Exact Solutions of the Paraxial Wave Dynamical Model

in Kerr Media with Truncated M-fractional Derivative

using the (G′/G, 1/G)-Expansion Method

PIM MALINGAM1..

ID , PAIWAN WONGSASINCHAI1..

ID , SEKSON SIRISUBTAWEE2,3..

ID,

SANOE KOONPRASERT2,3..

1Department of Mathematics, Faculty of Science and Technology

Rambhai Barni Rajabhat University, Chanthaburi 22000,

THAILAND

2Department of Mathematics, Faculty of Applied Science

King Mongkut's University of Technology North Bangkok, Bangkok 10800,

THAILAND

3Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400,

THAILAND

Abstract: The main purpose of this article is to use the (G′/G, 1/G)-expansion method to derive exact traveling

wave solutions of the paraxial wave dynamical model in Kerr media in the sense of the truncated M-fractional

derivative. To the best of the authors’ knowledge, the solutions of the model obtained using the expansion method

are reported here for the first time. The exact solutions are complex-valued functions expressed in terms of

hyperbolic, trigonometric, and rational functions. In order to show the physical interpretations of the solutions,

the magnitude of selected solutions is plotted in 3D, 2D, and contour plots for a range of values of the fractional-

order of the equation. With the aid of a symbolic software package, all of the obtained solutions are substituted

back into the relevant equation to verify their correctness. Obtaining the results by this technique confirms the

strength and efficacy of the method for generating a variety of exact solutions of the problems arising in applied

sciences and engineering.

Key-Words: - Exact solutions, Paraxial wave dynamical model, Kerr media, (G′/G, 1/G)-expansion method,

Truncated M-fractional derivative, Anti-soliton solution.

Received: February 26, 2023. Revised: November 23, 2023. Accepted: December 11, 2023. Published: December 31, 2023.

these dates in case of final acceptance, following strictly our data base and possible email

communication).

1 Introduction

Many physical phenomena arising in nature can

be modeled by nonlinear partial differential equa-

tions (NPDEs). For example, NPDEs have been

used as models in mechanics, [1], fiber optics, [2],

oceanography, [3], acoustics, [4], biology, [5], and

finance, [6]. Hence, finding solutions of NPDEs

has become important for modeling many real-world

problems. In particular, finding efficient techniques

for deriving exact traveling wave solutions of NPDEs

has attracted the interest of many researchers. Ef-

ficient methods that have been developed for ex-

tracting exact solutions of NPDEs include the mod-

ified (G′/G2)-expansion method, [7], the extended

generalized (G′/G)-expansion method, [8], the tanh-

coth method, [9], the Bäcklund transformations, [10],

the F-expansion method, [11], the modified auxil-

iary equation method, [12], and the two-variables

(G′/G, 1/G)-expansion method, [13].

One of the interesting PDEs in applied sciences

and engineering is the paraxial wave dynamical equa-

tion model for transmission of light through optical

fibers of Kerr media, [14], [15], [16], [17]. This

model can be used to explain wave dynamics in

optical fibers. This behavior includes optical soli-

tons consisting of non-diffractive spatiotemporal and

non-dispersive localized wave packets which transmit

through the fiber. The paraxial wave equation in Kerr

media can be written as, [15], [16], [17], [18],

i∂W

∂y +f

∂2W

∂t2+g

∂2W

∂x2+k|W|2W= 0,(1)

where W=W(x, y, t)is the Kerr term which rep-

resents the bound of the complex wave, and f, g, k

are real constants. Eq. (1) for a monochromatic beam

is equivalent to the nonlinear Schrödinger equation

(NLSE) of a quantum particle. If f, g in Eq. (1)

are such that fg < 0, Eq. (1) becomes a hyper-

bolic NLSE but if fg > 0, then Eq. (1) becomes

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an elliptic NLSE. In recent years, equation (1) has

been solved using various analytical methods for its

exact traveling wave solutions as follows. In [15]

the authors obtained solitons, elliptic function, and

other solutions of Eq. (1) via utilizing three analyti-

cal techniques, namely, the improved simple equation

method, the exp(−Φ(ζ))-expansion method and the

modified extended direct algebraic technique. Dis-

tinct types of structures for the obtained solutions

were depicted graphically. According to [16] some

singular, periodic, solitary wave, and rational so-

lutions of Eq. (1) were established using the Sar-

dar subequation method (SSM). The modulus, real

and imaginary plots of the solutions were demon-

strated for manifold implementations in many re-

search fields. In [17] the modified extended map-

ping technique was employed to assemble the soli-

tons, solitary waves, and rational solutions for Eq. (1).

The stability of Eq. (1) was studied via using modula-

tional instability (MI) analysis from which all soliton

solutions of the equation were verified to be stable and

exact. Moreover, the ϕ6model expansion technique

was utilized to obtain dark, bright, singular, bright-

dark, and periodic solitons for Eq. (1) as discussed

in [18]. The obtained solutions were expressed in

the form of trigonometric, hyperbolic, and exponen-

tial functions. In addition, a recent literature review

for solving Eq. (1) by using other different techniques

and solving the fractional complex paraxial wave dy-

namical model with Kerr media in the sense of the

conformable fractional derivative with respect to time

can be found in [19], and, [20], respectively.

However, a truncated M-fractional derivative,

[21], [22], [23], has recently attracted considerable

attention from many research scholars. Many par-

tial differential equations equipped with the truncated

M-fractional derivatives have been solved for their

exact traveling wave solutions which can be found

in [24], [25], [26], [27], [28]. An application of such

a derivative to the paraxial wave equation in Kerr me-

dia for formulating a novel equation is our major mo-

tivation. Consequently, it is very interesting to obtain

exact solutions of the resulting equation.

In this article, we adapt Eq. (1) by replacing its

classical partial derivatives with the truncated M-

fractional derivatives. The new equation can be writ-

ten as:

im∂β,γ

M,yW+f

2m∂β,γ

M,t m∂β,γ

M,tW

2m∂β,γ

M,x m∂β,γ

M,xW+k|W|2W= 0,

0< β ≤1,

(2)

where m∂β,γ

M,x,m∂β,γ

M,y and m∂β,γ

M,t are the truncated M-

fractional partial derivatives of order βwith respect

to x, y, and t, respectively, which will be defined

in section 2. The main aim of this paper is to use

the (G′/G, 1/G)-expansion method to extract exact

traveling wave solutions of Eq. (2) so that new solu-

tions and their physical behaviors are revealed here

for the first time. The remaining parts of this article

are organized as follows. Section 2 includes the defi-

nition of the truncated M-fractional derivative and its

characteristics. The main steps of the (G′/G, 1/G)-

expansion method are also described in this section.

The application of the expansion method to Eq. (2) is

described in section 3. Graphical representations of

chosen exact solutions are shown in section 4. The

conclusions of this research are discussed in the final

section.

2 Methoddology

In this section, a definition of the truncated M-

fractional derivative, its important properties and a

description of the (G′/G, 1/G)-expansion method

are presented. They are required for constructing ex-

act traveling wave solutions of Eq. (2).

2.1 Truncated M-fractional derivative and

its properties

Definition 2.1 The truncated Mittag-Leffler function

with one parameter is defined as, [21], [22], [23],

mEγ(z) =



n=0

Γ(γn + 1),(3)

where γ > 0and z∈C.

Definition 2.2 Let f: [0,∞)→Rbe a function.

Then, the truncated M-fractional derivative of fof

order βis defined by, [21], [22], [23],

mDβ,γ

M,tf(t) = lim

τ→0

ftmEγ(τt−β)−f(t)

τ,(4)

where 0< β ≤1and γ > 0. If the limit in (4) ex-

ists, then we say that the function fis β-truncated M-

fractional differentiable, or shortly, β-differentiable.

Moreover, if fis β-differentiable on (0, a), a >

0and limt→0+mDβ,γ

M,tf(t)exists, then we define

mDβ,γ

M,tf(0) = limt→0+mDβ,γ

M,tf(t).Some use-

ful properties of the truncated M-fractional derivative

are as follows, [21], [25], [29], [30], [31], [32]. Let

f(t), g(t)be β-differentiable functions for all t > 0,

β∈(0,1],and γ > 0. Then, we have

(1) mDβ,γ

M,t(λ) = 0,∀λ∈R.

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(2) mDβ,γ

M,t(af(t) + bg(t)) = amDβ,γ

M,tf(t)

+bmDβ,γ

M,tg(t),∀a, b ∈R.

(3) mDβ,γ

M,t (f(t)g(t)) = f(t)mDβ,γ

M,tg(t)

+g(t)mDβ,γ

M,tf(t).

(4) mDβ,γ

M,t f(t)

g(t)

=g(t)mDβ,γ

M,tf(t)−f(t)mDβ,γ

M,tg(t)

(g(t))2,

where g(t)= 0.

(5) mDβ,γ

M,t(f◦g)(t) = f′(g(t))mDβ,γ

M,tg(t)

when fis differentiable at g(t).

(6) If, in addition, fis differentiable,

then mDβ,γ

M,t(f(t)) = t1−β

Γ(γ+1)

df(t)

dt .

Utilizing the definition 2.2, the truncated M-fractional

partial derivative of u=u(x, t)with respect to t > 0

of order β∈(0,1] is defined as

m∂β,γ

M,tu(x, t)

= lim

τ→0

ux, t mEγ(τt−β)−u(x, t)

τ.

(5)

2.2 The (G′/G, 1/G)-expansion Method

Consider the following nonlinear partial differential

equation in three independent variables t, x, and y:

Fu, m∂α,γ

M,tu, m∂β,γ

M,xu, m∂δ,γ

M,yu, m∂α,γ

M,t m∂β,γ

M,xu,

m∂α,γ

M,t m∂δ,γ

M,yu,m∂β,γ

M,x m∂δ,γ

M,yu, ...= 0,

(6)

where 0< α, β, δ ≤1,and m∂α,γ

M,tu, m∂β,γ

M,xu, and

m∂δ,γ

M,yuare the truncated M-fractional partial deriva-

tives of a dependent variable uwith respect to inde-

pendent variables t, x, and y.Fis a polynomial of

the unknown function u=u(x, y, t)and its various

partial derivatives in which the highest order deriva-

tives and nonlinear terms are involved. Employing

the following traveling wave transformation, [30]

u(x, y, t) = U(ξ),

ξ= Γ(γ+ 1) kxβ

β+lyδ

δ+ctα

α+d,(7)

where k, l, c, and dare constants to be determined

later, we can reduce Eq. (6) to the following ODE in

U=U(ξ):

P(U, U′, U′′, . . .) = 0,(8)

where Pis a polynomial of U(ξ)and its various

derivatives in which the prime notation (′)repre-

sents the derivative with respect to ξ. Before we can

provide the key steps of the (G′/G, 1/G)-expansion

method, it is necessary to give the following informa-

tion, [13], [33], [34], [35], [36]. Consider the follow-

ing second-order linear ODE:

G′′(ξ) + λG(ξ) = µ, (9)

where λ, µ are constants. Denoting the functions ϕ

and ψas

ϕ(ξ) = G′(ξ)

G(ξ), ψ(ξ) = 1

G(ξ),(10)

we can transform equations (9) and (10) into the fol-

lowing system of two nonlinear ordinary differential

equations

ϕ′=−ϕ2+µψ −λ, ψ′=−ϕψ. (11)

The solutions of Eq. (9) can be separated into the fol-

lowing three cases.

Case 1: If λ < 0,then the general solution of (9) is

of the form

G(ξ) = A1sinh ξ√−λ

+A2cosh ξ√−λ+µ

λ,

(12)

and we have the following relationship

ψ2=−λ

λ2σ1+µ2ϕ2−2µψ +λ,(13)

where A1and A2are arbitrary constants and σ1=

1−A2

Case 2: If λ > 0,then the general solution of (9) can

be written as

G(ξ) = A1sin(ξ√λ) + A2cos ξ√λ+µ

λ,(14)

and we have the following associated relation

ψ2=λ

λ2σ2−µ2ϕ2−2µψ +λ,(15)

where A1and A2are arbitrary constants and σ2=

1+A2

Case 3: If λ= 0,then the general solution of (9) can

be displayed as

G(ξ) = µ

2ξ2+A1ξ+A2,(16)

and the corresponding relation is

ψ2=1

1−2µA2ϕ2−2µψ,(17)

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where A1and A2are arbitrary constants.

The main steps of the (G′/G, 1/G)-expansion

method, [13], [33], [34], [35], [36], can be described

as follows.

Step 1. Assume that the solution to Eq. (8) can be

expressed in terms of a polynomial of the two vari-

ables ϕand ψas follows

U(ξ) = a0+



j=1

ajϕj+



j=1

bjϕj−1ψ, (18)

where a0, aj,and bj, j = 1,2, . . . , N, are constants

to be found later with a2

N+b2

N= 0 and where the

functions ϕ=ϕ(ξ)and ψ=ψ(ξ)are implicitly as-

sociated to Eq. (9) through the relations in Eq. (10).

Step 2. We determine the value of the positive

integer Nin Eq. (18) by balancing the highest order

derivative and the nonlinear terms in Eq. (8). Denot-

ing the degree of U(ξ)by Deg[U(ξ)] = N, we can

calculate the degree of other terms in the equation us-

ing the following relations

Deg dqU(ξ)

dξq=N+q,

Deg (U(ξ))pdqU(ξ)

dξqs=N p +s(N+q).

(19)

Step 3. Replacing the solution form (18) with

the known value of Ninto Eq. (8) with the assistance

of Eq. (11) and Eq. (13), we can convert the function

Pin Eq. (8) into a polynomial in ϕand ψin which

the degree of ψis one. Equating each coefficient

of the resulting polynomial to zero, we obtain a sys-

tem of algebraic equations. These algebraic equations

can then be solved using the Maple software pack-

age to obtain values for the unknowns a0, aj, bj, j =

1,2, . . . , N, k, l, c, d, µ, λ(<0).Therefore, the exact

solutions of Eq. (6) can be obtained in terms of hyper-

bolic functions by using Eq. (12) and the transforma-

tion in Eq. (7).

Step 4. Similar to Step 3, substituting the result

from Eq. (18) into Eq. (8) with the aid of Eq. (11) and

Eq. (15) for λ > 0, we can obtain the exact solutions

of Eq. (6) by using the transformation (7). The ob-

tained exact solutions in this step are written in terms

of trigonometric functions.

Step 5. In the same manner as Step 3, substituting

the result from Eq. (18) into Eq. (8) with the aid of

Eq. (11) and Eq. (17) for λ= 0, we can obtain the

traveling wave solutions of Eq. (6) with the help of

the transformation (7). The resulting exact solutions

in this step are obtained in terms of rational functions.

Remark 1: Particularly, if the balance number N

in Step 2 is not a positive integer, then some special

transformations must be applied for U(ξ)in (8) so that

the equation (8) can be converted into a new equation

of a new variable. For example, if N=q

pis a fraction

in the lowest terms, then U(ξ) = Vq

p(ξ)is inserted in

(8). Consequently, the new equation, written in terms

of V(ξ), has a positive integer balance number, [37].

Remark 2: When applied, the (G′/G, 1/G)-

expansion method can provide three types of exact

solutions including hyperbolic, trigonometric, and ra-

tional function solutions.

Remark 3: The (G′/G, 1/G)-expansion method

can be reduced to the (G′/G)-expansion method by

some special setting, [37]. Thus, the (G′/G, 1/G)-

expansion method is more effective and more general

than the (G′/G)-expansion method.

3 Implementation of the

(G′/G, 1/G)-expansion Method

In this section, we obtain exact traveling wave solu-

tions of Eq. (2) by using the (G′/G, 1/G)-expansion

method. First, we assume that the exact solution of

(2) has the form

W(x, y, t) = U(χ)eiξ ,(20)

where Uis a real-valued function of χ,i=√−1,and

χ=Γ(γ+ 1)

β(d1xβ+d2yβ+ρtβ),

ξ=Γ(γ+ 1)

β(s1xβ+s2yβ+τtβ+ω),

(21)

where d1, d2, ρ, s1, s2, τ, and ωare real constants,

the order 0< β ≤1,and the parameter γ > 0.

Substituting the solution form (20) into the proposed

problem (2) and then separating the real and imagi-

nary parts of the resulting equation, we obtain the fol-

lowing equations:

Re: fρ2+gd2

1U′′ (χ)−fτ 2+gs2

1+ 2 s2U(χ)

+2 kU3(χ) = 0,(22)

Im: (d2+fρ τ +gd1s1)U′(χ)=0.(23)

Since U′(χ)in Eq. (23) is not zero, we have

d2=−(fρ τ +gd1s1).(24)

Next, balancing the terms U′′ and U3in Eq. (22) via

the formulas in (19), we obtain N= 1 and the solu-

tion form of Eq. (22) is then

U(χ) = a0+a1ϕ(χ) + b1ψ(χ),(25)

where a0,a1,and b1are constant coefficients which

will be determined later such that a2

1+b2

1= 0.As

explained in equations (12), (14), and (16), there are

three cases for the functions ϕ(χ)and ψ(χ)in (25)

depending on the sign of λ.

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Case 1 (Hyperbolic function solutions): If λ < 0,

we substitute Eq.(25) with Eqs.(11) and (13) into

Eq.(22) and then the left-hand side of Eq. (22) be-

comes a polynomial in ϕ(χ)and ψ(χ). Setting all of

the coefficients of this resulting polynomial to zero,

we obtain the following system of nonlinear algebraic

equations in λ, µ, ω, ρ, τ, a0, a1, b1, d1, s1,and s2:

ϕ3: 2 fλ4ρ2A4

1a1−4fλ4ρ2A2

1A2

2a1

+ 2 fλ4ρ2A4

2a1+ 2 gλ4A4

1a1d2

−4gλ4A2

1A2

2a1d2

1+ 2 gλ4A4

2a1d2

+ 2 kλ4A4

1a3

1−4kλ4A2

1A2

2a3

+ 2 kλ4A4

2a3

1+ 4 fλ2µ2ρ2A2

1a1

−4fλ2µ2ρ2A2

2a1+ 4 gλ2µ2A2

1a1d2

−4gλ2µ2A2

2a1d2

1+ 4 kλ2µ2A2

1a3

−4kλ2µ2A2

2a3

1−6kλ3A2

1a1b2

+ 6 kλ3A2

2a1b2

1+ 2 fµ4ρ2a1

+ 2 gµ4a1d2

1+ 2 kµ4a3

1−6kλ µ2a1b2

= 0,

ϕ2: 6 kλ4A4

1a0a2

1−12 kλ4A2

1A2

2a0a2

+ 6 kλ4A4

2a0a2

1+fλ3µ ρ2A2

1b1

−fλ3µ ρ2A2

2b1+gλ3µ A2

1b1d2

−gλ3µ A2

2b1d2

1+ 12 kλ2µ2A2

1a0a2

−12 kλ2µ2A2

2a0a2

1−6kλ3A2

1a0b2

+ 6 kλ3A2

2a0b2

1+fλ µ3ρ2b1

+gλ µ3b1d2

1+ 6 kµ4a0a2

1−4b3

1λ2kµ

−6kλ µ2a0b2

1= 0,

ϕ2ψ: 2 fλ4ρ2A4

1b1−4fλ4ρ2A2

1A2

2b1

+ 2 fλ4ρ2A4

2b1+ 2 gλ4A4

1b1d2

−4gλ4A2

1A2

2b1d2

1+ 2 gλ4A4

2b1d2

+ 6 kλ4A4

1a2

1b1−12 kλ4A2

1A2

2a2

1b1

+ 6 kλ4A4

2a2

1b1+ 4 fλ2µ2ρ2A2

1b1

−4fλ2µ2ρ2A2

2b1+ 4 gλ2µ2A2

1b1d2

−4gλ2µ2A2

2b1d2

1+ 12 kλ2µ2A2

1a2

1b1

−12 kλ2µ2A2

2a2

1b1−2kλ3A2

1b3

+ 2 kλ3A2

2b3

1+ 2 fµ4ρ2b1+ 2 gµ4b1d2

+ 6 kµ4a2

1b1−2kλ µ2b3

1= 0,

ϕ: 2 fλ5ρ2A4

1a1−4fλ5ρ2A2

1A2

2a1

+ 2 fλ5ρ2A4

2a1+ 2 gλ5A4

1a1d2

−4gλ5A2

1A2

2a1d2

1+ 2 gλ5A4

2a1d2

−fλ4τ2A4

1a1+ 2 fλ4τ2A2

1A2

2a1

−fλ4τ2A4

2a1−gλ4A4

1a1s2

+ 2 gλ4A2

1A2

2a1s2

1−gλ4A4

2a1s2

+ 6 kλ4A4

1a2

0a1−12 kλ4A2

1A2

2a2

0a1

+ 6 kλ4A4

2a2

0a1+ 4 fλ3µ2ρ2A2

1a1

−4fλ3µ2ρ2A2

2a1+ 4 gλ3µ2A2

1a1d2

−4gλ3µ2A2

2a1d2

1−2fλ2µ2τ2A2

1a1

+ 2 fλ2µ2τ2A2

2a1−2gλ2µ2A2

1a1s2

+ 2 gλ2µ2A2

2a1s2

1−6kλ4A2

1a1b2

+ 6 kλ4A2

2a1b2

1+ 12 kλ2µ2A2

1a2

0a1

−12 kλ2µ2A2

2a2

0a1−2λ4A4

1a1s2

+ 4 λ4A2

1A2

2a1s2−2λ4A4

2a1s2

+ 2 fλ µ4ρ2a1+ 2 gλ µ4a1d2

−fµ4τ2a1−gµ4a1s2

1−6kλ2µ2a1b2

+ 6 kµ4a2

0a1−4λ2µ2A2

1a1s2

+ 4 λ2µ2A2

2a1s2−2µ4a1s2= 0,

ϕψ :−3fλ4µ ρ2A4

1a1+ 6 fλ4µ ρ2A2

1A2

2a1

−3fλ4µ ρ2A4

2a1−3gλ4µ A4

1a1d2

+ 6 gλ4µ A2

1A2

2a1d2

1−3gλ4µ A4

2a1d2

+ 12 kλ4A4

1a0a1b1−24 kλ4A2

1A2

2a0a1b1

+ 12 kλ4A4

2a0a1b1−6fλ2µ3ρ2A2

1a1

+ 6 fλ2µ3ρ2A2

2a1−6gλ2µ3A2

1a1d2

+ 6 gλ2µ3A2

2a1d2

1+ 12 kλ3µ A2

1a1b2

−12 kλ3µ A2

2a1b2

1+ 24 kλ2µ2A2

1a0a1b1

−24 kλ2µ2A2

2a0a1b1−3fµ5ρ2a1

−3gµ5a1d2

1+ 12 kλ µ3a1b2

+ 12 kµ4a0a1b1= 0,

ψ:fλ5ρ2A4

1b1−2fλ5ρ2A2

1A2

2b1

+fλ5ρ2A4

2b1+gλ5A4

1b1d2

−2gλ5A2

1A2

2b1d2

1+gλ5A4

2b1d2

−fλ4τ2A4

1b1+ 2 fλ4τ2A2

1A2

2b1

−fλ4τ2A4

2b1−gλ4A4

1b1s2

+ 2 gλ4A2

1A2

2b1s2

1−gλ4A4

2b1s2

+ 6 kλ4A4

1a2

0b1−12 kλ4A2

1A2

2a2

0b1

+ 6 kλ4A4

2a2

0b1−2fλ2µ2τ2A2

1b1

+ 2 fλ2µ2τ2A2

2b1−2gλ2µ2A2

1b1s2

+ 2 gλ2µ2A2

2b1s2

1−2kλ4A2

1b3

+ 2 kλ4A2

2b3

1+ 12 kλ3µ A2

1a0b2

−12 kλ3µ A2

2a0b2

1+ 12 kλ2µ2A2

1a2

0b1

−12 kλ2µ2A2

2a2

0b1−2λ4A4

1b1s2

+ 4 λ4A2

1A2

2b1s2−2λ4A4

2b1s2

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−fλ µ4ρ2b1−gλ µ4b1d2

1−fµ4τ2b1

−gµ4b1s2

1+ 6 kλ2µ2b3

1+ 12 kλ µ3a0b2

+ 6 kµ4a2

0b1−4λ2µ2A2

1b1s2

+ 4 λ2µ2A2

2b1s2−2µ4b1s2= 0,

ϕ0:−fλ4τ2A4

1a0+ 2 fλ4τ2A2

1A2

2a0

−fλ4τ2A4

2a0−gλ4A4

1a0s2

+ 2 gλ4A2

1A2

2a0s2

1−gλ4A4

2a0s2

+ 2 kλ4A4

1a3

0−4kλ4A2

1A2

2a3

+ 2 kλ4A4

2a3

0+fλ4µ ρ2A2

1b1

−fλ4µ ρ2A2

2b1+gλ4µ A2

1b1d2

−gλ4µ A2

2b1d2

1−2fλ2µ2τ2A2

1a0

+ 2 fλ2µ2τ2A2

2a0−2gλ2µ2A2

1a0s2

+ 2 gλ2µ2A2

2a0s2

1−6kλ4A2

1a0b2

+ 6 kλ4A2

2a0b2

1+ 4 kλ2µ2A2

1a3

−4kλ2µ2A2

2a3

0−2λ4A4

1a0s2

+ 4 λ4A2

1A2

2a0s2−2λ4A4

2a0s2

+fλ2µ3ρ2b1+gλ2µ3b1d2

−fµ4τ2a0−gµ4a0s2

1−4b3

1λ3kµ

−6kλ2µ2a0b2

1+ 2 kµ4a3

−4λ2µ2A2

1a0s2+ 4 λ2µ2A2

2a0s2

−2µ4a0s2= 0.

(26)

Solving the above algebraic system using the Maple

package program, we get the following results.

Result 1

λ=λ, µ = 0, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±−fρ2+gd2

k, b1= 0, d1=d1,

d2=−(fρ τ +gd1s1), s1=s1,

s2=fλ ρ2+gλ d2

1−fτ2

2−gs2

(27)

where λ(<0), f, g, k, d1, ρ, s1, τ, ω are arbitrary

constants such that kfρ2+gd2

1<0. From Eqs.

(12), (20), (25), and (27), we get the solution of Eq.

(2) as:

W(x, y, t) =

±−fρ2+gd2

kA1cosh χ√−λ√−λ+A2sinh χ√−λ√−λ

A1sinh χ√−λ+A2cosh χ√−λ

×eiξ,

(28)

where A1, A2arbitrary constants and

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ+fλ ρ2+gλ d2

1−1

2fτ 2−1

2gs2

1yβ+τtβ+ω.

Result 2

λ=kb2

fρ2+gd2

1σ1

, µ = 0, ω =ω,

ρ=ρ, τ =τ, a0= 0, a1= 0, b1=b1,

d1=d1, d2=−(fρ τ +gd1s1),

s1=s1, s2=−σ1fτ2+gs2

1+kb2

2σ1

(29)

where σ1=A2

1−A2

2and f, g, k, d1, ρ, s1, τ, ω, b1,

A1, A2are arbitrary constants such that λ < 0. From

Eqs. (12), (20), (25), and (29), we obtain the exact

solution of Eq. (2) as follows:

W(x, y, t) =

A1sinh χ√−λ+A2cosh χ√−λ×eiξ,

(30)

where

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−σ1f τ 2+gs2

1+kb2

2σ1yβ+τtβ+ω.

Result 3

λ=4kb2

σ1fρ2+gd2

1, µ = 0, ω =ω, ρ =ρ,

τ=τ, a0= 0, a1=±−fρ2+gd2

4k,

b1=b1, d1=d1, d2=−(fρ τ +gd1s1), s1=s1,

s2=−σ1fτ2+gs2

1−2kb2

2σ1

(31)

where σ1=A2

1−A2

2and f, g, k, d1, ρ, s1, τ, ω, b1,

A1, A2are arbitrary constants such that λ < 0and

kfρ2+gd2

1<0. From Eqs. (12), (20), (25), and

(31), we obtain the exact solution of Eq. (2) as fol-

lows:

W(x, y, t) =

±−fρ2+gd2

4kA1cosh χ√−λ√−λ+A2sinh χ√−λ√−λ

A1sinh χ√−λ+A2cosh χ√−λ+µ

λ

+b1

A1sinh χ√−λ+A2cosh χ√−λ+µ

λ×eiξ ,

(32)

where

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−σ1f τ 2+gs2

1−2kb2

2σ1yβ+τtβ+ω.

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Result 4

λ=λ, µ =µ, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±−fρ2+gd2

4k,

b1=±(fρ2+gd2

1)λ2σ1+ (fρ2+gd2

1)µ2

4kλ ,

d1=d1, d2=−(fρ τ +gd1s1), s1=s1,

s2=1

4λfρ2+gd2

1−1

2fτ2+gs2

1,

(33)

where σ1=A2

1−A2

2and λ(<0), µ, f, g, k, d1, ρ,

s1, τ, ω, A1, A2are arbitrary constants such that

kfρ2+gd2

1<0and b1∈R. From Eqs. (12),

(20), (25), and (33), we obtain the exact solution of

Eq. (2) as follows:

W(x, y, t) =

±−fρ2+gd2

4kA1cosh χ√−λ√−λ+A2sinh χ√−λ√−λ

A1sinh χ√−λ+A2cosh χ√−λ+µ

λ

±





(f ρ2+gd2

1)λ2σ1+(f ρ2+gd2

1)µ2

4kλ

A1sinh χ√−λ+A2cosh χ√−λ+µ

λ



×eiξ ,

(34)

where

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ+1

4λfρ2+gd2

1−1

2fτ 2+gs2

1yβ+τ tβ+ω.

Case 2 (Trigonometric function solutions):

If λ > 0, we substitute Eq.(25) with Eqs.(11) and (15)

into Eq.(22), so that the left-hand side of Eq.(22) be-

comes a polynomial in ϕ(χ)and ψ(χ). Setting all of

the coefficients of this resulting polynomial to zero,

we obtain the following system of nonlinear algebraic

equations in λ, µ, ω, ρ, τ, a0, a1, b1, d1, s1,and s2:

ϕ3: 2 f λ4ρ2A4

1a1+ 4 fλ4ρ2A2

1A2

2a1

+ 2 fλ4ρ2A4

2a1+ 2 gλ4A4

1a1d2

+ 4 gλ4A2

1A2

2a1d2

1+ 2 gλ4A4

2a1d2

+ 2 kλ4A4

1a3

1+ 4 kλ4A2

1A2

2a3

+ 2 kλ4A4

2a3

1−4fλ2µ2ρ2A2

1a1

−4fλ2µ2ρ2A2

2a1−4gλ2µ2A2

1a1d2

−4gλ2µ2A2

2a1d2

1−4kλ2µ2A2

1a3

−4kλ2µ2A2

2a3

1+ 6 kλ3A2

1a1b2

+ 6 kλ3A2

2a1b2

1+ 2 fµ4ρ2a1

+ 2 gµ4a1d2

1+ 2 kµ4a3

1−6kλ µ2a1b2

1= 0,

ϕ2: 6 kλ4A4

1a0a2

1+ 12 kλ4A2

1A2

2a0a2

+ 6 kλ4A4

2a0a2

1−fλ3µ ρ2A2

1b1

−fλ3µ ρ2A2

2b1−gλ3µ A2

1b1d2

−gλ3µ A2

2b1d2

1−12 kλ2µ2A2

1a0a2

−12 kλ2µ2A2

2a0a2

1+ 6 kλ3A2

1a0b2

+ 6 kλ3A2

2a0b2

1+fλ µ3ρ2b1

+gλ µ3b1d2

1+ 6 kµ4a0a2

−4kλ2µ b3

1−6kλ µ2a0b2

1= 0,

ϕ2ψ: 2 fλ4ρ2A4

1b1+ 4 fλ4ρ2A2

1A2

2b1

+ 2 fλ4ρ2A4

2b1+ 2 gλ4A4

1b1d2

+ 4 gλ4A2

1A2

2b1d2

1+ 2 gλ4A4

2b1d2

+ 6 kλ4A4

1a2

1b1+ 12 kλ4A2

1A2

2a2

1b1

+ 6 kλ4A4

2a2

1b1−4fλ2µ2ρ2A2

1b1

−4fλ2µ2ρ2A2

2b1−4gλ2µ2A2

1b1d2

−4gλ2µ2A2

2b1d2

1−12 kλ2µ2A2

1a2

1b1

−12 kλ2µ2A2

2a2

1b1+ 2 kλ3A2

1b3

+ 2 kλ3A2

2b3

1+ 2 fµ4ρ2b1

+ 2 gµ4b1d2

1+ 6 kµ4a2

1b1

−2kλ µ2b3

1= 0,

ϕ: 2 fλ5ρ2A4

1a1+ 4 fλ5ρ2A2

1A2

2a1

+ 2 fλ5ρ2A4

2a1+ 2 gλ5A4

1a1d2

+ 4 gλ5A2

1A2

2a1d2

1+ 2 gλ5A4

2a1d2

−fλ4τ2A4

1a1−2fλ4τ2A2

1A2

2a1

−fλ4τ2A4

2a1−gλ4A4

1a1s2

−2gλ4A2

1A2

2a1s2

1−gλ4A4

2a1s2

+ 6 kλ4A4

1a2

0a1+ 12 kλ4A2

1A2

2a2

0a1

+ 6 kλ4A4

2a2

0a1−4fλ3µ2ρ2A2

1a1

−4fλ3µ2ρ2A2

2a1−4gλ3µ2A2

1a1d2

−4gλ3µ2A2

2a1d2

1+ 2 fλ2µ2τ2A2

1a1

+ 2 fλ2µ2τ2A2

2a1+ 2 gλ2µ2A2

1a1s2

+ 2 gλ2µ2A2

2a1s2

1+ 6 kλ4A2

1a1b2

+ 6 kλ4A2

2a1b2

1−12 kλ2µ2A2

1a2

0a1

−12 kλ2µ2A2

2a2

0a1−2λ4A4

1a1s2

−4λ4A2

1A2

2a1s2−2λ4A4

2a1s2

+ 2 fλ µ4ρ2a1+ 2 gλ µ4a1d2

−fµ4τ2a1−gµ4a1s2

1−6kλ2µ2a1b2

+ 6 kµ4a2

0a1+ 4 λ2µ2A2

1a1s2

+ 4 λ2µ2A2

2a1s2−2µ4a1s2= 0,

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ϕψ :−3fλ4µ ρ2A4

1a1−6fλ4µ ρ2A2

1A2

2a1

−3fλ4µ ρ2A4

2a1−3gλ4µ A4

1a1d2

−6gλ4µ A2

1A2

2a1d2

1−3gλ4µ A4

2a1d2

+ 12 kλ4A4

1a0a1b1+ 24 kλ4A2

1A2

2a0a1b1

+ 12 kλ4A4

2a0a1b1+ 6 fλ2µ3ρ2A2

1a1

+ 6 fλ2µ3ρ2A2

2a1+ 6 gλ2µ3A2

1a1d2

+ 6 gλ2µ3A2

2a1d2

1−12 kλ3µ A2

1a1b2

−12 kλ3µ A2

2a1b2

1−24 kλ2µ2A2

1a0a1b1

−24 kλ2µ2A2

2a0a1b1−3fµ5ρ2a1

−3gµ5a1d2

1+ 12 kλ µ3a1b2

+ 12 kµ4a0a1b1= 0,

ψ:fλ5ρ2A4

1b1+ 2 fλ5ρ2A2

1A2

2b1

+fλ5ρ2A4

2b1+gλ5A4

1b1d2

+ 2 gλ5A2

1A2

2b1d2

1+gλ5A4

2b1d2

−fλ4τ2A4

1b1−2fλ4τ2A2

1A2

2b1

−fλ4τ2A4

2b1−gλ4A4

1b1s2

−2gλ4A2

1A2

2b1s2

1−gλ4A4

2b1s2

+ 6 kλ4A4

1a2

0b1+ 12 kλ4A2

1A2

2a2

0b1

+ 6 kλ4A4

2a2

0b1+ 2 fλ2µ2τ2A2

1b1

+ 2 fλ2µ2τ2A2

2b1+ 2 gλ2µ2A2

1b1s2

+ 2 gλ2µ2A2

2b1s2

1+ 2 kλ4A2

1b3

+ 2 kλ4A2

2b3

1−12 kλ3µ A2

1a0b2

−12 kλ3µ A2

2a0b2

1−12 kλ2µ2A2

1a2

0b1

−12 kλ2µ2A2

2a2

0b1−2λ4A4

1b1s2

−4λ4A2

1A2

2b1s2−2λ4A4

2b1s2

−fλ µ4ρ2b1−gλ µ4b1d2

−fµ4τ2b1−gµ4b1s2

1+ 6 b3

1λ2kµ2

+ 12 kλ µ3a0b2

1+ 6 kµ4a2

0b1

+ 4 λ2µ2A2

1b1s2+ 4 λ2µ2A2

2b1s2

−2µ4b1s2= 0,

ϕ0:−fλ4τ2A4

1a0−2fλ4τ2A2

1A2

2a0

−fλ4τ2A24a0−gλ4A4

1a0s2

−2gλ4A2

1A2

2a0s2

1−gλ4A4

2a0s2

+ 2 kλ4A4

1a3

0+ 4 kλ4A2

1A2

2a3

+ 2 kλ4A4

2a3

0−fλ4µ ρ2A2

1b1

−fλ4µ ρ2A2

2b1−gλ4µ A2

1b1d2

−gλ4µ A2

2b1d2

1+ 2 fλ2µ2τ2A2

1a0

+ 2 fλ2µ2τ2A2

2a0+ 2 gλ2µ2A2

1a0s2

+ 2 gλ2µ2A2

2a0s2

1+ 6 kλ4A2

1a0b2

+ 6 kλ4A2

2a0b2

1−4kλ2µ2A2

1a3

−4kλ2µ2A2

2a3

0−2λ4A4

1a0s2

−4λ4A2

1A2

2a0s2−2λ4A4

2a0s2

+fλ2µ3ρ2b1+gλ2µ3b1d2

−fµ4τ2a0−gµ4a0s2

1−4b3

1λ3kµ

−6kλ2µ2a0b2

1+ 2 kµ4a3

0+ 4 λ2µ2A2

1a0s2

+ 4 λ2µ2A2

2a0s2−2µ4a0s2= 0.

(35)

Solving the above algebraic system using the Maple

package program, we get the following results:

Result 1

λ=λ, µ = 0, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±−fρ2+gd2

k, b1= 0, d1=d1,

d2=−(fρ τ +gd1s1), s1=s1,

s2=fλ ρ2+gλ d2

1−1

2fτ2−1

2gs2

(36)

where λ(>0), f, g, k, d1, ρ, s1, τ, ω are arbitrary

constants such that kfρ2+gd2

1<0. From Eqs.

(14), (20), (25), and (36), we get the solution of Eq.

(2) as follows:

W(x, y, t) = ±−fρ2+gd2

×

A1cos χ√λ√λ−A2sin χ√λ√λ

A1sin χ√λ+A2cos χ√λ



×eiξ,

(37)

where A1, A2re arbitrary constants and

χ=Γ(γ+ 1)

βd1xβ−(fρτ +gd1s1)yβ+ρtβ

ξ=Γ(γ+ 1)

×s1xβ+fλρ2+gλ d2

1−1

2fτ 2−1

2gs2

1yβ+τtβ+ω.

Result 2

λ=−kb2

σ2fρ2+gd2

1, µ = 0, ω =ω, ρ =ρ,

τ=τ, a0= 0, a1= 0, b1=b1, d1=d1,

d2=−(fρ τ +gd1s1), s1=s1,

s2=−σ2fτ2+gs2

1−kb2

2σ2

(38)

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where σ2=A2

1+A2

2and f, g, k, d1, ρ, s1, τ, ω, b1,

A1, A2are arbitrary constants such that λ > 0. From

Eqs. (14), (20), (25), and (38), we obtain the exact

solution of Eq. (2) as follows:

W(x, y, t) = b1

A1sin χ√λ+A2cos χ√λ×eiξ,

(39)

where

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

×s1xβ−σ2fτ2+gs2

1−kb2

2σ2

yβ+τ tβ+ω.

Result 3

λ=λ, µ =µ, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±−fρ2+gd2

4k,

b1=±−fρ2+gd2

1λ2σ2−(fρ2+gd2

1)µ2

4kλ ,

d1=d1, d2=−(fρ τ +gd1s1), s1=s1,

s2=λ

4fρ2+gd2

1−1

2fτ2+gs2

1,

(40)

where σ2=A2

1+A2

2and λ(>0), µ, f, g, k, d1, ρ,

s1, τ, ω, A1, A2are arbitrary constants such that

kfρ2+gd2

1<0and b1∈R. From Eqs. (14),

(20), (25), and (40), we obtain the exact solution of

Eq. (2) as follows:

W(x, y, t)

=±−fρ2+gd2

×



A1cos χ√λ√λ−A2sin χ√λ√λ

A1sin χ√λ+A2cos χ√λ+µ

λ



±−(fρ2+gd2

1)λ2σ2−(fρ2+gd2

1)µ2

4kλ

A1sin χ√λ+A2cos χ√λ+µ

λ×eiξ

(41)

where

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

×s1xβ+λ

4fρ2+gd2

1−1

2fτ 2+gs2

1yβ+τ tβ+ω.

Case 3 (Rational function solutions): If λ= 0,

we substitute Eq.(25) with Eqs.(11) and (17) into

Eq.(22), so that the left-hand side of Eq. Eq.(22) be-

comes a polynomial in ϕ(χ)and ψ(χ). Setting all of

the coefficients of this resulting polynomial to zero,

we obtain the following system of nonlinear algebraic

equations in µ, ω, ρ, τ, a0, a1, b1, d1, s1,and s2:

ϕ3: 8 f µ2ρ2A2

2a1−8fµ ρ2A2

1A2a1

+ 2 fρ2A4

1a1+ 8 gµ2A2

2a1d2

−8gµ A2

1A2a1d2

1+ 2 gA4

1a1d2

+ 8 kµ2A2

2a3

1−8kµ A2

1A2a3

+ 2 kA4

1a3

1−12 kµ A2a1b2

+ 6 kA2

1a1b2

1= 0,

ϕ2: 24 kµ2A2

2a0a2

1−24 kµ A2

1A2a0a2

+ 6 kA4

1a0a2

1+ 2 fµ2ρ2A2b1

−fµ ρ2A2

1b1+ 2 gµ2A2b1d2

−gµ A2

1b1d12−12 kµ A2a0b2

+ 6 kA2

1a0b2

1−4kµ b3

1= 0,

ϕ2ψ: 8 f µ2ρ2A2

2b1−8fµ ρ2A2

1A2b1

+ 2 fρ2A4

1b1+ 8 gµ2A2

2b1d2

−8gµ A2

1A2b1d2

1+ 2 gA4

1b1d2

+ 24 kµ2A2

2a2

1b1−24 kµ A2

1A2a2

1b1

+ 6 kA4

1a2

1b1−4kµ A2b3

+ 2 kA2

1b3

1= 0,

ϕ:−4fµ2τ2A2

2a1+ 4 fµ τ 2A2

1A2a1

−fτ2A4

1a1−4gµ2A2

2a1s2

+ 4 gµ A2

1A2a1s2

1−gA4

1a1s2

+ 24 kµ2A2

2a2

0a1−24 kµ A2

1A2a2

0a1

+ 6 kA4

1a2

0a1−8µ2A2

2a1s2

+ 8 µ A2

1A2a1s2−2A4

1a1s2= 0,

ϕψ :−12 fµ3ρ2A2

2a1+ 12 fµ2ρ2A2

1A2a1

−3fµ ρ2A4

1a1−12 gµ3A2

2a1d2

+ 12 gµ2A2

1A2a1d2

1−3gµ A4

1a1d2

+ 48 kµ2A2

2a0a1b1−48 kµ A2

1A2a0a1b1

+ 12 kA4

1a0a1b1+ 24 kµ2A2a1b2

−12 kµ A2

1a1b2

1= 0,

ψ:−4fµ3ρ2A2b1+ 2 fµ2ρ2A2

1b1

−4fµ2τ2A2

2b1+ 4 fµ τ 2A2

1A2b1

−fτ2A4

1b1−4gµ3A2b1d2

+ 2 gµ2A2

1b1d2

1−4gµ2A2

2b1s2

+ 4 gµ A2

1A2b1s2

1−gA4

1b1s2

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+ 24 kµ2A2

2a2

0b1−24 kµ A2

1A2a2

0b1

+ 6 kA4

1a2

0b1+ 24 kµ2A2a0b2

−12 kµ A2

1a0b2

1+ 8 kµ2b3

−8µ2A2

2b1s2+ 8 µ A2

1A2b1s2

−2A4

1b1s2= 0,

ϕ0:−4fµ2τ2A2

2a0+ 4 fµ τ 2A2

1A2a0

−fτ2A4

1a0−4gµ2A2

2a0s2

+ 4 gµ A2

1A2a0s2

1−gA4

1a0s2

+ 8 kµ2A2

2a3

0−8kµ A2

1A2a3

+ 2 kA4

1a3

0−8µ2A2

2a0s2

+ 8 µ A2

1A2a0s2−2A4

1a0s2= 0.

(42)

Solving the above algebraic system using the Maple

package program, we get the following results.

Result 1

µ= 0, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±−fρ2+gd2

k, b1= 0, d1=d1,

d2=−(fρ τ +gd1s1), s1=s1,

s2=−1

2fτ2+gs2

1,

(43)

where f, g, k, d1, ρ, s1, τ, ω are arbitrary constants

such that kfρ2+gd2

1<0. From Eqs. (16), (20),

(25), and (43), we obtain the exact solution of Eq. (2)

as follows:

W(x, y, t) = ±−f ρ2+gd2

kA1

A1χ+A2×eiξ,(44)

where A1, A2are arbitrary constants and

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−1

2fτ2+gs2

1yβ+τ tβ+ω.

Result 2

µ= 0, ω =ω, ρ =ρ, τ =τ, a0= 0, a1= 0,

b1=±−fρ2+gd2

kA1, d1=d1,

d2=−(fρ τ +gd1s1), s1=s1,

s2=−1

2fτ2+gs2

1,

(45)

where f, g, k, d1, ρ, s1, τ, ω, A1are arbitrary con-

stants such that kfρ2+gd2

1<0. From Eqs. (16),

(20), (25), and (45), the solution of Eq. (2) is:

W(x, y, t) = ±−f ρ2+gd2

kA1

A1χ+A2×eiξ,(46)

where A2is an arbitrary constant and

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−1

2fτ2+gs2

1yβ+τ tβ+ω.

Result 3

µ= 0, ω =ω, ρ =ρ, τ =τ, a0= 0,

a1=±1

2−fρ2+gd2

k, b1=a1A1,

d1=d1, d2=−(fρ τ +gd1s1), s1=s1,

s2=−1

2fτ2+gs2

1,

(47)

where f, g, k, d1, ρ, s1, τ, ω, A1are arbitrary con-

stants such that kfρ2+gd2

1<0. From Eqs. (16),

(20), (25), and (47), we obtain the exact solution of

Eq. (2) as follows:

W(x, y, t) = ±−f ρ2+gd2

kA1

A1χ+A2×eiξ,(48)

where A2is arbitrary constant and

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−1

2fτ2+gs2

1yβ+τ tβ+ω.

Result 4

µ=fρ2+gd2

1A2

1+ 4 kb2

2A2fρ2+gd2

1, ω =ω, ρ =ρ,

τ=τ, a0= 0, a1=±1

2−fρ2+gd2

b1=b1, d1=d1, d2=−(fρ τ +gd1s1),

s1=s1, s2=−1

2fτ2−1

2gs2

(49)

where f, g, k, d1, ρ, s1, τ, ω, b1, A1, A2are arbi-

trary constants such that kfρ2+gd2

1<0. From

Eqs. (16), (20), (25), and (49), we get the solution of

Eq. (2) as follows:

W(x, y, t) = ΦΨ + Ω

Θeiξ,(50)

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where

Φ = χfρ2+gd2

1A2

1+ 2 A1A2f ρ2+gd2

1+ 4 χ kb2

1,

Ψ = −fρ2+gd2

Ω = 4 b1A2f ρ2+gd2

1,

Θ = χ2fρ2+gd2

1A2

1+ 4 A1χ A2f ρ2+gd2

1

+4fρ2+ 4 gd2

1A2

2+ 4 χ2kb2

χ=Γ(γ+ 1)

βd1xβ−(fρ τ +gd1s1)yβ+ρ tβ,

ξ=Γ(γ+ 1)

βs1xβ−1

2fτ 2+gs2

1yβ+τ tβ+ω.

4 Graphs of Some Exact Solutions

In this section, we show graphs of some of the

exact traveling wave solutions of the truncated M-

fractional paraxial wave dynamical model in Kerr me-

dia in Eq. (2) that we obtained using the (G′/G, 1/G)-

expansion method. In particular, we show the exact

solutions through 3D, 2D, and contour plots for the

following range of fractional-order values: β= 0.9,

β= 0.8,and β= 0.6. The exact traveling wave

solutions in Eq. (28) and Eq. (37) have been chosen

to demonstrate how their physical behavior changes

in terms of 3D, 2D, and contour plots when values

of the fractional-order βare altered. All figures were

obtained using the Maple software package.

In Figure 1 (Appendix), magnitudes of the exact

traveling wave solution W(x, y, t)in (28) are plotted

on the domain

D1={(x, y, t)|06x660, y = 1,and 06t630}

for the 3D plots and on the domain

D2={(x, y, t)|06x660, y = 1,and t= 1}

for the 2D graphs. In addition, contour plots, which

represent a 3D surface by plotting (x, t)contours for

a range of fixed |W|values, are also illustrated. The

following parameter values: λ=−1, µ = 0,a0= 0,

b1= 0,f= 0.8,g= 2,k= 0.8,d1= 2,ρ= 0.5,

s1= 5,τ= 2,ω= 3,A1= 3,A2= 5,and γ= 1.5

are used in this figure. In particular, Figures 1 (a)-(c),

(d)-(f), and (g)-(i) (Appendix) show the 3D, 2D, and

contour graphs of magnitudes of the exact solution

W(x, y, t)in (28) calculated at β= 0.9, β = 0.8,and

β= 0.6, respectively. As can be observed from the

3D graphs of Figure 1 (Appendix), the physical be-

havior of the magnitude of solution (28) can be char-

acterized as an anti-soliton solution. In Figure 1 (Ap-

pendix), it is worth noticing that the singular point of

|W(x, t)|can be moved as the value of the fractional-

order βis changed. This is because the denomi-

nator term A1sinh χ√−λ+A2cosh χ√−λin

Eq. (28) can be zero depending upon the value of

β, which is embedded in χ. For the given parame-

ter values as mentioned above, the singular point is

x≈12.7918 when β= 0.9and t= 1 as shown in

Figure 1 (b) (Appendix).

In Figure 2 (Appendix), magnitudes of the exact

traveling wave solution W(x, y, t)in (37) are plotted

on the domain

D3={(x, y, t)|06x610, y = 1,and 06t610}

for the 3D plots and on the domain

D4={(x, y, t)|06x610, y = 1,and t= 1}

for the 2D graphs. Contour plots, which represent a

3D surface by plotting (x, t)contours for a range of

fixed |W|values, are also shown. The following pa-

rameter values: λ= 1, µ = 0,a0= 0,b1= 0,

f= 1,g= 0.5,k= 1,d1= 2,ρ= 0.1,s1= 5,

τ= 0.5,ω= 1,A1= 3,A2= 5,and γ= 0.8

are used in this figure. In particular, Figures 2 (a)-

(c), (d)-(f), and (g)-(i) (Appendix) display the 3D, 2D,

and contour plots of magnitudes of the exact solution

W(x, y, t)in (37) calculated at β= 0.9, β = 0.8,

and β= 0.6, respectively. As can be observed from

the 3D graphs of Figure 2 (Appendix), the magnitude

of solution (37) can be categorized as a singularly pe-

riodic wave solution. In Figure 2 (Appendix), it is

worth observing that the singular point of |W(x, t)|

can be changed when the value of the fractional-order

βis varied. This is because the denominator term

A1sin χ√λ+A2cos χ√λin Eq. (37) can be

zero depending upon the value of β, which appears

in χ. Specifically, the singular point of |W(x, t)|

is obtained when χ√λ=−arctan A2

A1. For the

given parameter values as described above, some of

the singular points are, for instance, x≈2.1327 and

x≈4.0164 when β= 0.9and t= 1 as shown in

Figure 2 (b) (Appendix).

5 Conclusions

In this paper, the paraxial wave dynamical model

in Kerr media with truncated M-fractional deriva-

tives given in (2) has been symbolically solved

to obtain exact traveling wave solutions using the

(G′/G, 1/G)-expansion method. Since the equation

has complex-valued solutions, we wrote exact solu-

tions as the product of a real function U(χ)and eiξ as

shown in (20). The algebraic manipulations required

to obtain the exact solutions of the function U(χ)

were carried out using the Maple software package.

We found that exact solutions for U(χ)can be written

in terms of either hyperbolic functions, trigonometric

functions, or rational functions. From these solutions

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for U(χ), we finally obtained the exact traveling wave

solutions of the equation (2) via Eq. (20) and the trans-

formation (21).

In [38] the authors used the modified simple

equation method (MSEM) and the auxiliary equa-

tion method (AEM) to find exact solutions of the M-

fractional paraxial wave equation with Kerr media.

Their governing equation was equipped with higher-

order truncated M-fractional partial derivatives with

respect to tand x. This is slightly different from

Eq. (2) in which the composite of the truncated M-

fractional derivatives of order less than one is used.

In addition, the fractional-order βwas not inserted as

an exponent of the independent variables x, y, and

tin their traveling wave transformation. However,

the real function U(χ)of their solutions expressed in

terms of the exponential functions were found. In [39]

the truncated time M-fractional paraxial wave equa-

tion in kerr media was explored for some optical so-

lutions. The unified scheme was implemented to ob-

tain exact traveling wave solutions of the proposed

equation. As a result, the solutions were expressed in

terms of hyperbolic, trigonometric, and rational func-

tions with some free parameters. Roughly compar-

ing our results to the obtained solutions in [38], [39],

some of the exact solutions obtained in this article

have not been derived in any previous work because

equation (2) and the used method are not the same as

in the referred literature.

From our results, the 3D, 2D, and contour plots

of magnitudes of selected solutions have been plotted

for a range of values of fractional-order βusing the

Maple package in order to understand the effects of

changing the fractional-order on the physical behav-

ior of chosen solutions. From Figure 1 and Figure 2

(Appendix), an anti-soliton solution and a singularly

periodic wave solution have been found. Finally, with

the assistance of Maple, all of the exact solutions have

been verified by substituting them back into the orig-

inal equation to check their correctness. In summary,

since the (G′/G, 1/G)-expansion method is an ex-

tension of the (G′/G)-expansion method and its rel-

evant methods, [40], the advantage of the proposed

method is that it is more productive, efficient, and re-

liable for generating exact traveling wave solutions of

nonlinear real-world problems modeled by NPDEs.

This work could be improved by using different frac-

tional order values for the truncated M-fractional par-

tial derivatives with respect to x, y, and t. A promis-

ing future work would be to compare the fractional

equation and solutions developed in this article with

real data obtained from physical phenomena.

Acknowledgments:

The authors are grateful to anonymous referees for

their valuable comments and several constructive

suggestions, which have significantly improved this

manuscript.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Pim Malingam and Paiwan Wongsasinchai: Con-

ceptualization, data curation, funding acquisition,

investigation, methodology, software, visualization,

writing-original draft, and writing-review and editing.

Sekson Sirisubtawee (Corresponding author): Con-

ceptualization, data curation, formal analysis, in-

vestigation, methodology, project administration,

resources, supervision, validation, visualization,

writing-original draft, and writing-review and editing.

Sanoe Koonprasert: Supervision, validation, and

writing-review and editing.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

Pim Malingam (pim.m@rbru.ac.th) would like to ac-

knowledge the financial support from the Department

of Mathematics, Faculty of Science and Technology,

Rambhai Barni Rajabhat University, Chanthaburi.

Conflicts of Interest

The authors declare that they have no conflicts of in-

terest.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

Appendix

Figures described in section 4 are shown in this sec-

tion.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.53

Pim Malingam, Paiwan Wongsasinchai,

Sekson Sirisubtawee, Sanoe Koonprasert

E-ISSN: 2224-2856

511

Volume 18, 2023

(a) 3D (b) 2D (c) contour

(d) 3D (e) 2D (f) contour

(g) 3D (h) 2D (i) contour

Fig. 1: Graphs for |W(x, y, t)|where W(x, y, t)is expressed in (28) and obtained using the (G′/G, 1/G)-

expansion method: (a)-(c) when β= 0.9; (d)-(f) when β= 0.8; (g)-(i) when β= 0.6.

(a) 3D (b) 2D (c) contour

(d) 3D (e) 2D (f) contour

(g) 3D (h) 2D (i) contour

Fig. 2: Graphs for |W(x, y, t)|where W(x, y, t)is expressed in (37) and obtained using the (G′/G, 1/G)-

expansion method: (a)-(c) when β= 0.9; (d)-(f) when β= 0.8; (g)-(i) when β= 0.6.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.53

Pim Malingam, Paiwan Wongsasinchai,

Sekson Sirisubtawee, Sanoe Koonprasert

E-ISSN: 2224-2856

512

Volume 18, 2023