The symbolic computation with the ansatz function and the logarithmic transformation method are used to obtain a formula for certain exact solutions of the

Keywords:
02.30Xx soliton solutions;
(3 + 1) Zakharov–Kuznetsov equation;
the logarithmic transformation method

Many physical phenomena originate in various fields of engineering and science, such as fluid dynamics, fiber optics, quantum mechanics, plasma, and physics. These physical phenomena were constructed in the form of nonlinear equations [1,2, 3,4,5, 6,7]. To understand the behavior of a phenomenon, we need to solve nonlinear equations that describe the phenomenon, which is often challenging. Many methods have been developed over the years to solve such nonlinear equations, many of which are based on assumptions. Investigating the exact solutions for these nonlinear equations is a major area of concern for many mathematicians and physicists because of their important role in understanding the behavior of nonlinear physical phenomena. The Zakharov–Kuznetsov (Z–K) equation supports stable solitary waves, see ref. [8,9]. This makes the Z–K equation a very attractive model equation for investigating of vortices in geophysical owe. The Z–K equation determines weak non-linear behavior of ion sound waves that contain electrons with hot and cold temperatures in a regular magnetic field, see ref. [9]. With regard to the exact solutions of the Z–K and

Some new methods and important developments in the search for analysis have been done to investigate wave solutions for partial non-linear differential equations. The results of this manuscript may be completely complementary to in existence manuscripts of literature such as: direct algebraic method that is extended and modified; techniques’ Seadawy and extended mapping method to find solutions for some nonlinear partial differential equations [5]; showing the bi-directional propagation of small-amplitude long-capillary gravity waves on the surface of shallow water [22]; bright and dark solitons, solitary wave solutions of higher-order non-linear differential equations and the elliptic function [23]; the higher-order non-linear differential equations with cubic quintic nonlinearity [24]; solitary wave solutions to the nonlinear-modified KdV dynamical equation [25]; modified equal–width equations and dispersive traveling wave solutions of the equal–width [26]; the exact travelling wave solutions of the SRLW equation and modified Liouville equation [27]; fourth-order nonlinear Ablowitz–Kaup–Newell–Segur water wave dynamical equation [28]; nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation [29]; the third-order NLS equation [30]; approximate solutions of nonlinear Parabolic equation by using modified variational iteration algorithm [31]; new solitary wave solutions of nonlinear Nizhnik–Novikov–Vesselov equation [32]; and Hirota bilinear method was used to study multiple soliton interactions [33].

The purpose of this work is to find traveling wave solutions of

(1)
u
t
+
μ
u
u
x
+
1
2
u
x
x
x
+
1
2
(
δ
+
1
)
u
x
y
y
+
u
x
z
z
=
0
,

the function
Consider the solution in its complex form as follows:

(2)
u
(
x
,
y
,
z
,
t
)
=
ϕ
(
ξ
)
,
ξ
=
k
x
+
ν
y
+
λ
z
+
ω
t
,

where
Substituting from equation (2) into equation (1) and separating the imaginary and real parts lead to the ordinary differential equation as:

(3)
k
(
δ
+
1
)
ν
2
+
k
2
+
2
λ
2
ϕ
″
(
ξ
)
+
k
μ
ϕ
(
ξ
)
2
+
2
ω
ϕ
(
ξ
)
=
0

For exact solutions to equation (1), we use the logarithmic transformation and function method for equation (3). Introduction to the set of basic equations is procured in Section 3. A conclusion is carried out in Section 4.

Solutions for multiple waves and fractional phenomena logarithmic transformation can be shown as

(4)
ϕ
(
ξ
)
=
2
log
(
f
(
ξ
)
)
ξ

equation (
3) has the form:
(5)
k
(
δ
+
1
)
ν
2
+
k
2
f
(
ξ
)
2
f
(
3
)
(
ξ
)
+
2
k
(
δ
+
1
)
ν
2
+
k
2
+
2
λ
2
f
′
(
ξ
)
3
+
2
k
μ
f
(
ξ
)
f
′
(
ξ
)
2
+
2
ω
f
(
ξ
)
2
f
′
(
ξ
)
−
3
k
(
δ
+
1
)
ν
2
+
k
2
+
2
λ
2
f
(
ξ
)
f
′
(
ξ
)
f
″
(
ξ
)
=
0

In this subsection, for the next three waves hypothesis, multiple wave solutions can be expressed for nonlinear equation (5) as follows:

(6)
f
(
ξ
)
=
b
0
cosh
(
a
1
ξ
+
a
2
)
+
b
1
cos
(
a
3
ξ
+
a
4
)
+
b
2
cosh
(
a
5
ξ
+
a
6
)
,

where the real constants
**Case 1**

(7)
a
1
=
ω
2
λ
k
2
−
6
λ
2
4
,
a
5
=
a
1
,
a
3
=
0
,
b
1
=
b
0
2
+
b
2
2
,
ν
=
4
λ
2
−
k
2
δ
+
1
,

substituting from (
6), (
7), and (
4). By taking
(8)
u
11
(
x
,
y
,
z
,
t
)
=
τ
b
0
sinh
a
2
+
γ
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
+
b
2
sinh
a
6
+
γ
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
b
0
cosh
a
2
+
γ
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
+
b
2
cosh
a
6
+
γ
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
+
b
0
2
+
b
2
2
cos
a
4

**Case 2**

(9)
a
1
=
1
2
,
a
3
=
−
i
2
,
a
5
=
0
,
ω
=
−
λ
2
k
2
−
6
λ
2
,
δ
=
−
k
2
+
4
λ
2
−
ν
2
ν
2
,
b
0
=
i
b
1
,

substituting from (
6), (
9), and (
4), we have:
(10)
u
12
(
x
,
y
,
z
,
t
)
=
2
b
1
i
sinh
a
2
+
k
x
+
t
ω
+
ν
y
+
λ
z
2
+
sinh
k
x
+
t
ω
+
ν
y
+
λ
z
2
+
i
a
4
b
2
cosh
a
6
+
b
1
i
cosh
a
2
+
k
x
+
t
ω
+
ν
y
+
λ
z
2
+
cosh
k
x
+
t
ω
+
ν
y
+
λ
z
2
+
i
a
4

Figure 2
**Case 3**

(11)
a
1
=
0
,
a
3
=
k
δ
λ
2
ν
2
+
2
λ
4
+
λ
2
ν
2
,
a
5
→
i
k
δ
λ
2
ν
2
+
2
λ
4
+
λ
2
ν
2
,
b
1
=
i
b
2
,
ω
=
−
4
k
2
δ
ν
2
+
2
λ
2
+
ν
2

substituting from (
6), (
11), and (
4). By taking
(12)
u
13
(
x
,
y
,
z
,
t
)
2
i
b
2
γ
3
sin
a
4
+
γ
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
−
i
sin
γ
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
−
i
a
6
b
0
cosh
a
2
+
b
2
i
cos
a
4
+
γ
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
+
cos
γ
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
−
i
a
6

Figure 3
The hypothesis of a double exponential function takes the form:

(13)
f
(
ξ
)
=
b
1
e
a
1
ξ
+
a
2
+
b
2
e
a
3
ξ
+
a
4
,

where the real constants
(14)
a
1
=
2
3
,
a
3
=
2
3
,
μ
=
−
6
3
ν
2
+
12
ν
2
−
11
3
+
22
,

substitution from (
13), (
14) and (
4), we obtain
(15)
u
2
(
x
,
y
,
z
,
t
)
=
2
2
e
a
2
b
1
+
e
a
4
b
2
e
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
3
3
e
a
2
b
1
+
e
a
4
b
2
e
2
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
3

Figure 4
The Homoclinic breather approach function takes the form:

(16)
f
(
ξ
)
=
e
ξ
1
(
−
p
)
+
b
0
cos
ξ
3
p
1
+
b
1
e
ξ
2
p
,

where the real constants
(17)
a
5
→
i
6
μ
λ
2
p
1
,
a
1
→
3
μ
λ
2
p
,
a
3
→
−
3
μ
λ
2
p
,
ω
→
9
k
μ
2
λ
2
,

substitution from (
16) and (
17) using (
4). By taking
(18)
u
3
(
x
,
y
,
z
,
t
)
=
2
γ
3
6
b
0
e
a
2
p
+
3
γ
3
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
sinh
6
γ
3
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
−
i
a
6
p
1
−
3
b
1
e
a
2
+
a
4
p
−
3
b
0
e
a
2
p
+
3
γ
3
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
cosh
6
γ
3
(
k
x
+
t
ω
+
ν
y
+
λ
z
)
−
i
a
6
p
1
+
b
1
e
a
2
+
a
4
p
+
1

Figure 5
In this work, the multi-wave solution method has been used successfully to obtain solution the

Taif University Researchers Supporting Project number (TURSP-2020/165), Taif University, Taif, Saudi Arabia.

**Conflict of interest:** Authors state no conflict of interest.

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