Exact solutions for the conformable fractional coupled nonlinear Schrödinger equations with variable coefficients

Abstract

In this work, the general algebraic method is proposed for finding some solutions of a conformable fractional coupled nonlinear Schrödinger equation with variable coefficients arising in inhomogeneous fibers with two orthogonal polarization states. With the aid of symbolic computations, many types of new soliton pulse and periodic pulse solutions including complex doubly periodic solutions, solitary wave solutions, and trigonometric function solutions are obtained. Some 3D and 2D numerical simulations about these solutions are portraited, which show the novelty and visibility of the dynamical structure and propagation behavior of the corresponding model. Moreover, we found the W-shaped soliton pulse and the periodic wave pulse can be effectively controlled by changing the relative parameters of the frequency coefficients and orders, which will help us to have a better understanding about the internal structure of this model.

Keywords

conformable derivative fractional coupled nonlinear Schrödinger equations W-shaped soliton pulse general algebraic method exact solutions

Introduction

Since Newton and Leibniz founded calculus, the development of natural science has entered a new era, many natural phenomena can be described by partial differential equation models, but in recent years, people realized that more and more problems of engineering science and mathematical physics need to be descripted by fractional differential equations(FDEs), such as chaotic oscillations,¹ engineering and physics ,² two-scale thermal science ,³ optical fibers, ⁴ and so on.^5–6 For better understanding and explaining the physical meaning of these models, searching for analytic solutions including approximate solutions and exact solutions of these FDEs plays an important and significant role in these areas. In the last two decades, many powerful methods have been proposed by scholars to handle this subject, such as homotopy analysis method(HAM),⁷ homotopy perturbation method(HPM),⁸ Adomian’s decomposition method(ADM),⁹ and variational iteration method(VIM)¹⁰ to find the approximate solutions of FDEs. Bäcklund transformation method,¹¹ Darboux transformation,¹² and Hirota bilinear method¹³ can be used to find the N-soliton solutions. Improved F-expansion method,¹⁴ projective Riccati equations method,¹⁵ Jacobi elliptic function expansion method,¹⁶ $G^{'} / G$ -expansion method,¹⁷ exp-function method^18–19 etc.,²⁰ can be used to find doubly periodic solutions, solitary wave solutions, and trigonometric function solutions of these models.

Up to now, scholars have built many types of definitions about the fractional derivative, the most classic definitions are

Riemann–Liouville fractional derivative²¹

D_{t}^{α} y (t) = {\begin{cases} \frac{1}{Γ (n - α)} \frac{d^{n}}{d t^{n}} \int_{0}^{t} {(t - ξ)}^{n - α - 1} y (ξ) d ξ, n - 1 < α < n, n \in N, \\ \frac{d^{(n)} y (t)}{d t^{n}}, α = n \in N . \end{cases}

Caputo fractional derivative²²

D_{t}^{α} y (t) = {\begin{cases} \frac{1}{Γ (n - α)} \int_{0}^{t} {(t - ξ)}^{n - α - 1} y^{(n)} (ξ) d ξ, n - 1 < α < n, n \in N, \\ \frac{d^{(n)} y (t)}{d t^{n}}, α = n \in N . \end{cases}

Jumaries’s fractional derivative²³

D_{t}^{α} y (t) = {\begin{cases} \frac{1}{Γ (1 - α)} \frac{d}{d t} \int_{0}^{t} {(t - ξ)}^{- α} [y (ξ) - y (0)] d ξ, 0 < α < 1, \\ {(y^{(n)} (t))}^{(α - n)}, n \leq α < n + 1, n \in N^{+} . \end{cases}

He’s fractional derivative²⁴

D_{t}^{α} y (t) = {\begin{cases} \frac{1}{Γ (n - α)} \frac{d^{n}}{d t^{n}} \int_{t_{0}}^{t} {(ξ - t)}^{n - α - 1} [y_{0} (ξ) - y (ξ)] d ξ, n - 1 < α < n, n \in N, \\ \frac{d^{(n)} y (t)}{d t^{n}}, α = n \in N . \end{cases}

Furthermore, Atangana fractional derivative,²⁵ M-fractional derivative,²⁶ Atangana–Baleanu derivative, ²⁷ and the conformable fractional derivative which will be utilized in this article,^28,29 etc ³⁰ are built recently, the geometrical explanation on fractional calculus has been discussed by He in Ref. [31].

We consider the conformable fractional coupled nonlinear Schrödinger equation in inhomogeneous fibers (CFCNLS) in the form^32–34

{\begin{cases} i D_{t}^{α} u + f (t) D_{x}^{2 β} u + 2 g (t) ({| u |}^{2} + {| v |}^{2}) u = 0, t > 0,0 < α, β \leq 1, \\ i D_{t}^{α} v + f (t) D_{x}^{2 β} v + 2 g (t) ({| u |}^{2} + {| v |}^{2}) v = 0 . \end{cases}

(1)

Here,

u = u (x, t), D_{x}^{2 β} u = D_{x}^{β} (D_{x}^{β} u),

v = v (x, t), D_{x}^{2 β} v = D_{x}^{β} (D_{x}^{β} v), i = \sqrt{- 1}

u

and

v

are two complex-valued functions represent the slowly varying amplitudes of two orthogonally polarized optical pulses, the coeffificients

f (t)

and

g (t)

are the group velocity dispersion and the Kerr nonlinearity coefficients. When

α = β = 1

, two class of solitary wave solutions of equation (1) are studied by using the modified sine-Gordon equation method in Ref. [32], the interactions of anti-dark soliton solutions are discussed through the bilinear Hirota operators in Ref. [33] and the asymmetric M-shaped and W-shaped soliton pulse are generated in Ref. [34]. Let us review some basic definitions and properties of the fractional calculus theory which will be used further in this paper.^28–29

Definition for a function $y (t) : [0, \infty) \to R$ .We defined the conformable fractional derivative operator and integral operator of order $α$ as²⁸

D_{t}^{α} y (t) = \lim_{ε \to 0} \frac{y (t + ε t^{1 - α}) - y (t)}{ε}, t > 0, α \in (0,1] .

I_{t}^{α} y (t) = \int_{0}^{t} τ^{α - 1} y (τ) d τ, t > 0, α \in (0,1] .

Also, we have the following important properties^28,29:

\begin{array}{l} (1) D_{t}^{α} y (t) = t^{1 - α} y^{'} (t) . \\ (2) D_{t}^{α} (a y (t) + b z (t)) = a D_{t}^{α} y (t) + b D_{t}^{α} z (t), \forall a, b \in R . \\ (3) D_{t}^{α} (y (t) z (t)) = y (t) D_{t}^{α} z (t) + z (t) D_{t}^{α} y (t) . \\ (4) D_{t}^{α} (y (t) / z (t)) = [z (t) D_{t}^{α} y (t) - y (t) D_{t}^{α} z (t)] / z^{2} (t) . \\ (5) D_{t}^{α} (y \circ z) (t) = t^{1 - α} y' (z (t)) z' (t) . \end{array}

The paper is organized as follows: In Section 2, we introduce the general algebraic method based on the sub-ode method. In Section 3, some new exact solutions of the CFCNLS are found and discussed by utilizing the proposed method. Finally, the conclusion is presented in Section 4.

The general algebraic method (GAM)

A brief description of the GAM is presented as follows:

Step 1. Consider the following nonlinear fractional partial differential equation

E (u, u_{t}^{α}, u_{x}^{β}, u_{x}^{2 β}, u u_{x}^{β}, \dots) = 0 .

(2)

Step 2. Using the fractional complex transformation^35–37

u (x, t) = u (ξ), ξ = \frac{k_{1}}{β} x^{β} + \frac{c_{1}}{α} t^{α} .

(3)

Here,

k_{1}, c_{1}

are constants or functions of

x, t

and equation. (2) can be converted to an ordinary differential equation(ODE) as follows:

O (u, u_{ξ}, u_{ξ ξ}, u u_{ξ}, \dots) = 0 .

(4)

Step 3. Assume that equation (4) has the following solution

u (ξ) = \sum_{i = 0}^{n} A_{i} φ^{i} (ξ) + \sum_{i = 1}^{n} A_{- i} φ^{- i} (ξ) .

(5)

Here,

n

is a balance number,

A_{i}

A_{- i}

and variable function

ξ = ξ (x, t)

are determined later. And

φ (ξ)

is a solution of the following auxiliary ODE:

φ^{' 2} (ξ) = \sum_{i = 0}^{4} a_{i} φ^{i} (ξ) .

(6)

Step 4. We assume that equation (6) have the following solutions:

φ (ξ) = b_{0} + b_{1} e + b_{2} f + b_{3} g + b_{4} h + b_{- 1} e^{- 1} + b_{- 2} f^{- 1} + b_{- 3} g^{- 1} + b_{- 4} h^{- 1} .

(7)

Here,

b_{i} (i = 0, \pm 1, \dots, \pm 4)

are constants of to be determined later, and the four functions

$e = e (ξ), f = f (ξ), g = g (ξ), h = h (ξ)$ are expressed as the follows³⁸:

e = \frac{1}{Z}, f = \frac{F}{Z}, g = \frac{F^{2}}{Z}, h = \frac{F^{'}}{Z}, Z = p + q F + r F^{2} + l F^{'} .

(8)

Here,

F'^{2} = A + B F^{2} + C F^{4} + 2 D F + 2 E F^{3} .

Step 5

After finding the new solutions of equation (6) by equations (7) and (8), let us substitute Equations (6) and (5) in equation (4), setting the coefficients of $φ^{i} (ξ)$ and $φ^{j} (ξ) \sqrt{\sum_{i = 0}^{4} a_{i} φ^{i} (ξ)}$ $(i, j = 0, \pm 1, \pm 2, \dots)$ to zero yield a set of algebraic equations(AEs) for $A_{i}$ , $A_{- i}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4},$ $k_{1}$ and $c_{1}$ . After solving the AEs and substituting each of the solutions $φ (ξ)$ along with (3) in equation (2), we can get the solutions of equation (2).

Remark 1. In equation (8), the functions $F, e, f, g, h$ satisfy the restricted relation (6) (7) and (8a–8d) mentioned in i³⁸, if we select $p = r = l = 0, q = 1, h = \frac{F^{'}}{F}$ , this method turns to the $\frac{G^{'}}{G}$ method^39–44, if we select $p = q = l = 0, r = 1, h = \frac{F^{'}}{F^{2}}$ , it turns to the $\frac{G^{'}}{G^{2}}$ method⁴⁵, if we select $p = r = 0, q = 1, h = \frac{F^{'}}{F + l F^{'}}$ , it turns to the $\frac{G^{'}}{G + μ G^{'}}$ method,⁴⁶ and if we select $a_{0} = α^{2} {(L n A)}^{2}, a_{1} = 2 α β {(L n A)}^{2}, a_{2} = (2 α σ + β^{2}) {(L n A)}^{2}, a_{3} = 2 β σ {(L n A)}^{2},$ $a_{4} = σ^{2} {(L n A)}^{2},$ it turns to the direct algebraic method.⁴⁷

In the following, we will use this method to solve the CFCNLS.

Exact solutions to the CFCNLS

Exact solutions

If we let $D_{t}^{α} u = \frac{\partial^{α} u}{\partial t^{α}}, D_{x}^{2 β} u = \frac{\partial^{2 β} u}{\partial x^{2 β}}$ , equation (1) can be rewritten as follows

{\begin{cases} i u_{t}^{α} + f (t) u_{x}^{2 β} + 2 g (t) ({| u |}^{2} + {| v |}^{2}) u = 0, t > 0,0 < α, β \leq 1, \\ i v_{t}^{α} + f (t) v_{x}^{2 β} + 2 g (t) ({| u |}^{2} + {| v |}^{2}) v = 0 . \end{cases}

(9)

We can give the following complex variable transformation^48–50:

u = ϕ (ξ) e^{i η}, v = ψ (ξ) e^{i η},

(10)

ξ = \frac{k_{1}}{β} x^{β} + \int c_{1} t^{α - 1} d t, η = \frac{k_{2}}{β} x^{β} + \int c_{2} t^{α - 1} d t .

(11)

Here, constants

k_{1}, k_{2}, a n d c_{1} = c_{1} (t), c_{2} = c_{2} (t)

are to be determined later.

Substituting equations (10) and (11) in equation (9), separate the real part and the imaginary part, we obtain

{\begin{cases} k_{1}^{2} f ϕ_{ξ ξ} - (c_{2} + k_{2}^{2} f) ϕ + 2 g (ϕ^{3} + ψ^{2} ϕ) = 0, \begin{array}{c} (12.1) \end{array} \\ k_{1}^{2} f ψ_{ξ ξ} - (c_{2} + k_{2}^{2} f) ψ + 2 g (ψ^{3} + ϕ^{2} ψ) = 0, \begin{array}{c} (12.2) \end{array} \\ (c_{1} + 2 k_{1} k_{2} f) ϕ_{ξ} = 0, \begin{array}{c} (12.3) \end{array} \\ (c_{1} + 2 k_{1} k_{2} f) ψ_{ξ} = 0 . \begin{array}{c} (12.4) \end{array} \end{cases}

(12)

Where

ϕ_{ξ ξ} = \frac{d^{2} ϕ (ξ)}{d ξ^{2}}, ϕ_{ξ} = \frac{d ϕ (ξ)}{d ξ}, ψ_{ξ ξ} = \frac{d^{2} ψ (ξ)}{d ξ^{2}}, ψ_{ξ} = \frac{d ψ (ξ)}{d ξ}, f = f (t), g = g (t)

We assume that equation (12) have the following solutions:

\begin{array}{l} ϕ = ϕ (ξ) = A_{0} + A_{1} φ (ξ) + A_{- 1} φ^{- 1} (ξ) = A_{0} + A_{1} φ + A_{- 1} φ^{- 1}, \\ ψ = ψ (ξ) = B_{0} + B_{1} φ (ξ) + B_{- 1} φ^{- 1} (ξ) = B_{0} + B_{1} φ + B_{- 1} φ^{- 1} . \end{array}

(13)

Here,

φ'^{2} = \sum_{i = 0}^{4} a_{i} φ^{i}, φ = φ (ξ)

and

φ^{'} = \frac{d φ (ξ)}{d ξ}

Substituting both equations (6) and (13) in equation (12) and setting the coefficients of $φ^{i}$ to zero yields an AEs with respect to the unknowns $A_{0}, A_{1}, A_{- 1}, B_{0}, B_{1}, B_{- 1}, a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, k_{1}, k_{2}, c_{1}, f, g$ , and $c_{2}$ as follows:

\begin{array}{l} φ^{0} : 2 g A_{0}^{3} + 12 g A_{- 1} A_{0} A_{1} + 4 g A_{1} B_{- 1} B_{0} + 2 g A_{0} B_{0}^{2} + 4 g A_{0} B_{- 1} B_{1} + 4 g A_{- 1} B_{0} B_{1} - A_{0} c_{2} \\ + \frac{1}{2} f a_{3} A_{- 1} k_{1}^{2} + \frac{1}{2} f a_{1} A_{1} k_{1}^{2} - f A_{0} k_{2}^{2} = 0, \\ φ^{1} : 6 g A_{0}^{2} A_{1} + 6 g A_{- 1} A_{1}^{2} + 2 g A_{1} B_{0}^{2} + 4 g A_{1} B_{- 1} B_{1} + 4 g A_{0} B_{0} B_{1} + 2 g A_{- 1} B_{1}^{2} - A_{1} c_{2} \\ + f a_{2} A_{1} k_{1}^{2} - f A_{1} k_{2}^{2} = 0, \\ φ^{2} : 6 g A_{0} A_{1}^{2} + 4 g A_{1} B_{0} B_{1} + 2 g A_{0} B_{1}^{2} + \frac{3}{2} f a_{3} A_{1} k_{1}^{2} = 0, \\ φ^{3} : 2 g A_{1}^{3} + 2 g A_{1} B_{1}^{2} + 2 f a_{4} A_{1} k_{1}^{2} = 0, \\ φ^{- 1} : 6 g A_{- 1} A_{0}^{2} + 6 g A_{- 1}^{2} A_{1} + 2 g A_{1} B_{- 1}^{2} + 4 g A_{0} B_{- 1} B_{0} + 2 g A_{- 1} B_{0}^{2} + 4 g A_{- 1} B_{- 1} B_{1} - A_{- 1} c_{2} \\ + f a_{2} A_{- 1} k_{1}^{2} - f A_{- 1} k_{2}^{2} = 0, \\ φ^{- 2} : 6 g A_{- 1}^{2} A_{0} + 2 g A_{0} B_{- 1}^{2} + 4 g A_{- 1} B_{- 1} B_{0} + \frac{3}{2} f a_{1} A_{- 1} k_{1}^{2} = 0, \\ φ^{- 3} : 2 g A_{- 1}^{3} + 2 g A_{- 1} B_{- 1}^{2} + 2 f a_{0} A_{- 1} k_{1}^{2} = 0, \end{array}

\begin{array}{l} φ^{0} : 4 g A_{0} A_{1} B_{- 1} + 2 g A_{0}^{2} B_{0} + 4 g A_{- 1} A_{1} B_{0} + 2 g B_{0}^{3} + 4 g A_{- 1} A_{0} B_{1} + 12 g B_{- 1} B_{0} B_{1} - B_{0} c_{2} \\ + \frac{1}{2} f a_{3} B_{- 1} k_{1}^{2} + \frac{1}{2} f a_{1} B_{1} k_{1}^{2} - f B_{0} k_{2}^{2} = 0, \\ φ^{1} : 2 g A_{1}^{2} B_{- 1} + 4 g A_{0} A_{1} B_{0} + 2 g A_{0}^{2} B_{1} + 4 g A_{- 1} A_{1} B_{1} + 6 g B_{0}^{2} B_{1} + 6 g B_{- 1} B_{1}^{2} - B_{1} c_{2} \\ + f a_{2} B_{1} k_{1}^{2} - f B_{1} k_{2}^{2} = 0, \\ φ^{2} : 2 g A_{1}^{2} B_{0} + 4 g A_{0} A_{1} B_{1} + 6 g B_{0} B_{1}^{2} + \frac{3}{2} f a_{3} B_{1} k_{1}^{2} = 0, \\ φ^{3} : 2 g A_{1}^{2} B_{1} + 2 g B_{1}^{3} + 2 f a_{4} B_{1} k_{1}^{2} = 0, \\ φ^{- 1} : 2 g A_{0}^{2} B_{- 1} + 4 g A_{- 1} A_{1} B_{- 1} + 4 g A_{- 1} A_{0} B_{0} + 6 g B_{- 1} B_{0}^{2} + 2 g A_{- 1}^{2} B_{1} + 6 g B_{- 1}^{2} B_{1} - B_{- 1} c_{2} \\ + f a_{2} B_{- 1} k_{1}^{2} - f B_{- 1} k_{2}^{2} = 0, \\ φ^{- 2} : 4 g A_{- 1} A_{0} B_{- 1} + 2 g A_{- 1}^{2} B_{0} + 6 g B_{- 1}^{2} B_{0} + \frac{3}{2} f a_{1} B_{- 1} k_{1}^{2} = 0, \\ φ^{- 3} : 2 g A_{- 1}^{2} B_{- 1} + 2 g B_{- 1}^{3} + 2 f a_{0} B_{- 1} k_{1}^{2} = 0 . \end{array}

After solving the above AEs along with equations (10) and (11), we could determine the following solutions:

Family 1 $ϕ (ξ) = ψ (ξ) = C$ , $c_{2} = 4 C^{2} g (t) - k_{2}^{2} f (t)$ .

We find the following trivial solution of equation (9)

u_{0} = v_{0} = C e^{i {\frac{k_{2}}{β} x^{β} + \int [4 C^{2} g (t) - k_{2}^{2} f (t)] t^{α - 1} d t}} .

Family 2 $c_{1} + 2 k_{1} k_{2} f = 0$ .

In this situation, we produce the following solutions of the above AEs:

Case 1

\begin{array}{l} a_{1} = a_{3} = 0, A_{0} = B_{0} = 0, f = λ g, B_{1} = \pm \sqrt{- A_{1}^{2} - λ a_{4} k_{1}^{2}}, k_{2} = \pm k_{1} \sqrt{\frac{a_{2}}{2}}, \\ c_{1} = \mp k_{1}^{2} λ \sqrt{2 a_{2}} g, c_{2} = \frac{a_{2} k_{1}^{2} λ g}{2} \end{array}

Case 2

\begin{array}{l} a_{1} = a_{3} = 0, A_{0} = B_{0} = 0, f = λ g, c_{2} = (a_{2} k_{1}^{2} - 6 k_{1}^{2} \sqrt{a_{0} a_{4}} - k_{2}^{2}) λ g \\ A_{1} = \overset{\pm}{\pm} k_{1} \sqrt{\frac{a_{4} λ}{2}}, A_{- 1} = \overset{\pm}{\pm} k_{1} \sqrt{\frac{a_{0} λ}{2}}, B_{1} = \pm k_{1} \sqrt{- \frac{3 a_{4} λ}{2}}, B_{- 1} = \pm k_{1} \sqrt{- \frac{3 a_{0} λ}{2}} . \end{array}

Thus, the following solutions of equation (9) are deduced:

Type 1

{\begin{cases} u_{1} = A_{1} φ [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{2 a_{2}} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{a_{2}}{2}} x^{β} + \frac{a_{2} k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]}, \\ v_{1} = \pm \sqrt{- A_{1}^{2} - λ a_{4} k_{1}^{2}} φ [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{2 a_{2}} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{a_{2}}{2}} x^{β} + \frac{a_{2} k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]} . \end{cases}

Type 2

{\begin{cases} u_{2} = [\pm k_{1} \sqrt{\frac{a_{4} λ}{2}} φ (ξ_{2}) \pm k_{1} \sqrt{\frac{a_{0} λ}{2}} φ^{- 1} (ξ_{2})] e^{i [\frac{k_{2}}{β} x^{β} + (a_{2} k_{1}^{2} - 6 k_{1}^{2} \sqrt{a_{0} a_{4}} - k_{2}^{2}) λ \int g (t) t^{α - 1} d t]}, \\ v_{2} = [\pm k_{1} \sqrt{- \frac{3 a_{4} λ}{2}} φ (ξ_{2}) \pm k_{1} \sqrt{- \frac{3 a_{0} λ}{2}} φ^{- 1} (ξ_{2})] e^{i [\frac{k_{2}}{β} x^{β} + (a_{2} k_{1}^{2} - 6 k_{1}^{2} \sqrt{a_{0} a_{4}} - k_{2}^{2}) λ \int g (t) t^{α - 1} d t]}, \\ ξ_{2} = \frac{k_{1}}{β} x^{β} - 2 k_{1} k_{2} λ \int g (t) t^{α - 1} d t . \end{cases}

Here, $φ_{i}$ is an arbitrary solution of the auxiliary equation $φ_{i}'^{2} = a_{0} + a_{2} φ_{i}^{2} + a_{4} φ_{i}^{4}$ in $u_{1}, u_{2}, v_{1}, v_{2}$ , the coefficients $a_{0}, a_{2}, a_{4}$ are arbitrary constants, we can find some new solutions of equation (6) by using equations (7) and (8), and many types of $φ_{i}$ have been found in a large number of papers such as Refs. [51, 52], and if we select different sets of coefficients, then the corresponding solutions of Equation (9) can be found. Let us take some typical cases for example.

Set 1 For $a_{0} = 1 - m^{2}, a_{2} = 2 m^{2} - 1, a_{4} = - m^{2},$ $φ_{1.1} = c n ξ$ , we have

{\begin{cases} u_{1.1} = A_{1} c n [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{4 m^{2} - 2} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{2 m^{2} - 1}{2}} x^{β} + \frac{(2 m^{2} - 1) k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]}, \\ v_{1.1} = \pm \sqrt{λ m^{2} k_{1}^{2} - A_{1}^{2}} c n [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{4 m^{2} - 2} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{2 m^{2} - 1}{2}} x^{β} + \frac{(2 m^{2} - 1) k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]} . \end{cases}

Obviously, $u_{1.1}$ is degenerated to the famous bell-shape-like solitary wave solutions when $α = β = 1, m = 1$ .

{\begin{cases} u_{1 . 1_{m = 1}} = A_{1} \sec h [k_{1} x \mp k_{1}^{2} λ \sqrt{2} \int g (t) d t] e^{i [\pm k_{1} \sqrt{\frac{1}{2}} x + \frac{k_{1}^{2} λ}{2} \int g (t) d t]}, \\ v_{1 . 1_{m = 1}} = \pm \sqrt{λ k_{1}^{2} - A_{1}^{2}} \sec h [k_{1} x \mp k_{1}^{2} λ \sqrt{2} \int g (t) d t] e^{i [\pm k_{1} \sqrt{\frac{1}{2}} x + \frac{k_{1}^{2} λ}{2} \int g (t) d t]} . \end{cases}

Set 2 For $a_{0} = m^{2} - 1, a_{2} = 2 - m^{2}, a_{4} = - 1,$ $φ_{1.2} = d n ξ$ , we have

{\begin{cases} u_{1.2} = A_{1} d n [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{2 (2 - m^{2})} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{2 - m^{2}}{2}} x^{β} + \frac{(2 - m^{2}) k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]}, \\ v_{1.2} = \pm \sqrt{λ k_{1}^{2} - A_{1}^{2}} d n [\frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{2 (2 - m^{2})} \int g (t) t^{α - 1} d t] e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{2 - m^{2}}{2}} x^{β} + \frac{(2 - m^{2}) k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t]} . \end{cases}

Set 3 For $a_{0} = \frac{1 - m^{2}}{4}, a_{2} = \frac{1 + m^{2}}{2}, a_{4} = \frac{1 - m^{2}}{4},$ $φ_{1.3} = n c ξ \pm s c ξ$ , we have

{\begin{cases} u_{1.3} = A_{1} (n c ξ_{1.3} + s c ξ_{1.3}) e^{i [\pm \frac{k_{1}}{β} \sqrt{1 + m^{2}} x^{β} + \frac{(1 + m^{2}) k_{1}^{2} λ}{4} \int g (t) t^{α - 1} d t]}, \\ v_{1.3} = \pm \sqrt{- A_{1}^{2} - \frac{λ (1 - m^{2})}{4} k_{1}^{2}} (n c ξ_{1.3} + s c ξ_{1.3}) e^{i [\pm \frac{k_{1}}{β} \sqrt{1 + m^{2}} x^{β} + \frac{(1 + m^{2}) k_{1}^{2} λ}{4} \int g (t) t^{α - 1} d t]} \\ ξ_{1.3} = \frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{1 + m^{2}} \int g (t) t^{α - 1} d t . \end{cases}

Selecting $α = β = 1, A_{1} = 1, k_{1} = 1, λ = - 8, m = 0$ in the above solution, thus

u_{1.3, m = 0} = v_{1.3, m = 0} = [\frac{1}{\cos [x \pm 8 \int g (t) d t]} + \tan [x \pm 8 \int g (t) d t] e^{i [\pm x - 2 \int g (t) d t]}

Set 4 For $a_{0} = \frac{1}{4}, a_{2} = \frac{1}{2} - m^{2}, a_{4} = \frac{1}{4},$ $φ_{1.4} = \frac{s n ξ}{1 \pm c n ξ}$ , we have

{\begin{cases} u_{1.4} = \frac{A_{1} s n ξ_{1.4}}{1 \pm c n ξ_{1.4}} e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{1 - 2 m^{2}}{4}} x^{β} + \frac{(1 - 2 m^{2}) k_{1}^{2} λ}{4} \int g (t) t^{α - 1} d t]}, \\ v_{1.4} = \frac{\pm \sqrt{- A_{1}^{2} - \frac{1}{4} λ k_{1}^{2}} s n ξ_{1.4}}{1 \pm c n ξ_{1.4}} e^{i [\pm \frac{k_{1}}{β} \sqrt{\frac{1 - 2 m^{2}}{4}} x^{β} + \frac{(1 - 2 m^{2}) k_{1}^{2} λ}{4} \int g (t) t^{α - 1} d t],} \\ ξ_{1.4} = \frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ \sqrt{1 - 2 m^{2}} \int g (t) t^{α - 1} d t . \end{cases}

Set 5 For $a_{0} = 1, a_{2} = - 1 - 6 m - m^{2}, a_{4} = 4 m {(1 + m)}^{2},$ $φ_{1.5} = \frac{s n ξ}{1 + m s n^{2} ξ}$ , we have

{\begin{cases} u_{1.5} = \frac{A_{1} s n ξ_{1.3}}{1 + m s n^{2} ξ_{1.3}} e^{i η_{1.5}}, \\ v_{1.5} = \frac{\pm i \sqrt{A_{1}^{2} + λ 4 m {(1 + m)}^{2} k_{1}^{2}} s n ξ_{1.3}}{1 + m s n^{2} ξ_{1.3}} e^{i η_{1.5}}, \\ ξ_{1.5} = \frac{k_{1}}{β} x^{β} \mp k_{1}^{2} λ i \sqrt{2 (1 + 6 m + m^{2})} \int g (t) t^{α - 1} d t, \\ η_{1.5} = \pm i \frac{k_{1}}{β} \sqrt{\frac{1 + 6 m + m^{2}}{2}} x^{β} - \frac{(1 + 6 m + m^{2}) k_{1}^{2} λ}{2} \int g (t) t^{α - 1} d t . \end{cases}

Set 6 For $a_{0} = 1, a_{2} = - m^{2} - 1, a_{4} = m^{2}, φ_{2.1} = s n ξ$ , we have

{\begin{cases} u_{2.1} = (\pm k_{1} m \sqrt{\frac{λ}{2}} s n ξ_{2.1} \pm k_{1} \sqrt{\frac{λ}{2}} n s ξ_{2.1}) e^{i [\frac{k_{2}}{β} x^{β} + ((- m^{2} - 1) k_{1}^{2} - 6 k_{1}^{2} m - k_{2}^{2}) λ \int g (t) t^{α - 1} d t]}, \\ v_{2.1} = (\pm k_{1} m i \sqrt{\frac{3 λ}{2}} s n ξ_{2.1} \pm k_{1} i \sqrt{\frac{3 λ}{2}} s n ξ_{2.1}) e^{i [\frac{k_{2}}{β} x^{β} + ((- m^{2} - 1) k_{1}^{2} - 6 k_{1}^{2} m - k_{2}^{2}) λ \int g (t) t^{α - 1} d t]}, \\ ξ_{2.1} = \frac{k_{1}}{β} x^{β} - 2 k_{1} k_{2} λ \int g (t) t^{α - 1} d t . \end{cases}

Remark 2

If we let

$A_{1} = B_{1}, \sqrt{λ k_{1}^{2} - A_{1}^{2}} = E_{1}, k_{1} = μ, \sqrt{2} k_{1} = 2 α, f = λ g = β (t), u_{1.1, m = 1}, v_{1.1, m = 1}$ turn to the solution (14) mentioned in Ref. [32] which made a clerical error, then

{\begin{cases} u_{[32]} = B_{1} \sec h [μ (x \mp 2 α \int β (t) d t)] e^{i [\pm α x + α^{2} \int β (t) d t]}, \\ v_{[32]} = E_{1} \sec h [μ (x \mp 2 α \int β (t) d t)] e^{i [\pm α x + α^{2} \int β (t) d t]} . \end{cases}

Case 3

\begin{array}{l} a_{1} = \frac{a_{2} a_{3}}{2 a_{4}} - \frac{a_{3}^{3}}{8 a_{4}^{2}}, A_{0} = \pm B_{0} \sqrt{\frac{a_{3}^{3}}{16 a_{1} a_{4}^{2}} - 1}, A_{1} = \pm B_{0} \sqrt{\frac{a_{3}}{a_{1}} - \frac{16 a_{4}^{2}}{a_{3}^{2}}}, B_{1} = \frac{4 a_{4}}{a_{3}} B_{0}, f = λ g, \\ c_{1} = \mp 2 B_{0} k_{2} \sqrt{\frac{- λ a_{3}}{a_{1} a_{4}}} g, c_{2} = (\frac{2 B_{0}^{2} a_{3}^{3}}{16 a_{1} a_{4}^{2}} - 2 B_{0}^{2} - λ k_{2}^{2}) g, k_{1} = \pm B_{0} \sqrt{\frac{- a_{3}}{λ a_{1} a_{4}}} . \end{array}

We find the following solutions of equation (9):

Type 3

{\begin{cases} u_{3} = [\pm B_{0} \sqrt{\frac{a_{3}^{3}}{16 a_{1} a_{4}^{2}} - 1} \pm B_{0} \sqrt{\frac{a_{3}}{a_{1}} - \frac{16 a_{4}^{2}}{a_{3}^{2}}} φ (ξ_{3})] e^{i [\frac{k_{2}}{β} x^{β} + (\frac{2 B_{0}^{2} a_{3}^{3}}{16 a_{1} a_{4}^{2}} - 2 B_{0}^{2} - λ k_{2}^{2}) \int g (t) t^{α - 1} d t]}, \\ v_{3} = [B_{0} + \frac{4 a_{4}}{a_{3}} B_{0} φ (ξ_{3})] e^{i [\frac{k_{2}}{β} x^{β} + (\frac{2 B_{0}^{2} a_{3}^{3}}{16 a_{1} a_{4}^{2}} - 2 B_{0}^{2} - λ k_{2}^{2}) \int g (t) t^{α - 1} d t],} \\ ξ_{3} = \pm \frac{B_{0}}{β} \sqrt{\frac{- a_{3}}{λ a_{1} a_{4}}} x^{β} \mp 2 B_{0} k_{2} \sqrt{\frac{- λ a_{3}}{a_{1} a_{4}}} \int g (t) t^{α - 1} d t . \end{cases}

Set 7 For $a_{0} = \frac{1}{4} (1 - m^{2}), a_{1} = m^{2} - 1, a_{2} = 2 - m^{2}, a_{3} = - 2, a_{4} = 1, φ_{3.1} = \frac{c n ξ}{1 \pm s n ξ + c n ξ}$ , thus

{\begin{cases} u_{3.1} = \pm B_{0} \sqrt{\frac{2 m^{2} - 1}{2 - 2 m^{2}}} (1 - \frac{2 c n ξ_{3.1}}{1 \pm s n ξ_{3.1} + c n ξ_{3.1}}) e^{i [\frac{k_{2}}{β} x^{β} - (B_{0}^{2} (2 - \frac{1}{1 - m^{2}}) + k_{2}^{2} λ)) \int g (t) t^{α - 1} d t]}, \\ v_{3.1} = (B_{0} - \frac{2 B_{0} s n ξ_{3.1}}{1 - s n ξ_{3.1} + c n ξ_{3.1}}) e^{i [\frac{k_{2}}{β} x^{β} -) B_{0}^{2}) 2 - \frac{1}{1 - m^{2}}) + k_{2}^{2} λ)) \int g) t) t^{α - 1} d t]}, \\ ξ_{3.1} = \pm \frac{B_{0}}{β} \sqrt{\frac{2}{λ (m^{2} - 1)}} x^{β} - 2 B_{0} k_{2} \sqrt{\frac{2 λ}{m^{2} - 1}} \int g (t) t^{α - 1} d t . \end{cases}

Selecting $α = β = 1, k_{2} = 1, B_{0} = 1, λ = - 8, m = \frac{\sqrt{3}}{2}, g_{1} (t) = 1, g_{2} (t) = \cos t$ , we get

u_{3.1, m = \sqrt{3} / 2} = [1 - \frac{2 c n [\pm x - 16 \int g (t) d t]}{1 \pm s n [\pm x - 16 \int g (t) d t] + c n [\pm x - 16 \int g (t) d t]}] e^{i [x + 10 \int g (t) d t]} .

Set 8 For $a_{0} = \frac{1}{4}, a_{1} = 1, a_{2} = 2 - m^{2}, a_{3} = 2 - 2 m^{2}, a_{4} = 1 - m^{2}, φ_{3.2} = \frac{s n ξ}{1 - s n ξ + c n ξ}$ , we have

{\begin{cases} u_{3.2} = \pm i \sqrt{2 + 2 m^{2}} B_{0} (\frac{1}{2} + \frac{s n ξ_{3.2}}{1 - s n ξ_{3.2} + c n ξ_{3.2}}) e^{i [\frac{k_{2}}{β} x^{β} -) B_{0}^{2}) 1 + m^{2}) + k_{2}^{2} λ)) \int g (t) t^{α - 1} d t]}, \\ v_{3.2} =) B_{0} + \frac{2 B_{0} s n ξ_{3.2}}{1 - s n ξ_{3.2} + c n ξ_{3.2}}) e^{i [\frac{k_{2}}{β} x^{β} -) B_{0}^{2}) 1 + m^{2}) + k_{2}^{2} λ)) \int g) t) t^{α - 1} d t]}, \\ ξ_{3.2} = \pm \frac{B_{0}}{β} \sqrt{\frac{2}{- λ}} x^{β} - 2 B_{0} k_{2} \sqrt{- 2 λ} \int g (t) t^{α - 1} d t . \end{cases}

Set 9 For $a_{0} = 1, a_{1} = - 4, a_{2} = 8 - 4 m^{2}, a_{3} = 8 m^{2} - 8, a_{4} = 4 - 4 m^{2}, φ_{3.3} = \frac{c n ξ}{c n ξ \pm s n ξ d n ξ}$ , we have

{\begin{cases} u_{3.3} = i B_{0} \sqrt{2 + 2 m^{2}} (\frac{1}{2} - \frac{c n ξ_{3.3}}{c n ξ_{3.3} \pm s n ξ_{3.3} d n ξ_{3.3}}) e^{i [\frac{k_{2}}{β} x^{β} - (B_{0}^{2} (1 + m^{2}) + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]}, \\ v_{3.3} = (B_{0} - 2 B_{0} \frac{c n ξ_{3.3}}{c n ξ_{3.3} \pm s n ξ_{3.3} d n ξ_{3.3}}) e^{i [\frac{k_{2}}{β} x^{β} - (B_{0}^{2} (1 + m^{2}) + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]}, \\ ξ_{3.3} = - \frac{B_{0}}{β \sqrt{- 2 λ}} x^{β} + \sqrt{- 2 λ} B_{0} k_{2} \int g (t) t^{α - 1} d t . \end{cases}

Set 10 For $a_{0} = 1, a_{1} = - 4 m, a_{2} = 8 m^{2} - 4, a_{3} = 8 m - 8 m^{3}, a_{4} = 4 m^{4} - 4 m^{2}, φ_{3.4} = \frac{d n ξ}{d n ξ \pm m^{2} s n ξ c n ξ}$ , we have

{\begin{cases} u_{3.4} = i B_{0} \sqrt{2 m^{2} + 2} (\frac{1}{2 m} - \frac{d n ξ_{3.4}}{d n ξ_{3.4} \pm m^{2} s n ξ_{3.4} c n ξ_{3.4}}) e^{i [\frac{k_{2}}{β} x^{β} - (B_{0}^{2} (\frac{1}{m^{2}} + 1) + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]}, \\ v_{3.4} = (B_{0} - \frac{2 B_{0} m d n ξ_{3.4}}{d n ξ_{3.4} \pm m^{2} s n ξ_{3.4} c n ξ_{3.4}}) e^{i [\frac{k_{2}}{β} x^{β} - (B_{0}^{2} (\frac{1}{m^{2}} + 1) + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]} \\ ξ_{3.4} = - \frac{B_{0}}{β m \sqrt{- 2 λ}} x^{β} + \frac{B_{0} k_{2} \sqrt{- 2 λ}}{m} \int g (t) t^{α - 1} d t . \end{cases}

Selecting $α = β = 1, k_{2} = 1, B_{0} = 1, λ = - 2, m = 0.5$ , we get

u_{3.4, m = 0.5} = [1 - \frac{d n (- x + 4 \int g (t) d t)}{d n (- x + 4 \int g (t) d t) + 0.25 s n (- x + 4 \int g (t) d t) c n (- x + 4 \int g (t) d t)}] e^{i [x - 3 \int g (t) d t]} .

Set 11 For $a_{0} = 1, a_{1} = - 4 \sqrt{1 - m^{2}}, a_{2} = 8 - 4 m^{2}, a_{3} = - 8 \sqrt{1 - m^{2}}, a_{4} = 4 - 4 m^{2},$ $φ_{3.5} = \frac{s n ξ}{\sqrt{1 - m^{2}} s n ξ + c n ξ d n ξ}$ , we have

{\begin{cases} u_{3.5} = B_{0} \sqrt{\frac{4 m^{2} - 2}{1 - m^{2}}} (\frac{1}{2} - \frac{\sqrt{1 - m^{2}} s n ξ_{3.5}}{\sqrt{1 - m^{2}} s n ξ_{3.5} + c n ξ_{3.5} d n ξ_{3.5}}) e^{i [\frac{k_{2}}{β} x^{β} - (\frac{(1 - 2 m^{2}) B_{0}^{2}}{1 - m^{2}} + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]}, \\ v_{3.5} = (B_{0} - \frac{2 B_{0} \sqrt{1 - m^{2}} s n ξ_{3.5}}{\sqrt{1 - m^{2}} s n ξ_{3.5} + c n ξ_{3.5} d n ξ_{3.5}}) e^{i [\frac{k_{2}}{β} x^{β} - (\frac{(1 - 2 m^{2}) B_{0}^{2}}{1 - m^{2}} + k_{2}^{2} λ) \int g (t) t^{α - 1} d t]} \\ ξ_{3.5} = \frac{B_{0}}{β \sqrt{- 2 λ (1 - m^{2})}} x^{β} + \frac{B_{0} k_{2} \sqrt{- 2 λ}}{\sqrt{1 - m^{2}}} \int g (t) t^{α - 1} d t . \end{cases}

Remark 3. All of these solutions have been checked by mathematical software, and they are noted for the first time to our knowledge.

We simulated some of these solutions in Figure (1), (2), (3), (4), (5), (6), and (7).

Figure 1.

The modulus of $u_{1.1}$ at $α = β = 1, A_{1} = 1, k_{1} = 1, λ = \frac{8}{3}, m = \frac{\sqrt{3}}{2}, g (t) = 1$ and $t = 0.1$ .

Figure 2.

The modulus of $u_{1.1}$ at $α = \frac{1}{3}, β = \frac{1}{2}, g (t) = t, A_{1} = k_{1} = 1, λ = \frac{8}{3}, m = \frac{\sqrt{3}}{2}$ and $x = 1$ .

Figure 3.

The modulus of $u_{1.1}$ at $α = β = 1, A_{1} = 1, k_{1} = 1, λ = \frac{8}{3}, m = 1, g (t) = 1$ and $t = 0.1$ .

Figure 4.

The real part of $u_{1.3}$ at $α = β = 1, A_{1} = 1, k_{1} = 1, λ = - 8, m = 0, g (t) = 1$ and $t = 0.1$ .

Figure 5.

The modulus of $u_{3.1}$ at $β = 1, k_{2} = 1, B_{0} = 1, λ = - 8, m = \frac{\sqrt{3}}{2}, g (t) = 1$ and $t = 0.1$ .

Figure 6.

The imaginary part of $u_{3.1}$ at $β = 1, k_{2} = 1, B_{0} = 1, λ = - 8, m = \frac{\sqrt{3}}{2}, g (t) = \cos t$ and $t = 0.1$ .

Figure 7.

The modulus of $v_{3.4}$ at $α = β = 1, k_{2} = 1, B_{0} = 1, λ = - 2, m = 0.5$ and $t = 0.1$ .

Results and discussion

After utilizing the GAM, we get many types of similar solutions of equation (1), and some structure of these solutions are simulated in Figure (1), (2), (3), (4), (5), (6), and (7). The visualization can help us to better understand the dynamic behavior and propagation property of these soliton pulse and periodic pulse solutions, and the doubly periodic pulse solution of equation (1) is shown in Figure 1, which can be degenerated to a bell-soliton solution when the modulus $m = 1$ is exhibited in Figure 3. We find that the Jacobi-like elliptic function solution $u_{1.3}$ is degenerated to an unbounded trigonometric-like function solutions when the modulus $m \to 0$ in Figure 4. Solution $u_{3.1}$ in Figures 5 and 6 shows that the evolution of wave signal obeys a stable and periodic state when the frequency $k_{2} = 1$ , while the amplitude of solution $v_{3.4}$ possesses periodic W-shaped feature are shown in Figure 7. Some changes of the modulus for $u_{1.1}$ and $v_{3.4}$ are simulated in Figure 8 at different selection of the order $α, β$ and the kerr nonlinear coefficient $g (t)$ . We select $g (t) = t, A_{1} = k_{1} = 1, λ = \frac{8}{3}, m = \frac{\sqrt{3}}{2}, t = 1$ for $u_{1.1}$ in Figure (8a) and $= β = 1, k_{2} = 1,$ $B_{0} = 1, λ = - 2, m = 0.5$ , $t = 0.1, g_{1} (t) = 1, g_{2} (t) = t$ for $v_{3.4}$ in Figure (8b), which show that the wave shape varies dramatically for different $α, β$ , but keeps stability for the different $g (t)$ .

Figure 8.

The change of modulus for $u_{1.1}$ and $v_{3.4}$ .

Conclusion

In conclusion, many types of new exact soliton pulse and periodic pulse solutions for the CFCNLS have been found after utilizing the GAM. Some propagation behavior of these solutions are discussed and simulated, and the graphs of which show that these doubly periodic wave solutions, solitary wave solutions, and single periodic solutions are propagated through different pattern which were controlled by the parameters of the two components. The efficient and significant method can be used by many other nonlinear models such as vmKdV equation, Ginzburg-Landau equation, and Burgers-BBM equation.

Footnotes

Author contribution

Baojian Hong: Completed all of the study, carried out the results and drafted the paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the practical innovation training program projects for the university students of Jiangsu Province (Grant No. 202211276054Y), Natural science research projects of Institutions in Jiangsu Province (Grant No. 18KJB110013) and Nanjing Institute of Technology (Grant No. ZKJ201513, YZKC2019086). The author wish to express his sincere appreciation to the editors and the anonymous referees for their valuable comments and suggestions.

Data availability

The author affirmed availability of data and material in deriving the solutions mentioned in this manuscript.

ORCID iD

Baojian Hong

References

Tavazoei

Haeri

Jafari

, et al. Some applications of fractional calculus in suppression of chaotic oscillations. IEEE Transactions on Industrial Electronics 2008; 55(11): 4094‐4101.

Hilfer

. Applications of Fractional Calculus in Physics. Orlando: Academic Press, 1999.

. Seeing with a single scale is always unbelieving: from magic to two-scale fractal. Thermal Science 2021; 25(2B): 1217‐1219.

Oqielat

Ajou

Zhour

, et al. Series solutions for nonlinear time-fractional Schrödinger equations: Comparisons between conformable and Caputo derivatives. Alexandria Engineering Journal 2020; 59: 2101‐2114.

Yang

. Solitary waves of the variant Boussinesq-Burgers equation in a fractal dimensional space. Fractals 2022; 30(3): 2250056.

Qian

. A. fractal approach to the diffusion process of red ink in a saline water. Thermal Science 2022; 26(3B): 2447‐2451.

Masjedi

Weaver

. Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method. Applied Mathematical Modelling 2022; 103: 516‐542.

Gepreel

. The homotopy perturbation method applied to the nonlinear fractional KolmogorovPetrovskii- Piskunov equations. Applied Mathematics Letters 2011; 24: 1428‐1434.

Song

Wang

. A new improved Adomian decomposition method and its application to fractional differential equations. Applied Mathematical Modelling 2013; 37(3): 1590‐1598.

10.

. A modifified Li-He’s variational principle for plasma. International Journal of Numerical Methods for Heat Fluid Flow 2021; 31(5): 1369‐1372.

11.

Hong

. Bäcklund transformation and n-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients. International Journal of Nonlinear Science 2006; 1(2): 3‐10.

12.

Matveev

Salle

. Darboux Transformations and Solitons. Berlin, Heidelberg: Springer Verlag, 1991.

13.

. Application of Hirotas bilinear formalism to the Toeplitz lattice-some special , Ma WX. soliton-like solutions. Physics Letters A 2002; 293: 161‐165.

14.

Bashar

Islam

SMR

. Exact solutions to the (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation by using modifified simple equation and improve F-expansion methods. Physics Open 2020; 5: 100027.

15.

Hong

Tian

. New explicit exact solutions for the generalized coupled Hirota Satsuma KdV system. Computers and Mathematics with Application 2007; 53: 1181‐1190.

16.

Hong

. New Jacobi elliptic functions solutions for the variable-coeffificient mKdV equation. Applied mathematics and computation 2009; 215(8): 2908‐2913.

17.

Mohanty

Kravchenko

Dev

. Exact traveling wave solutions of the Schamel Burgers equation by using generalized-improved and generalized (G'/G) expansion methods. Results in Physics 2022; 33: 105124.

18.

. Exp-function Method for Fractional Differential Equations. International Journal ofNonlinear Sciences and Numericalsimulation 2013; 14(6): 363‐366.

19.

Zhang

. Generalized solitary solution and compacton-like solution of the Jaulent-Miodek equations using the Exp-function method. Physics Letter A 2008; 372: 1044‐1047.

20.

Jaradat

Alomari

Abudayah

, et al. An approximate analytical solution of the nonlinear Schrödinger equation with harmonic oscillator using homotopy perturbation method and Laplace-Adomian decomposition method. Advances in Mathematical Physics 2018; 2018: 1‐11.

21.

Haq

. Partial-approximate controllability of semi-linear systems involving two Riemann-Liouville fractional derivatives. Chaos, Solitons and Fractals 2022; 157: 111923.

22.

Caputo

. Linear models of dissipation whose Q is almost frequency independent: part II. Geophysical Journal International 1967; 13: 529‐539.

23.

Guner

Atik

Kayyrzhanovich

. New exact solution for space-time fractional differential equations via (G'/G)-expansion method. Optik 2017; 130: 696‐701.

24.

. A tutorial review on fractal spacetime and fractional calculus. International Journal ofTheoretical Physics 2014; 53: 3698‐3718.

25.

Atangana

Baleanu

Alsaedi

. Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal. Open Physics 2016; 14: 145‐149.

26.

Yao

Manzoor

Zafar

, et al. Exact soliton solutions to the Cahn-Allen equation and Predator-Prey model with truncated M-fractional derivative. Results in Physics 2022; 37: 105455.

27.

Tajadodi

. A Numerical approach of fractional advection-diffusion equation with Atangana. Baleanu derivative. Chaos, Solitons and Fractals 2020; 130: 109527.

28.

Khalil

Al Horani

Yousef

Sababheh

. A new definition of fractional derivative. Journal of Computational and Applied Mathematics 2014; 264: 65‐70.

29.

Seadawy

Ali

Raddadi

. Exact and explicit traveling wave solution to the time fractional phi-four and (2+1) dimensional CBS equations using the modified extended tanh function method in mathematical physics. Partial Differential Equations in Applied Mathematics 2021; 4: 100039.

30.

. A new fractal derivation. Thermal Science 2011; 15(suppl.1): 145‐147.

31.

. Fractal calculus and its geometrical explanation. Results in Physics 2018; 10: 272‐276.

32.

El-Shiekh

Gaballah

. Solitary wave solutions for the variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system using modified sine-Gordon equation method. Journal of Ocean Engineering and Science 2020; 5: 180‐185.

33.

Zhang

Yang

, et al. Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers. Nonlinear Dynamics 2018; 94: 1351‐1360.

34.

Liu

Zhao

Duan

, et al. Asymmetric W-shaped and M-shaped soliton pulse generated from a weak modulation in an exponential dispersion decreasing fiber. Chinese Physics B 2017; 26: 120503.

35.

. Fractional complex transformation for fractional differential equation. Mathematical and Computational Applications 2010; 15: 970‐973.

36.

Elagan

. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics Letters A 2012; 376: 257‐259.

37.

Darvishi

Najafifi

Wazwaz

. Comformable space-time fractional nonlinear (1+1)- dimensional Schrödinger-type models and their traveling wave solutions. Chaos Solitons and Fractals 2021; 150: 111187.

38.

Hong

. New exact Jacobi elliptic function solutions for the coupled Schrödinger- Boussinesq equations. Journal of Applied Mathematics 2013; 2013: 170835

39.

Baleanu

Ug˘urlu

Kilic

, et al. Improved -expansion method for the time- fractional biological population model and Cahn-Hilliard equation. Journal of Computational and Nonlinear Dynamics 2015; 10(5): 051016.

40.

Mohanty

Kravchenko

Dev

. Exact traveling wave solutions of the Schamel Burgers equation by using generalized-improved and generalized (G'/G)-expansion methods. Results in Physics 2022; 33: 105124.

41.

Samir

Badra

Seadawy

, et al. Exact wave solutions of the fourth order non-linear partial differential equation of optical fiber pulses by using different methods. Optik-Internationa Journal for Light and Electron Optics 2021; 230: 166313.

42.

Yokus

Durur

Ahmad

, et al. Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G'/G, 1/G)-expansion and (1/G')-expansion techniques, Results in Physics 2020; 19: 103409

43.

Shallal

Ali

Raslan

, et al. Exact solutions of the conformable fractional EW and MEW equations by a new generalized expansion method. Journal of Ocean Engineering andScience 2020; 5(3): 223‐229.

44.

Sirisubtawee

Koonprasert

Sungnul

, et al. Exact traveling wave solutions of the space-time fractional complex Ginzburg-Landau equation and the space-time fractional Phi-4 equation using reliable methods. Advances in Difference Equations 2019; 2019: 219.

45.

Owyed

Abdou

Abdel-Aty

, et al. Optical solitons solutions for perturbed time fractional nonlinear Schrödinger equation via two strategic algorithms. AIMS Mathematics 2020; 5(3): 2057‐2070.

46.

Ali

Ahmad

Baili

, et al. Exact analytical wave solutions for space-time variable-order fractional modified equal width equation. Results in Physics 2022; 33: 105216.

47.

Tian

Liu

. Direct algebraic method for solving fractional Fokas equation. Thermal Science 2021; 25: 111‐120.

48.

Tang

. Exact solutions to conformable time-fractional Klein-Gordon equation with high-order nonlinearities. Results in Physics 2020; 18: 16.

49.

Tajadodi

Khan

Irshad

, et al. Exact solutions of conformable fractional differential equations. Results in Physics 2021; 22: 103916.

50.

Javeed

Saif

Waheed

, et al. Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers. Results in Physics 2018; 9: 1275‐1281.

51.

Ebaid

Khaled

. New types of exact solutions for nonlinear Schrödinger equation with cubic nonlinearity. Journal of Computational and Applied Mathematics 2011; 235: 1984‐1992.

52.

Ebaid

Aly

. Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions. Wave Motion 2012; 49: 296‐308.