^w, (5) wherep₁, p₂,p₃,p₄andq₁,q₂,q₃,q₄are real/complex constants for obtaining the travelling wave solution of eq. (3) which is given by

(w)= A0+ N k=1

Ak(w)^k+ N k=1

Bk(w)^−k, (6) where A₀,A_k,B_k (1 ≤ k ≤ N) and p_n,q_n (1 ≤ n ≤ N) are constants to be determined, such that solution (6) satisfies eq. (4). Here, positive integerN needs to be computed using balancing principle.

Step3: By substituting eq. (6) into eq. (4) and collect- ing all terms, the following polynomial equation will be obtained:

P(Z1,Z2,Z3,Z4)=0 (7) in terms of Z_i = e^qⁱ^w for i = 1, . . . ,4. Setting each coefficient of P to zero, we derive a set of algebraic equations for pn, qn (1 ≤ n ≤ 4) and α, β, γ, c, A₀, A₁, B₁, using the computer algebra software Mathematica/Maple.

Step 4: In this step, we obtained the solution of algebraic equations, and using these non-trivial solutions in (6), one can obtain new dispersive soliton solutions of eq. (3).

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3. Some exact solitary wave solutions of eq. (3) via GERF method

In this section, we shall apply the generalised exponential function method to obtain the exact solitary wave solutions of the JM equation. Making use of the wave transformation u(x,y,z,t) = F(w) with w=αx+β y+γz−c t in eq. (3), we get the desired ordinary differential equation as

F(w)(βch−2α²βk F(w)+γk(α+β+γ ))

=α³βF⁽⁴⁾(w). (8) Integrating eq. (8), and neglecting the constant of inte- gration, we have

F(w)(βch+γk(α+β+γ ))+α³(−β)F⁽³⁾(w)

−α²βk F(w)² =0. (9)

3.1 Implementations of GERF method to the reduced eq.(9)

For simplicity, balancing principle used on terms F² and F⁽³⁾ in eq. (9) which yields N +3 = 2(N +1) implies N = 1. Employing N = 1 along with eqs (5) and (6) in eq. (9), one obtains

F(w)= A0+ A1(w)+ B1

(w). (10)

As we pursue, we shall construct the exact solitary wave solutions by employing the GERF method in §2 as follows:

Family1

For [p1,p2,p3,p4] = [i,−i,1,1] and[q1,q2,q3,q4]

= [i,−i,i,−i], eq. (5) yields

(w)=−tan(w), where w=αx+βy+γ z−c t. (11) For obtaining the values of parameters, we require to solve algebraic equations with the help of symbolic computation viaMathematica, and the following set of solutions can be furnished as

Case1.1

α =0, c= −γk(β+γ )

βh . (12)

We substitute these values in (10) and (11), so that we obtain

F(w)= A₀ −A₁ tan(w)−B₁ cot(w). (13) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)=A0 −A1 tan

γkt(β+γ )

βh +βy+γz

−B₁ cot

γkt(β+γ )

βh +βy+γz

. (14) Case1.2

β =0, γ =α, A1 = ±6α

k , B1 = ±6α

k . (15) We substitute these values in (10) and (11), so that we obtain

F(w)= A₀ ±6αtan(w)

k ±6αcot(w)

k . (16)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0 +6αtan(ct−αx+αz)

k

−6αcot(ct−αx+αz)

k . (17)

Similar solutions are also given as

u(x,y,z,t)= A0 +6αtan(ct−αx+αz) k

+6αcot(ct−αx+αz)

k , (18)

u(x,y,z,t)= A₀ −6αtan(ct−αx+αz) k

−6αcot(ct−αx+αz)

k , (19)

u(x,y,z,t)= A0 −6αtan(ct−αx+αz) k

+6αcot(ct−αx+αz)

k . (20)

Case1.3

α =0, c= −γk(β+γ )

βh , A1 = 6α k , B1 = −6α

k . (21)

F(w)= A0 −6αtan(w)

k −6αcot(w)

k . (22)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= − A1tan

γkt(β+γ )

βh +βy+γz

−B1cot

γkt(β+γ )

βh +βy+γz

+A0. (23)

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Family2

For [p1,p2,p3,p4] = [2 + i,2 − i,1,1] and [q1,q2,q3,q4] = [−i,i,−i,−i], eq. (5) yields

(w)=2+tan(w). (24)

The following cases are planned:

Case2.1

α =0, c= −γk(β+γ )

βh . (25)

F(w)= A0 +A1(tan(w)+2)+ B1

tan(w)+2. (26) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)=A0+A1

tan

γkt(β+γ )

βh +βy+γz

+2

+ B₁

tan

γkt(β+γ )

βh +βy+γz

+2. (27)

Case2.2

c= −4α³β+αγk+γk(β+γ )

βh , A1=0,

B1 = 30α

k . (28)

F(w)= A₀ + 30α

k(tan(w)+2). (29) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)

= A0 + 30α

k

tan

_t(^4α³^β+αγ^k+γ^{k(β+γ )})

βh +ξ

+2 , (30) whereξ =αx+βy+γz.

Case2.3

c= −4α³β+αγk+γk(β+γ )

βh ,

A₁ = −6α

k , B₁=0. (31)

F(w)= A0− 6α(tan(w)+2)

k . (32)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0

− 6α k

tan

t

4α³β+αγk+γk(β+γ )

βh +ξ

+2

. (33) Family3

For [p₁,p₂,p₃,p₄] = [1,−1,1,1] and [q1,q2,q3,q4] = [1,−1,1,−1], eq. (5) yields

(w)=tanh(w). (34)

Case3.1

c= −−4α³β+αγk+γk(β+γ )

βh ,

A1= 6α

k ,B1 =0. (35)

F(w)= A0 +6αtanh(w)

k . (36)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)

= A₀ +

6αtanh t

−4α³β+αγk+γk(β+γ )

βh +ξ

k .

(37) Case3.2

c= −−16α³β+αγk+γk(β +γ )

βh ,

A1= 6α

k , B1 = 6α

k . (38)

F(w)= A₀ +6αtanh(w)

k +6αcoth(w)

k . (39)

+ 6α k tanh

t

−16α³β+αγk+γk(β+γ )

βh +ξ

+ 6α k coth

t

−16α³β+αγk+γk(β+γ )

βh +ξ

. (40)

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Case3.3

c= −−4α³β+αγk+γk(β+γ )

βh ,

A1 =0, B1= 6α

k . (41)

F(w)= A₀ +6αcoth(w)

k . (42)

+6α k coth

t

−4α³β+αγk+γk(β+γ )

βh +ξ

. (43) Family4

For [p1,p2,p3,p4] = [2,1,1,1] and [q₁,q₂,q₃,q₄] = [1,0,1,0], eq. (5) yields

(w)= 1+2e^w

1+e^w . (44)

Case4.1

α =0, c= −γk(β+γ )

βh . (45)

F(w)= A0 + A₁(2e^w+1)

e^w+1 + B₁(e^w+1)

2e^w+1 . (46) Thus, JM equation (3) has exact soliton solution as

u(x,y,z,t)=A₀+ A₁

2 exp

γkt

ββ+γh +βy+γz

+1

exp γkt

β+γ

βh +βy+γz

+1

+ B₁

exp

γkt β+γ

βh +βy+γz

+1

2 exp γkt

ββ+γh +βy+γz

+1

. (47)

Case4.2

c= α³β−αγk−βγk−γ²k

βh ,

A₁ =0, B₁ = −12α

k . (48)

F(w)= A0 −12α (e^w+1)

k(2e^w+1) . (49) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0

− 12α

exp

−^t^α³^β−αγ^k_β^−βγ_h ^k^−γ²^k +ξ

+1 k

2 exp

−^t(^α³^β−αγ^k^−βγ^k^−γ²^k)

βh +ξ

+1 .

(50) Case4.3

c= α³β−αγk−βγk−γ²k

βh , A1 = 6α

k , B1=0. (51) We substitute these values in (10) and (44), so that we obtain

F(w)= A0 +6α (2e^w+1)

k(e^w+1) . (52)

+ 6α

2 exp

−^t^α³^β−αγ^k_β^−βγ_h ^k^−γ²^k +ξ

+1 k

exp

−^t(^α³^β−αγ^k−βγ^k−γ²^k)

βh +ξ

+1 .

(53) Family5

For [p1,p2,p3,p4] = [−2 + i,−2 − i,1,1] and [q1,q2,q3,q4] = [−i,i,−i,i], eq. (5) yields

(w)=tan(w)−2. (54)

Case5.1

α =0, c = −γk(β+γ )

βh . (55)

F(w)= A0 +A1tan(w)−2A1 + B1

tan(w)−2. (56) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0+A1 tan

γkt(β+γ )

βh +βy+γz

−2A1 − B₁ 2−tan

γkt(β+γ )

βh +βy+γz . (57)

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Case5.2

c= −4α³β+αγk+γk(β+γ )

βh ,

A1 =0, B1 = 30α

k . (58)

F(w)= A0 + 30α

k(tan(w)−2). (59) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0 − 30α

k(tan(ct−αx−βy−γz)+2). (60) Case5.3

c= −4α³β+αγk+γk(β+γ )

βh ,

A1 = −6α

k , B1=0. (61)

F(w)= A₀ +12α

k − 6αtan(w)

k . (62)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= 12α

k +A₀

−α k tan

t

4α³β +αγk+γk(β+γ )

βh +ξ

.

(63) Family6

For [p1,p2,p3,p4] = [−3,−2,1,1] and [q1,q2,q3,q4] = [0,1,0,1], eq. (5) yields

(w)= −3−2e^w

1+e^w . (64)

Case6.1

α =0, c= −γk(β+γ )

βh , A0,A1 and B1 arbitrary.

(65) We substitute these values in (10) and (64), so that obtain F(w)= A0 +A1

− 1 e^w+1 −2

− B1(e^w+1) 2e^w+3 .

(66) Thus, JM equation (3) has exact soliton solution as

u(x,y,z,t)= A0

+A1

− 1

exp_γ_kt_{(β+γ )}

βh +βy+γz +1−2

+B1

1 4 exp_γ_kt_{(β+γ )}

βh +βy+γz +6−1 2

. (67)

Case6.2

c= α³β−αγk−γk(β +γ )

βh , A1 =0,

B₁ = −36α

k ,A₀, α, β andγ arbitrary. (68) We substitute these values in (10) and (64), so that we obtain

F(w)= A₀ +36α (e^w+1)

k(2e^w+3) . (69) Thus, JM equation (3) has exact soliton solution as

u(x,y,z,t)= A0+ 12α k

×

⎛

⎝ 1 3 exp

_t(^α³^β−αγ^k^−γ^k^{(β+γ )})

βh −ξ

+2 +1

⎞

⎠. (70)

Case6.3

c= α³β−αγk−γk(β +γ )

βh , A1 = 6α

k , B1=0. (71) We substitute these values in (10) and (64), so that we obtain

F(w)= A0 +6 kα

− 1 e^w+1−2

. (72)

Thus, JM equation (3) has exact soliton solution as

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u(x,y,z,t)= A0+6α k

×

⎛

⎝− 1 exp

−^t(^α³^β−αγ^k^−γ^k^{(β+γ )})

βh +ξ

+1

−2

⎞

⎠. (73) Family7

For[p1,p2,p3,p4] = [1,−3,−1,1] and [q1,q2,q3, q4] = [1,−1,1,−1], eq. (5) yields

(w)=coth(w)−2. (74)

Case7.1

α =0, c= −γk(β+γ )

βh ,A0,A1 and B1 arbitrary.

(75) We substitute these values in (10) and (74), so that we obtain

F(w)= A0 +A1(coth(w)−2)+ B1

coth(w)−2. (76) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A₁

×

coth

γkt(β+γ )

βh +βy+γz

−2

+ B₁

coth

γkt(β+γ )

βh +βy+γz

−2 +A0. (77)

Case7.2

c= −−4α³β+αγk+γk(β+γ )

βh ,

A₁ =0, B₁ = −18α

k ,A₀, α, β andγ arbitrary. (78) We substitute these values in (10) and (74), so that we obtain

F(w)= A0 − 18α

k(coth(w)−2). (79) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0

− 18α

k

coth

_t(^−4α³^β+αγ^k+γ^{k(β+γ )})

βh +ξ

−2 .

(80)

Case7.3

c= −−4α³β+αγk+γk(β+γ )

βh ,

A₁= 6α

k , B₁=0. (81)

F(w)= A0 +6α(coth(w)−2)

k . (82)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0+ 6α

k

×

coth t

βh +ξ

−2

. (83) Case7.4

α =0, c= −γk(β+γ )

βh , A1=0. (84)

F(w)= A0 + B₁

coth(w)−2. (85)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0 + B1

coth

γkt(β+γ )

βh +βy+γz −2

. (86) Family8

For [p1,p2,p3,p4] = [1,1,−1,1] and [q1,q2, q₃,q₄] = [1,−1,1,−1], eq. (5) yields

(w)= −coth(w). (87)

Case8.1

α =0, c = −γk(β+γ )

βh ,A₀,A₁ and B₁ arbitrary. (88) We substitute these values in (10) and (87), so that we obtain

F(w)= A0 −A1coth(w)−B1coth(w). (89) Thus, JM equation (3) has exact soliton solution as

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u(x,y,z,t)=A₀−A₁coth

γkt(β+γ )

βh +βy+γz

−B1tanh

γkt(β +γ )

βh +βy+γz

. (90) Case8.2

c= −−4α³β+αγk+γk(β+γ )

βh ,

A1 = −6α

k , B1=0, α, β andγ arbitrary. (91) We substitute these values in (10) and (87), so that we obtain

F(w)= A0 +6αcoth(w)

k . (92)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0 + 6α

k

×coth t

−4α³β+αγk+γk(β +γ )

βh +ξ

. (93) Case8.3

c= −−16α³β +αγk+γk(β+γ )

βh ,

A1 = −6α

k , B1= −6α

k . (94)

F(w)= A0+ 6αtanh(w)

k +6αcoth(w)

k . (95)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A₀

+6α k tanh

t

−16α³β+αγk+γk(β+γ )

βh +ξ

+6α k coth

t

βh +ξ

. (96)

Family9

For [p1,p2,p3,p4] = [−2,−1,1,1] and [q1,q2, q3,q4] = [0,1,0,1], eq. (5) yields

(w)= −2−e^w

1+e^w , (97)

wherew=αx+βy+γ z−c t. The following cases are planned:

Case9.1

A1=0, B1 = −12α k , c= α³β−αγk−γk(β +γ )

βh , and A0 arbitrary. (98) We substitute these values in (10) and (97), so that we obtain

F(w)=A0+A1

− 1 e^w+1−1

+B1

1 e^w+2−1

. (99) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0

− 12α

k

exp

−^t(^α³^β−αγ^k^−γ^k^{(β+γ )})

βh +αx+βy+γz

+2 + 12α

k . (100)

Case9.2

c= α³β−αγk−γk(β +γ )

βh ,

A1= 6α

k , B1 =0, α, β and γ arbitrary. (101) We substitute these values in (10) and (97), so that we obtain

F(w)= A0+6α k

− 1 e^w+1−1

. (102)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A₀ − 6α

k − 6α

k

exp

−^t(^α³^β−αγ^k^−γ^k^{(β+γ )})

βh +αx+βy+γz +1

. (103)

Family10

For[p1,p2,p3,p4] = [1,0,1,1]and[q1,q2,q3,q4] = [1,0,1,0], eq. (5) yields

(w)= e^w

1+e^w, (104)

wherew=αx+βy+γ z−c t.

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Case10.1 A1 = 6α

k , B1 =0,

c= α³β−αγk−γk(β+γ )

βh , and A0 arbitrary. (105) We substitute these values in (10) and (104), so that we obtain

F(w)= A0+ 6αe^w

k(e^w+1). (106)

Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0+ 6αe^α^x^+β^y^+γ^z

k

exp

t(^α³^β−αγ^k−γ^{k(β+γ )})

βh

+exp(αx +βy+γz)

. (107)

Case10.2

c= −γk(β+γ )

βh , A1 =B1,

α =0, A0, β and γ arbitrary. (108) We substitute these values in (10) and (104), so that we obtain

F(w)= A0+B1e^−w+ B1e^w

e^w+1 +B1. (109) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A0

+B1exp

−γkt(β+γ )

βh +β(−y)−γz

+

B1exp

γkt(β+γ )

βh +βy+γz

exp

γkt(β+γ )

βh +βy+γz+1

+B1. (110)

Case10.3

α =0, c= −γk(β +γ ) βh ,

B₁ =0,A₀,A₁, β andγ arbitrary. (111) We substitute these values in (10) and (104), so that we obtain

F(w)= A0+ A₁e^w

e^w+1. (112)

Thus, JM equation (3) has exact soliton solution as

u(x,y,z,t)= A0+

A₁exp

γkt(β+γ )

βh +βy+γz

exp

γkt(β+γ )

βh +βy+γz

+1 .

(113) Family11

For [p1, p2,p3,p4] = [4i,0,1,−1] and [q1,q2, q₃,q₄] = [0,0,2i,−2i], eq. (5) yields

(w)= 1

sin(w)cos(w), (114)

wherew=αx+βy+γ z−c t.

Case11.1 α =0, A1=0,

c = −γk(β+γ )

βh and A₀,B₁, γ, β arbitrary. (115) We substitute these values in (10) and (114), so that we obtain

F(w)= A0+A1csc(w)sec(w)+B1sin(w)cos(w).

(116) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A₀+B₁sin

γkt(β+γ )

βh +βy+γz

×cos

γkt(β+γ )

βh +βy+γz

. (117) Case11.2

α =0, A1=0, c = −γk(β+γ )

βh and A₀,B₁, γ, β arbitrary. (118) We substitute these values in (10) and (114), so that we obtain

F(w)= A0+A1csc(w)sec(w). (119) Thus, JM equation (3) has exact soliton solution as u(x,y,z,t)= A₀+A₁csc

γkt(β +γ )

βh +βy+γz

(10)

Figure 1. Three types of dynamic behaviours of the multisoliton profile foruwith parameters A0=1, A1 =2 B1 =0.1, α=1, β=0.5, γ =3,h=1.

Figure 2. Three types of dynamic behaviours of the multisoliton profile foruwith parameters A0=1,A1=2,B1=0.1, α=1, β=0.5, γ =3,h=1.

Figure 3. Three types of dynamic behaviours of multisoliton profile for u with parameters A0 = 1, A1 = 2, B1=0.1, α=1, β=0.5, γ =3,h =1.

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Figure 4. Three types of dynamic behaviours of multisoliton profile for u with parameters A0 = 1, A1 = 2, B1=0.1, α=1, β=0.5, γ =3,h =1.

Figure 5. Two types of dynamic behaviours of multisoliton profile foru using solution (80) with parameters A0 = 1, A1=2,B1=0.1, α=1, β=0.5, γ =3,h =1.

Figure 6. Three types of dynamic behaviour of multisoliton profile for u with parameters A0 = 1, A1 = 2, B1=0.1, α=1, β=0.5, γ =3,h =1 atx=1,y=1.

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Figure 7. Two types of dynamic behaviours of multisoliton profile foru with parameters A0 = 1, A1 = 2,B1 = 0.1, α=1, β=0.5, γ =3,h=1 atx =1,y=1.

×sec

γkt(β +γ )

βh +βy+γz

. (120) 4. Results and discussion

Numerous exact solutions are obtained using the generalised exponential function method of an extended Jimbo–Miwa equation. These solutions are noval and different from previous findings. Such solutions are explained via three-dimensional plots in figures 1–7 for different values of parameters A0 = 1, A1 = 2,B1 = 0.1, α = 1, β = 0.5, γ = 3,h = 1 at x = 1,y = 1. These new results contain some spe- cial arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

In figure 1, various types of derived solutions (14), (17) and (23) are analysed using three-dimensional plots. In figure2, multisoliton profiles of solutions (27), (30) and (33) are provided for better understanding of dynamic behaviour. In figure3, kink-type soliton profiles are exhibited in solutions (37), (40) and (43). In figure4, three-dimensional plots of solutions (57), (60) and (63) are given. In figure 5, multisoliton solutions are give via three-dimensional plot for solution (80). In figure6, distinguished types of kink soliton solution are provided by the plots for solutions (90), (93) and (96).

In figure7, two different types of soliton solutions are shown for solutions (117) and (120).

However, this diversification gives a new dimension in the study of the Jimbo–Miwa equation to construct modelling waves in ferromagnetic media.

5. Conclusion

The ‘new generalised first extended (3+1)-dimensional Jimbo–Miwa equation (3)’, which is the three- dimensional equation (in space) proposed by Khalique and Moleleki [37], was used to find some solitary wave solutions. The new generalised extended rational function method proposed by Ghanbari and Inc, known as GERF method, is used to obtain new analytical wave solutions of Jimbo–Miwa equation. The ordinary differential equation was then solved and its general solution has been obtained. These derived solutions are entirely different from the rational solutions presented by Khalique and Moleleki [37] and other related research work. It is our belief that the results obtained in this study will urge experimental and theoretical scien- tists for manipulating solitons in other areas of applied mathematics, optics, mathematical physics and engineering to study complex phenomenon. To analyse the behaviour of equation physically, the obtained solutions are extended with numerical simulation. Thus, elastic multisoliton, positon and kink profiles of solutions are presented to make this research physically sound.

Acknowledgements

This work is funded by Science and Engineering Research Board (SERB), DST, India, under project scheme MATRICS via grant No. MTR/2020/000531.

The author, Sachin Kumar, has received this research grant.

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