EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5751
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Abundant Optical Solutions for The Stochastic
Heisenberg Ferromagnetic Equation Used in a

Ferromagnet via Two Different Methods

Wael W. Mohammed1,∗, Naveed Iqbal1, Hijeh M. Alshammari1,
Mohamed S. Algolam1

1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440,
Saudi Arabia

Abstract. In this study, we look at the stochastic Heisenberg Ferromagnetic equation (SHFE)
perturbed in the Itô sense by multiplicative Brownian motion. The SHFE is transformed into a dif-
ferent Heisenberg Ferromagnetic equation with random variable coefficients (HFE-RVCs) utilizing
the proper transformation. To provide novel solutions for trigonometric functions, hyperbolic func-
tions, and rational functions for HFE-RVCs, we employ the generalizing Riccati equation mapping
technique (GREM-method) and Jacobi elliptic functions (JEF-method). The solutions of SHFE
can then be obtained. For the first time, we postulate that the solution to the Heisenberg Ferro-
magnetic equation is stochastic, in contrast to earlier research that suggested it was deterministic.
Additionally, we investigate the impact of multiplicative Brownian motion on the exact solutions
of the SHFE by offering several graphical representations.

2020 Mathematics Subject Classifications: 35A20, 60H15, 60H10, 83C15, 35Q51

Key Words and Phrases: White noise, stochastic solutions, Jacobi elliptic functions method,
random variable coefficients

1. Introduction

The Heisenberg ferromagnetic equation has important implications for the behavior
of ferromagnetic materials. It predicts phenomena such as spontaneous magnetization,
where the atoms’ magnetic moments align parallel to each other even in the absence of
an external magnetic field. It also explains the existence of magnetic domains and how
they can change their alignment under the influence of an external field. Additionally,
the Heisenberg equation helps scientists understand the temperature dependence of mag-
netism and phenomena such as the Curie temperature, which represents the transition
from ferromagnetic to paramagnetic behavior [3, 4].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5751

Email addresses: w.mohammed@uoh.edu.sa (W. W. Mohammed), n.iqbal@uoh.edu.sa (N. Iqbal),
wwelhadad@gmail.com (H. M. Alshammari), moh.algolam@uoh.edu.sa (M. S. Algolam)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 2 of 15

On the other side, random fluctuations play a crucial role in the behavior of ferromag-
netic materials as described by the Heisenberg ferromagnetic equation. These fluctuations
arise from the probabilistic nature of quantum mechanics and lead to variations in the
alignment of atomic magnetic moments, both in the absence and presence of an external
magnetic field. This means that even when the system is in its lowest energy state, there
will still be some degree of variation in the magnetic moments’ orientation. These fluctu-
ations can give rise to interesting phenomena, such as spontaneous magnetization and the
formation of magnetic domains.

It seems more relevant here the stochastic Heisenberg Ferromagnetic equation (SHFE)
is taken into consideration as follows:

idU + [ρ1Uxx + ρ2Uyy + ρ3Uxy − ρ4 |U|2 U ]dt = iνUdβ, (1)

where U is a stochastic complex function of the variables t, x and y; ν is the amplitude of
noise and β(t) is Brownian motion in one variable t; ρ2, ρ2, ρ3 and ρ4 are real functions
of the variable t and they are defined as

ρ1 = λ4(θ + θ2), ρ2 = λ4(θ1 + θ2), ρ3 = 2λ4θ2, ρ4 = 2λ4K,

where λ is the lattice parameter with the interaction coefficients θ, θ1, θ2, while K is the
anisotropic parameter .

Because of the relevance of Heisenberg Ferromagnetic equation, many researchers have
acquired the exact solutions for this equation without stochastic term by employing dif-
ferent methods such as F-expansion method [6], generalized Riccati mapping [9], modified
exp-function expansion [2], Jacobi elliptic functions [12], auxiliary ordinary differential
equation [13], sub-ODE [11], Hirota bilinear [5, 14], and sine-Gordon expansion and the
modified exp(−Φ(ξ))-expansion function methods [10]. However, many authors have stud-
ied the SHFE (1) by utilizing different methods including (G′/G)-expansion method [1],
Jacobi elliptic function method [7] and mapping method [8].

This work aims to obtain accurate solutions to the SHFE (1). In order to do this,
we apply an appropriate transformation to change the SHFE into another HFE-RVCs.
Following that, we use the JEF and GREMmethods to obtain the solutions for HFE-RVCs.
Finally, we may obtain the stochastic solutions of SHFE by applying the transformation
that was applied. Because of the significance of the Heisenberg Ferromagnetic equation
(1) in the behavior of ferromagnetic materials, these obtained solutions are essential for
comprehending a number of challenging physical processes. Using MATLAB tools, we
present some graphics to illustrate the impact of the stochastic term.

The rest of this paper is organized as follows: In Section 2, we derive HFE-RVCs from
SHFE (1) and determine the solutions of HFE-RVCs. In Section 3, we obtain the solutions
to SHFE (1). We discuss how Brownian motion affects the solutions in Section 4. Lastly,
we present the conclusions.



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 3 of 15

2. HFE-RVCs and Its Solutions

In this section, we derive the HFE-RVCs. By applying the transformation

U(t, x, y) = W(t, x, y) exp[iφ(t, x, y) + νβ(t)− 1

2
ν2t], (2)

we get HFE-RVCs as follows

iWt −Wφt + ρ1[Wxx + 2iφxWx + iφxxW − φ2
xW]

+ρ2[Wyy + 2iφyWy + iφyyW − φ2
yW] + ρ3[Wxy + iφxyW

+iφxWy + iφyWx − φxφyW]−A(t)W3 = 0,
(3)

where we applied the Itô derivatives, A(t) = ρ4e
2νβ(t)−ν2t and W is a stochastic real

function.

2.1. GREM-method

The GREM-method described in [15] is used here. To attain the solutions of the
HFE-RVCs (3), we suppose the solutions of Eq. (3) in the special forms:

W(t, x, y) =
N∑
k=0

αk(t)Fk(µ), µ = k1x+ k2y +

∫ t

0
λ(τ)dτ, (4)

and
φ(t, x, y) = ψ0(t) + ψ1(t)x+ ψ2(t)y. (5)

where α0, α1,....., αm−1, αm, λ, ψ0, ψ1, and ψ2, are functions of t and αm ̸= 0, and F
fulfills

F ′ = sF2 + rF + p, (6)

with s, r, and p are constants. First, let us balance Wxx with W3 in Eq. (3) in order to
determine the value of N as follows

N + 2 = 3N =⇒ N = 1.

Rewriting Eq. (4) as
W(t, x, y) = α0(t) + α1(t)F(µ). (7)

We differentiate Eqs. (7) and (5) with regards to t, x and y as follows

Wt = (
·
α0 + pα1λ) + (

·
α1 + α1rλ)F + sλα1F2,

Wx = k1α1[sF2 + rF + p], Wy = k2α1[sF2 + rF + p],

Wxx = k21α1[2s
2F3 + 3srF2 + (2sp+ r2)F + rp],

Wyy = k22α1[2s
2F3 + 3srF2 + (2sp+ r2)F + rp],

Wxy = k1k2α1[2s
2F3 + 3srF2 + (2sp+ r2)F + rp],



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 4 of 15

W3 = α3
1F3 + 3α0α

2
1F2 + 3α2

0α1F + α3
0, (8)

and

φt =
·
ψ0 +

·
ψ1x+

·
ψ2y, φx = ψ1, φy = ψ2, φxx = φyy = φxy = 0. (9)

Substituting Eqs. (5), (7), (8) and (9) into Eq. (3). Then, by setting each of Fk’s
coefficients to zero, we gain

F0 :
·
α0 + pλα1 + 2pk1ρ1ψ1α1 + 2pk2ρ1ψ2α1 + pk2ρ3ψ1α1 + pk1ρ3ψ2α1 = 0,

F1 :
·
α1 + rλα1 + 2rk1ρ1ψ1α1 + 2rk2ρ1ψ2α1 + rk2ρ3ψ1α1 + rk1ρ3ψ2α1 = 0,

F2 : sλα1 + 2sk1ρ1ψ1α1 + 2sk2ρ1ψ2α1 + sk2ρ3ψ1α1 + sk1ρ3ψ2α1 = 0,

xF0 : α0

·
ψ1 = 0,

yF0 : α0

·
ψ2 = 0,

xF1 : α1

·
ψ1 = 0,

yF1 : α1

·
ψ2 = 0,

F0 : −α0

·
ψ0 + prα1(k

2
1ρ1 + k22ρ2 + k1k2ρ3)− ψ2

1ρ1α0 − ψ2
2ρ2α0 − ψ1ψ2ρ3α0 −Aα3

0 = 0,

F1 : α1[
·
ψ0 + (2sp+ r2)(k21ρ1 + k22ρ2 + k1k2ρ3)− ψ2

1ρ1 − ψ2
2ρ2 − ψ1ψ2ρ3 − 3Aα2

0] = 0,

F2 : 3srα1(k
2
1ρ1 + k22ρ2 + k1k2ρ3)− 3Aα0α

2
1 = 0,

and
F3 : 2s2α1(k

2
1ρ1 + k22ρ2 + k1k2ρ3)−Aα3

1 = 0.

By solving these equations, we have

α0(t) = r = 0, α1 = ℓ, k21ρ1 + k22ρ2 + k1k2ρ3 =
ℓ2A(t)

2s2
, (10)

λ(t) = −2k1ρ1ψ1 − 2k2ρ2ψ2 − k2ρ3ψ1 − k1ρ3ψ2, (11)

and

ψ1 = ℏ1, ψ2 = ℏ2, ψ0(t) = −pℓ
2

s

∫ t

0
A(τ)dτ +

∫ t

0
(ℏ21ρ1 + ℏ22ρ2 + ℏ1ℏ2ρ3)dτ, (12)

where ℏ1, ℏ2 and ℓ are constants. For simplicity, let ℏ1 = k1 and ℏ2 = k2 in order to
obtain

λ(t) = −ℓ
2A(t)

s2
, ψ0(t) = (

ℓ2

2s2
− pℓ2

s
)

∫ t

0
A(τ)dτ,



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 5 of 15

and
α0(t) = 0, α1 = ℓ, ψ1 = k1, ψ2 = k2, r = 0.

Hence, by utilizing Eqs (7) and (10), the solution of the HFE-RVCs (3) is

W(t, x, y) = ℓF(µ), µ = k1x+ k2y −
ℓ2ρ4
s2

∫ t

0
e2νβ(τ)−ν2τdτ. (13)

Depending on p and s, there exist many sets to find the solution F of Eq. (6) as
follows:

Set I: If ps < 0, hence Eq. (6) possess the solutions:

F1(µ) = −
√

−p
s

tanh(
√
−psµ),

F2(µ) = −
√

−p
s

coth(
√
−psµ),

F3(µ) = −
√

−p
s

(
coth(

√
−4psµ)± csch(

√
−4psµ)

)
,

F4(µ) =
−1

2

√
−p
s

(
tanh(

1

2

√
−psµ) + coth(

1

2

√
−psµ)

)
.

Consequently, HFE-RVCs (3) has the hyperbolic function solutions:

W1(t, x, y) = −ℓ
√

−p
s

tanh(
√
−psµ), (14)

W2(t, x, y) = −ℓ
√

−p
s

coth(
√
−psµ), (15)

W3(t, x, y) = −ℓ
√

−p
s

(
coth(

√
−4psµ)± csch(

√
−4psµ)

)
, (16)

W4(t, x, y) = −1

2
ℓ

√
−p
s

(
tanh(

1

2

√
−psµ) + coth(

1

2

√
−psµ)

)
. (17)

Set II: If ps > 0, hence Eq. (6) possess the solutions:

F5(µ) =

√
p

s
tan(

√
psµ),

F6(µ) = −
√
p

s
cot(

√
psµ),

F7(µ) =

√
p

s

(
tan(

√
4psµ)± sec(

√
4psµ)

)
,



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 6 of 15

F8(µ) = −
√
p

s

(
cot(

√
4psµ)± csc(

√
4psµ)

)
,

F9(µ) =
1

2

√
p

s

(
tan(

1

2

√
psµ)− cot(

1

2

√
psµ)

)
,

Consequently, HFE-RVCs (3) has the trigonometric function solutions:

W5(t, x, y) = ℓ

√
p

s
tan(

√
psµ), (18)

W6(t, x, y) = −ℓ
√
p

s
cot(

√
psµ), (19)

W7(t, x, y) = ℓ

√
p

s

(
tan(

√
4psµ)± sec(

√
4psµ)

)
, (20)

W8(t, x, y) = ℓ1

√
p

s

(
cot(

√
4psµ)± csc(

√
4psµ)

)
, (21)

W9(t, x, y) =
1

2
ℓ1

√
p

s

(
tan(

1

2

√
psµ)− cot(

1

2

√
psµ)

)
. (22)

Set III: If p = 0 and s ̸= 0, hence Eq. (6) possess the the rational function solution

F10(µ) =
−1

sµ
.

Consequently, the HFE-RVCs (3) has the solution

W10(t, x, y) = − ℓ

sµ
, (23)

where µ = k1x+ k2y − ℓ2ρ4
s2

∫ t
0 e

2νβ(τ)−ν2τdτ.

2.2. JEF-method

Assuming that the solutions to Eq. (3) with N = 1 have the following type

W(t, x, y) = a0(t) + a1(t)J(µ), (24)

where J(µ) can be any of the elliptic functions cn(ωµ, ň), sn(ωµ, ň) or dn(ωµ, ň). We
differentiate Eq. (24) with regards to t, x and y to get

Wt =
·
a0 +

·
α1J + ωλa1J

′, Wx = ωk1a1J
′, Wy = ωk2a1J

′,

Wxx = a1k
2
1ω

2J ′′ = k21a1B1J + k21a1B2J
3,

Wyy = k22a1B1J + k22a1B2J
3,

Wxy = k1k2a1B1J + k1k2a1B2J
3,



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 7 of 15

W3 = a31J
3 + 3a0a

2
1J

2 + 3a20a1J + a30, (25)

where B1 and B2 are constants that will be determined later and rely on ω, and ň. Plugging
Eqs. (25) and (9) into Eq. (3). Then, by setting each of Jk’s coefficients to zero, we gain

J0 :
·
a0 = 0,

J1 :
·
a1 = 0,

J ′ : ωa1[λ+ 2ρ1ψ1k1 + 2ρ2ψ2k2 + ρ3ψ1k2 + ρ3ψ2k1] = 0,

xJ0 : a0
·
φ1 = 0,

yJ0 : a0
·
φ2 = 0,

xJ1 : a1
·
φ1 = 0,

yJ1 : a1
·
φ2 = 0,

J0 : −a0
·
φ0 − φ2

1ρ1a0 − φ2
2ρ2a0 − φ1φ2ρ3a0 −Aa30 = 0,

J1 : −a1
·
φ0 + ρ1a1k

2
1B1 + ρ2a1k

2
2B1 + ρ3a1k1k2B1

−ρ1a1φ2
1 − ρ2a1φ

2
2 − ρ3a1φ1φ2 − 3Aa20a1 = 0,

J2 : 3Aa0a
2
1 = 0,

and
J3 : ρ1a1k

2
1B2 + ρ2a1k

2
2B2 + ρ3a1k1k2B2 −Aa31 = 0.

Solving these equations, yields

a0(t) = 0, a1 = ℏ, ρ1k21 + ρ2k
2
2 + ρ3k1k2 =

ℏ2A(t)
B2

, φ1 = k1, φ2 = k2, (26)

and

φ0 =
ℏ2(B1 − 1)

B2

∫ t

0
A(τ)dτ, λ(t) =

−2ℏ2

B2
A(t),

where ℏ is a constant. Therefore, the solutions of the HFE-RVCs (3), using Eqs (24) and
(26), are

W(t, x, y) = ℏJ(µ), µ = k1x+ k2y −
2ℏ2ρ4
B2

∫ t

0
e2νβ(τ)−ν2τdτ. (27)

In the following, we define J(µ) as:
Set 1: When J(µ) = sn(ωµ, ň), Eq. (27) has the form

W(t, x, y) = ℏ[sn(k1ωx+ ωk2y −
2ωℏ2ρ4
B2

∫ t

0
e2νβ(τ)−ν2τdτ, ň)], (28)

where
B1 = −ω2(1 + ň2) and B2 = 2ω2ň2.



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 8 of 15

Set 2: When J(µ) = cn(ωµ, ň), Eq. (27) has the form

W(t, x, y) = ℏ[cn(k1ωx+ ωk2y −
2ωℏ2ρ4
B2

∫ t

0
e2νβ(τ)−ν2τdτ, ň)], (29)

where
B1 = ω2(1− 2ň2) and B2 = −2ω2ň2.

Set 3: When J(µ) = dn(ωµ, ň), Eq. (27) has the form

W(t, x, y) = ℏ[dn(k1ωx+ ωk2y −
2ωℏ2ρ4
B2

∫ t

0
e2νβ(τ)−ν2τdτ, ň)], (30)

where
B1 = ω2(2− ň2) and B2 = −2ω2.

3. Exact Solutions of SHFE

The solution of SHFE (1) is obtained by putting Eq. (13) into Eq. (2) as

U(t, x, y) = W(t, x, y) exp[iφ(t, x, y) + νβ(t)− 1

2
ν2t], (31)

where

φ(t, x, y) = k1x+ k2y + (
ρ4ℓ

2

2s2
− pρ4ℓ

2

s
)

∫ t

0
e2νβ(τ)−ν2τdτ.

3.1. GREM-Method

Substituting Eqs (18)-(23) into (31), we get the solutions of the SHFE (1) as:

U1(t, x, y) = ℓ

√
−p
s

(
tanh[

√
−ps(k1x+ k2y −

ℓ2ρ4
s2

∫ t

0
e2νβ(τ)−ν2τdτ)]

)
e[iφ+νβ(t)− 1

2
ν2t],

(32)

U2(t, x, y) = ℓ

√
−p
s

(
coth[

√
−ps(k1x+ k2y −

ℓ2ρ4
s2

∫ t

0
e2νβ(τ)−ν2τdτ)]

)
e[iφ+νβ(t)− 1

2
ν2t],

(33)

U3(t, x, y) = ℓ
√

−p
s

(
coth[

√
−4ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]

+csch[
√
−4ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]
)
e[iφ+νβ(t)− 1

2
ν2t],

(34)

U4(t, x, y) = −1
2ℓ
√

−p
s

(
tanh[12

√
−ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]

+ coth[12
√
−ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]
)
e[iφ+νβ(t)− 1

2
ν2t],

(35)

for ps < 0,

U5(t, x, y) = ℓ

√
p

s
tan[

√
ps(k1x+ k2y −

ℓ2ρ4
s2

e2νβ(τ)−ν2τdτ)]e[iφ+νβ(t)− 1
2
ν2t], (36)



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 9 of 15

U6(t, x, y) = −ℓ
√
p

s
cot[

√
ps(k1x+ k2y −

ℓ2ρ4
s2

∫ t

0
e2νβ(τ)−ν2τdτ)]e[iφ+νβ(t)− 1

2
ν2t], (37)

U7(t, x, y) = ℓ
√

p
s

(
tan[

√
4ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]

− sec[
√
ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]
)
e[iφ+νβ(t)− 1

2
ν2t],

(38)

U8(t, x, y) = ℓ
√

p
s

(
cot[

√
4ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]

+ csc[
√
4ps(k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]
)
e[iφ+νβ(t)− 1

2
ν2t]

(39)

U9(t, x, y) =
1
2ℓ
√

p
s

(
tan[

√
ps
2 (k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]

− cot[
√
ps
2 (k1x+ k2y − ℓ2ρ4

s2

∫ t
0 e

2νβ(τ)−ν2τdτ)]
)
e[iφ+νβ(t)− 1

2
ν2t],

(40)

for ps > 0, and

U10(t, x, y) =
( −1

s(k1x+ k2y − ℓ2ρ4
s2

∫ t
0 e

2νβ(τ)−ν2τdτ)

)
e[iφ+νβ(t)− 1

2
ν2t] for s ̸= 0. (41)

Remark 1. Eqs (32) and (33), with ν = 0 coincide with the solutions (45) and (46) that
reported in [6].

3.2. JEF-method

The solutions of the SHFE (1) are obtained by substituting Eqs (28)-(30) into (31) as:

U(t, x, y) = ℏ
(
sn(k1ωx+ ωk2y −

ℏ2ρ4
ωň2

∫ t

0
e2νβ(τ)−ν2τdτ, ň)

)
e[iφ+νβ(t)− 1

2
ν2t], (42)

U(t, x, y) = ℏ
(
cn(k1ωx+ ωk2y +

ℏ2ρ4
ωň2

∫ t

0
e2νβ(τ)−ν2τdτ, ň)

)
e[iφ+νβ(t)− 1

2
ν2t], (43)

and

U(t, x, y) = ℏ
(
dn(k1ωx+ ωk2y +

ℏ2ρ4
ω

∫ t

0
e2νβ(τ)−ν2τdτ, ň)

)
e[iφ+νβ(t)− 1

2
ν2t], (44)

where

φ0 = k1x+ k2y +
ℏ2(B1 − 1)

B2

∫ t

0
A(τ)dτ.

If ň→ 1, then the Eqs (42)-(44) become

U(t, x, y) = ℏ
(
tanh(k1ωx+ ωk2y −

ℏ2ρ4
ω

∫ t

0
e2νβ(τ)−ν2τdτ)

)
e[iφ+νβ(t)− 1

2
ν2t], (45)

and

U(t, x, y) = ℏ
(
sech(k1ωx+ ωk2y +

ℏ2ρ4
ω

∫ t

0
e2νβ(τ)−ν2τdτ)

)
e[iφ+νβ(t)− 1

2
ν2t]. (46)

Remark 2. If we put ν = 0 in Eqs (42)-(46), we have the same solutions that stated in
[12].



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 10 of 15

4. Discussion and impacts of noise

Discussion: In this paper, the exact solutions of the SHFE (1) were acquired. We
applied two methods such as the GREM-method and JEF-method which they provided
many types of solutions such as optical kink solution (32), optical singular solution (33),
optical singular periodic (36) and (37), elliptic solutions (42)-(44) and etc. One of the
key features of singular solitons in the Heisenberg ferromagnet equation is their ability
to maintain their shape and energy despite interacting with other solitons or external
perturbations. This stability arises from the non-linear nature of the equation, which
allows solitons to balance the competing effects of dispersion and nonlinearity. Solitons
in the Heisenberg ferromagnet equation often exhibit complex and interesting dynamics,
such as the formation of bound states or the reflection and transmission of solitons at
interfaces between different regions of the material.

Impacts of Brownian motion: the impacts of noise on the solutions to the Heisen-
berg ferromagnet equation can have significant implications for the design and performance
of magnetic devices. By understanding the impacts of noise on the solutions to the Heisen-
berg ferromagnet equation, scientists and engineers can work towards developing better
strategies for mitigating the effects of noise and improving the performance of magnetic
devices based on these solutions. Ultimately, addressing the challenges posed by noise
will be crucial in unlocking the full potential of the Heisenberg ferromagnet equation and
advancing the field of magnetic materials and technology.

Now, the effect of noise on the exact solutions of the SHFE (1) will be examined. We
provide some figures for different solutions with varying noise intensity ν. Graphs 1, 2 and
3 show the solutions U(t, x, y) presented in Eqs (43), (45) and (46) for distinct value of ν
as follows:



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 11 of 15

(a) ν = 0 (b) ν = 0.1

(c) ν = 0.4 (d) ν = 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time "t"

0

0.2

0.4

0.6

0.8

1

1.2

S
o

lu
ti
o

n

=0

=0.1

=0.4

=1

=2

(e) ν = 2 (f) ν = 0, 0.1, 0.4, 1, 2

Figure 1: (a-e) display 3D-shape of |U(x, y, t)| described in Eq. (43) with k1 = k2 = 1, p = s = 0.5, ℓ =
1, ρ4 = 1, x ∈ [−4, 4], y = 0 and t ∈ [0, 3] (f) exhibits 2D-shape of Eq. (43) with distinct ν



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 12 of 15

(a) ν = 0 (b) ν = 0.1

(c) ν = 0.4 (d) ν = 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time "t"

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S
o

lu
ti
o

n

=0

=0.1

=0.4

=1

=2

(e) ν = 2 (f) ν = 0, 0.1, 0.4, 1, 2

Figure 2: (a-e) display 3D-shape of |U(x, y, t)| presented in Eq. (43) with ň = ℏ = 0.5, k1 = k2 = 1,
ρ4 = 2, ω = 1, x ∈ [−4, 4], y = 0, and t ∈ [0, 4] (f) exhibits 2D-shape of Eq. (43) with different ν



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 13 of 15

(a) ν = 0 (b) ν = 0.1

(c) ν = 0.4 (d) ν = 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time "t"

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

S
o

lu
ti
o

n

=0

=0.1

=0.4

=1

=2

(e) ν = 2 (f) ν = 0, 0.1, 0.4, 1, 2

Figure 3: (a-e) display 3D-shape of |U(x, y, t)| presented in Eq (46) with k1 = k2 = 1, ω = 2, ρ4 = 1, x ∈
[−4, 4], y = 0 and t ∈ [0, 3] (f) shows 2D-shape of Eq. (46)

As seen in Figures 1-3, a wide variety of solutions, such as optical kink solutions, optical
singular solutions, optical periodic solutions, and others, arise when noise is disregarded
(i.e., ν = 0). After a few transit patterns, the surface flattens when noise is added at
ν = 0.1, 0.4, 1, 2. This finding demonstrates the stabilization of the SHFE Eq. (1)
solutions around zero due to multiplicative Brownian motion.



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 14 of 15

5. Conclusions

The stochastic Heisenberg ferromagnet equation (SHFE) (1) driven by multiplicative
noise in the Itô sense was examined in this study. We transformed the SHFE into another
Heisenberg ferromagnet equation with random variable coefficients (3) (HFE-RVCs) by
applying the proper transformation. We found novel stochastic exact solutions for HFE-
RVCs in the form of hyperbolic, trigonometric, rational, and elliptic functions utilizing two
various methods including the JEF-method and GREM-method, and then we acquired the
solutions of SHFE (1). Additionally, we generated a few earlier solutions, such the solutions
documented in [6, 12]. The obtained solutions are essential for comprehending a number
of challenging physical processes due to the significance of the Heisenberg ferromagnet
equation in the behavior of ferromagnetic materials. The impact of the stochastic term
on the stochastic exact solutions of SHFE was finally illustrated using a few graphics. In
future work, we can study SHFE (1) with additive noise.

Acknowledgements

This research has been funded by Scientific Research Deanship at the University of
Ha’il-Saudi Arabia through project number RG-24097.

References

[1] F. M. Al-Askar. The solitary wave solutions of the stochastic heisenberg ferromagnetic
spin chain equation using two different analytical methods. Frontiers in Physics,
11:1267673, 2023.

[2] H. Bulut, T. A. Sulaiman, and H. M. Baskonus. Dark, bright and other soliton
solutions to the heisenberg ferromagnetic spin chain equation. Physica A, 12:123,
2018.

[3] M. Daniel, L. Kavitha, and R. Amuda. Soliton spin excitations in an anisotropic
heisenberg ferromagnet with octupole-dipole interaction. Physical Review, 59:13774,
1999.

[4] X. Y. Gao. Variety of the cosmic plasmas: general variable-coefficient korteweg-de
vries-burgers equation with experimental/observational support. EPL, 110:15002,
2015.

[5] D. Y. Liu, B. Tian, Y. Jiang, X. Y. Xie, and X. Y. Wu. Analytic study on a (2+1)-
dimensional nonlinear schrödinger equation in the heisenberg ferromagnetism. Com-
putational and Applied Mathematics, 71:2001–2007, 2016.

[6] Y. L. Ma, B. Q. Li, and Y. Y. Fu. A series of the solutions for the heisenberg
ferromagnetic spin chain equation. Mathematical Methods in the Applied Sciences,
41:3316–3322, 2018.

[7] W. W. Mohammed, F. M. Al-Askar, C. Cesarano, T. Botmart, and M. El-Morshedy.
Wiener process effects on the solutions of the fractional (2+1)-dimensional heisenberg
ferromagnetic spin chain equation. Mathematics, 10:2043, 2022.



W. W. Mohammed et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5751 15 of 15

[8] W. W. Mohammed, F. Gassem, and R. Sidaoui. Random traveling wave equations for
the heisenberg ferromagnetic spin chain model and their optical stochastic solutions
in a ferromagnetic materials. Axioms, 13:864, 2024.

[9] A. R. Seadawy, N. Nasreen, D. Lu, and M. Arshad. Arising wave propagation in
nonlinear media for the (2+1)-dimensional heisenberg ferromagnetic spin chain dy-
namical model. Physica A, 538:122846, 2020.

[10] T. A. Sulaiman, T. Akturk, H. Bulut, and H. M. Baskonus. Investigation of various
soliton solutions to the heisenberg ferromagnetic spin chain equation. Journal of
Electromagnetic Waves and Applications, 32:1093, 2018.

[11] G. S. Tang, S. H. Wang, and G. W. Wang. Solitons and complexitons solutions of an
integrable model of (2+1)-dimensional heisenberg ferromagnetic spin chain. Nonlinear
Dynamics, 88:2319–2327, 2017.

[12] H. Triki and A. M. Wazwaz. New solitons and periodic wave solutions for the (2+1)-
dimensional heisenberg ferromagnetic spin chain equation. Journal of Electromagnetic
Waves and Applications, 30:788–794, 2016.

[13] Q. M. Wang, Y. T. Gao, C. Q. Su, B. Q. Mao, Z. Gao, and J. W. Yang. Dark
solitonic interaction and conservation laws for a higher-order (2+1)-dimensional non-
linear schrödinger-type equation in a heisenberg ferromagnetic spin chain with bilinear
and biquadratic interaction. Annals of Physics, 363:440–456, 2015.

[14] X. H. Zhao, B. Tian, D. Y. Liu, X. Y. Wu, J. Chai, and Y. J. Guo. Dark solitons
interaction for a (2+1)-dimensional nonlinear schrödinger equation in the heisenberg
ferromagnetic spin chain. Superlattices and Microstructures, 100:587–595, 2016.

[15] S. D. Zhu. The generalizing riccati equation mapping method in nonlinear evolution
equation: application to (2+1)-dimensional boiti-leon-pempinelle equation. Chaos,
Solitons and Fractals, 37:1335–1342, 2008.