Home Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods
Article Open Access

Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods

  • Wen-Hui Zhu , M. Raheel EMAIL logo and Jian-Guo Liu EMAIL logo
Published/Copyright: September 8, 2022
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 20 Issue 1

Abstract

This current research is about some new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger (NLS) system with novel truncated M-fractional derivative. The obtained results may be used in the description of the model in fruitful way. The novel derivative operator is applied to study the aforementioned model. The achieved results are in the form of dark, bright, and combo optical solitons. The achieved solutions are also verified by using the MATHEMATICA software. The obtained solutions are explained with different plots. Modified integration methods, Exp a function, extended ( G ′ ∕ G ) -expansion, and extended sinh-Gordon equation expansion method are applied to achieve the results. These exact solitons suggest that these methods are effective, straight forward, and reliable compared to other methods.

Keywords: integrable generalized NLS system; Exp a function method; extended (G′/G)-expansion method; extended sinh-Gordon equation expansion method; optical solitons

1 Introduction

Fractional calculus [1,2,3, 4,5,6, 7,8,9, 10,11,12] has become very popular due to its many applications in different areas of sciences. Many models have been made in the area of physical sciences and engineering that are representing the different phenomenon. For example, mostly naturally occurring phenomena are modeled in the form of nonlinear Schrödinger equations [13,14,15]. To determine the exact solutions of the models, a lot of schemes have been developed. Instantly, the modified extended tanh expansion scheme [16] has been applied to discuss the Biswas–Arshed model. Some different wave solutions of the perturbed Gerdjikov-Ivanov equation are gained with the help of the semi-inverse variational method [17]. Various solitons of the new coupled evolution equation were explained [18]. Distinct solitons are investigated by applying the sine-Gordon equation method [19]. Two types of soliton solutions have been obtained by using Exp ( − ϕ ( η ) ) -expansion and generalized Kudryashov methods in ref. [20]. New kinds of general solutions have been achieved in ref. [21]. The Sardar subequation method is used to gain the optical and some other wave solutions in ref. [22]. Three new types of wave solutions have been gained with the help of the modified exp ( − Ω ) -expansion method in ref. [23]. Different types of optical soliton solutions have been collected by using the collective variable method in ref. [24]. Similarly, other methods have been applied; generalized exponential rational function method [25,26,27], Liu’s extended trial function method [28], generalized unified method [29], sine-Gordon expansion method [30], enhanced modified simple equation method [31], unified method [32], extended tanh function method [33], Lie symmetry method [34], symbolic computational method, Hirota’s simple method and long wave method [35], Jacobi elliptic function expansion method [36], Elzaki transform decomposition method [37], m + 1 G -expansion method and adomian decomposition method [38], extended modified auxiliary equation mapping method [39], simplest equation method and Kudryashov’s new function method [40], modified simple equation method [41], modified Kudryashov simple equation method [42], first integral method [43], Bäcklund transformation method [44], extended Jacobi elliptic function expansion method [45], improved ( G ∕ G ) -expansion method, improved ( G ′ ∕ G ) -expansion method [46], and many more [47,48,49, 50,51].

Our concerning model is time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system. Different types of exact solitons have been found by various techniques as follows: optical soliton solutions have been calculated by using the extended modified auxiliary equation mapping method in ref. [39], some travelling wave solutions of the integrable generalized NLS system have been obtained in ref. [52], and various optical wave solutions have been achieved of this system with the help of Kudryashov method and it is modified form given in ref. [53].

In addition to these methods, there are three other methods: Exp a function method, extended ( G ′ ∕ G ) -expansion method, and extended sinh-Gordon equation expansion method (ShGEEM). These methods have been used to explain the many different models: the Tzitzéica like equations are investigated for their exact solitons by using the Exp a function method [54]. New kind of optical wave solutions of two nonlinear Schrödinger equations were searched by utilizing two analytical methods [55]. Similarly, this is used to investigate the roots of the other many nonlinear Schrödinger equations [56,57]. By using the extended ( G ′ ∕ G ) -expansion method, different optical solitons of the Biswas–Milovic equation are generated in ref. [58]; bell-shaped, kink-shaped, and periodic type solitons of the Pochhammer–Chree equations are derived with the help of this method [59]; and discrete and periodic type solitons of the Ablowitz–Ladik lattice system are found [60]. Similarly, the extended ShGEEM has been applied to determine the various wave solutions of different models in refs [61,62, 63,64].

The main task of this study is to research some new exact soliton solutions of the truncated M-fractional integrable generalized (2+1)-dimensional NLS system based on the Exp a function method, the extended ( G ′ ∕ G ) -expansion method, and extended ShGEEM.

This paper is organized as follows: Section 2 describes the truncated M-fractional derivative and its characteristics. Section 3 presents the demonstration and mathematically treatment of model. Section 4 gives the mathematical analysis of the time-fractional integrable generalized NLS system. Section 5 obtains a large number of exact wave solutions. Section 6 makes a conclusion.

2 Truncated M-fractional derivative

Definition. Let h ( t ) : [ 0 , ∞ ) → ℜ , then the truncated M-fractional derivative of h of order α is shown [65]:

(1) D M , t α , γ h ( t ) = lim τ → 0 h ( t E γ ( τ t 1 − α ) ) − h ( t ) τ , 0 < α < 1 , γ > 0 ,

where E γ ( ⋅ ) is a truncated Mittag-Leffler function of one parameter that is defined as ref. [66]:

(2) E γ ( z ) = ∑ j = 0 i z j Γ ( γ j + 1 ) , γ > 0 and z ∈ C .

Characteristics: Let 0 < α ≤ 1 , γ > 0 , a , b ∈ ℜ , and g , f , α -differentiable at a point t > 0 , then by ref. [65]:

(3) ( i ) D M , t α , γ ( a g ( t ) + b f ( t ) ) = a D M , t α , g ( t ) + b D M , t α , γ f ( t ) ,

(4) ( ii ) D M , t α , γ ( g ( t ) . f ( t ) ) = g ( t ) D M , t α , γ f ( t ) + f ( t ) D M , t α , γ g ( t ) ,

(5) ( iii ) D M , t α , γ g ( t ) f ( t ) = f ( t ) D M , t α , γ g ( t ) − g ( t ) D M , t α , γ f ( t ) ( f ( t ) ) 2 ,

(6) ( iv ) D M , t α , γ ( C ) = 0 , where h ( t ) = C is a constant .

(7) ( v ) D M , t α , γ g ( t ) = t 1 − α Γ ( γ + 1 ) d g ( t ) d t .

3 Description of strategies

3.1 Summary of Exp a function method

In this section, we demonstrate this method.

Suppose a nonlinear partial differential equation (NLPDE):

(8) G ( g , g 2 g t , g x , g t t , g x x , g x t , … ) = 0 .

This NLPDE shown in Eq. (8) changed in to nonlinear ordinary differential equation (ODE):

(9) Λ ( G , G ′ , G ″ , … ) = 0 ,

with the use of following wave transformations:

(10) g ( x , y , t ) = G ( τ ) , τ = a x + b y + r t ,

Let’s consider a root of Eq. (9) is given in refs [54,56,67,68]:

(11) G ( τ ) = α 0 + α 1 d τ + … + α m d m τ β 0 + β 1 d τ + … + β m d m τ , d ≠ 0 , 1 ,

where α i and β i ( 0 ≤ i ≤ m ) are unknowns that are found later. Natural number m is calculated with the help of the homogeneous balance scheme into Eq. (9). By inserting Eq. (11) into nonlinear Eq. (9), we obtain

(12) ℘ ( d τ ) = ℓ 0 + ℓ 1 d τ + … + ℓ t d t τ = 0 .

Considering ℓ i ( 0 ≤ i ≤ t ) in Eq. (12) to be equal to 0, a system of algebraic equations is achieved as follows:

(13) ℓ i = 0 , where i = 0 , … , t

with the aforementioned achieved results, we gain the nontrivial solitons of Eq. (8).

3.2 Explanation of the extended ( G ′ ∕ G ) -expansion scheme

This portion is about the key steps of the extended ( G ′ ∕ G ) -expansion scheme [46].

Step 1: Let’s assume the below NLPDE:

(14) G ( q , D M , t α , γ q , q 2 q x , q y , q y y , q x x , q x y , … ) = 0 ,

where q is a wave profile and depend on x and y and t . Let the following travelling wave transformations:

Step 2:

(15) q ( x , y , t ) = Q ( τ ) , τ = x − ν y + Γ ( γ + 1 ) α ( κ t α ) .

By substituting Eq. (15) into Eq. (14), we obtain the nonlinear ODE:

(16) Λ ( Q ( τ ) , Q 2 ( τ ) Q ′ ( τ ) , Q ″ ( τ ) , … ) = 0 ,

Step 3: Suppose the solutions of the Eq. (16) is of the structure:

(17) Q ( τ ) = ∑ i = − m m α i G ′ ( τ ) G ( τ ) i ,

In Eq. (17), α 0 and α i , ( i = ± 1 , ± 2 , ± 3 , … , ± m ) are undetermined that are found later. Note that α i ≠ 0 . By applying the homogenous balance principle into Eq. (16), we find the value of m . The function G = G ( τ ) satisfy the Riccati differential equation given as follows:

(18) d G G ″ − a G 2 − b G G ′ − c ( G ′ ) 2 = 0 ,

with a , b , c , and d are constants.

Step 4: Let Eq. (17) have solutions in the form given as follows:

Case 1: If b ≠ 0 and b 2 + 4 a d − 4 a c > 0 , then

(19) G ′ ( τ ) G ( τ ) = b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d .

Case 2: If b ≠ 0 and b 2 + 4 a d − 4 a c < 0 , then

(20) G ′ ( τ ) G ( τ ) = b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d .

Case 3: If b ≠ 0 and b 2 + 4 a d − 4 a c = 0 , then

(21) G ′ ( τ ) G ( τ ) = b 2 ( d − c ) + d D ( d − c ) ( C − D τ ) .

Case 4: If b = 0 and a d − a c > 0 , then

(22) G ′ ( τ ) G ( τ ) = a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d .

Case 5: If b = 0 and a d − a c < 0 , then

(23) G ′ ( τ ) G ( τ ) = a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d ,

where a , b , c , d , C 1 , and C 2 are the constants.

Step 5: Eq. (17) with Eq. (18) is inserted into Eq. (16) and summed up the coefficients of the same power of G ′ ( τ ) G ( τ ) . By taking each coefficient equal to 0, we gain the system of algebraic equations involving ν , κ , α i , ( i = 0 , ± 1 , ± 2 , … , ± m ) and other parameters.

Step 6:

By manipulating the aforementioned achieved system with the help of MATHEMATICA tool.

Step 7:

By inserting the aforementioned achieved results into Eq. (17), we obtain solitons of the nonlinear Eq. (14).

3.3 Demonstration of the extended ShGEEM

In this section, we brief the basic steps of the extended ShGEEM:

Step 1:

Consider the NLPDE given as follows:

(24) G ( q , D M , t α , γ q , q 2 q x , q y , q y y , q x x , q x y , … ) = 0 ,

where q is a wave profile and depends on x and y and t .

Let the following be travelling wave transformations:

(25) q ( x , y , t ) = Q ( τ ) , τ = x − ν y + Γ ( γ + 1 ) α ( κ t α ) .

By substituting Eq. (25) into Eq. (24), we obtain the following nonlinear ODE given as follows:

(26) Λ ( Q ( τ ) , Q 2 ( τ ) Q ′ ( τ ) , Q ″ ( τ ) , … ) = 0 .

Set 2:

Consider the solution of Eq. (26) follows:

(27) Q ( p ) = α 0 + ∑ i = 1 m ( β i sinh ( p ) + α i cosh ( p ) ) i ,

where α 0 , α i , β i ( i = 1 , 2 , 3 , … , m ) are unknown parameters that are found later and p is a new function of τ that satisfy the below equations:

(28) d p d τ = sinh ( w ) .

By using the homogeneous balance scheme into Eq. (26), we achieve the value of m . Eq. (28) is obtained from the following sinh-Gordon equation:

(29) q x t = κ sinh ( v ) .

Indisputable to ref. [63], we obtain the results from Eq. (28) follows:

(30) sinh p ( τ ) = ± csch ( τ ) or cosh p ( τ ) = ± coth ( τ )

and

(31) sinh p ( τ ) = ± ι sech ( τ ) or cosh p ( τ ) = ± tanh ( τ ) ,

where ι = − 1 ,

Step 3:

Inserting Eq. (27) along natural number m with Eq. (28) into Eq. (26) to achieve the algebraic expressions in p ′ k ( τ ) sinh l p ( τ ) cosh m p ( τ ) ( k = 0 , 1 ; l = 0 , 1 ; m = 0 , 1 , 2 , … ). Now take the each coefficient of p ′ k ( τ ) sinh l p ( τ ) cosh m p ( τ ) equal to zero, to gain the set of algebraic equations containing ν , κ , α 0 , α i and β i ( i = 1 , 2 , 3 , … , m ) .

Step 4:

By solving the achieved set of algebraic equations with the use of MATHEMATICA tool, we can obtain the values of parameters, ν , κ , α 0 , α i , and β i .

Step 5:

By using the gained results and Eqs. (30) and (31), we may obtain solutions of Eq. (24) follows:

(32) Q ( τ ) = α 0 + ∑ i = 1 m ( ± ι β i sech ( τ ) ± α i tanh ( τ ) ) i

and

(33) Q ( τ ) = α 0 + ∑ i = 1 m ( ± β i csch ( τ ) ± α i coth ( τ ) ) i .

4 Description and mathematical analysis of the model

Consider the following time-fractional integrable generalized NLS system given in ref. [52]:

(34) ι D M , t α , γ g + b 1 g x y + b 2 g h = 0 , b 3 h x + b 4 ( ∣ g ∣ 2 ) y = 0 ,

where g = g ( x , y , t ) is the complex-valued wave function and h = h ( x , y , t ) is the real-valued wave function. In Eq. (34), b i ( i = 1 , 2 , 3 , 4 ) are the parameters.

Let’s assume the following travelling wave transformation:

(35) g ( x , y , t ) = G ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α , h ( x , y , t ) = H ( τ ) , where τ = κ 1 x + λ 1 y + θ 1 Γ ( γ + 1 ) α t α ,

where κ i and λ i , i = 1 , 2 represent the speed of soliton and wave number, respectively, while θ 1 and θ 2 show the frequency of the soliton.

By substituting Eq. (35) into Eq. (34), we obtain the imaginary and real parts given as follows:

(36) ( θ 1 + b 1 ( κ 1 λ 2 + κ 2 λ 1 ) ) G ′ = 0 ,

(37) − θ 2 G − b 1 κ 2 λ 2 G + b 1 κ 1 λ 1 G ″ + b 2 G H = 0 ,

and

(38) b 3 κ 1 H ′ + b 4 λ 1 ( G 2 ) ′ = 0 .

By integrating Eq. (38) once and taking constant of integration equal to zero, we obtain:

(39) H ( τ ) = − λ 1 b 4 κ 1 b 3 G ( τ ) 2 ,

and from Eq. (36), we obtain

(40) θ 1 = − b 1 ( κ 1 λ 2 + κ 2 λ 1 ) .

By substituting Eq. (39) into Eq. (37), we obtain

(41) ( θ 2 + b 1 κ 2 λ 2 ) G − b 1 κ 1 λ 1 G ″ + λ 1 b 2 b 4 κ 1 b 3 G 3 = 0 .

By using the homogenous balance scheme into Eq. (41), we obtain m = 1 .

5 Exact wave solutions

5.1 Solution to the Exp a function method

For m = 1 , Eq. (11) reduces into:

(42) G ( τ ) = α 0 + α 1 d τ β 0 + β 1 d τ .

By using Eq. (42) into Eq. (41) and solving the system of equations, we achieve the following solution sets:

Set 1:

(43) α 0 = b 1 b 3 β 0 κ 1 log ( d ) 2 b 2 b 4 , α 1 = − b 1 b 3 β 1 κ 1 log ( d ) 2 b 2 b 4 , β 0 = β 0 , β 1 = β 1 , θ 2 = 1 2 ( b 1 κ 1 λ 1 ( − log 2 ( d ) ) − 2 b 1 κ 2 λ 2 ) .

From Eqs. (43), (42), and (35), we obtain

(44) g ( x , y , t ) = b 1 b 3 κ 1 log ( d ) 2 b 2 b 4 β 0 − β 1 d τ β 0 + β 1 d τ × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(45) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 b 1 b 3 κ 1 log ( d ) 2 b 2 b 4 β 0 − β 1 d τ β 0 + β 1 d τ 2 .

As an explanation, the dynamic properties of Eq. (45) are demonstrated in Figure 1.

Figure 1 Solution (45) with b 1 = b 2 = b 3 = b 4 = β 1 = κ 1 = λ 1 = γ = 1 {b}_{1}={b}_{2}={b}_{3}={b}_{4}={\beta }_{1}={\kappa }_{1}={\lambda }_{1}=\gamma =1 , θ 1 = β 0 = 2 {\theta }_{1}={\beta }_{0}=2 , d = e d=e , y = 0 y=0 , (a) α = 0.7 \alpha =0.7 , (b) α = 0.8 \alpha =0.8 , and (c) α = 0.9 \alpha =0.9 .
Figure 1

Solution (45) with b 1 = b 2 = b 3 = b 4 = β 1 = κ 1 = λ 1 = γ = 1 , θ 1 = β 0 = 2 , d = e , y = 0 , (a) α = 0.7 , (b) α = 0.8 , and (c) α = 0.9 .

Set 2:

(46) α 0 = − b 1 b 3 β 0 κ 1 log ( d ) 2 b 2 b 4 , α 1 = b 1 b 3 β 1 κ 1 log ( d ) 2 b 2 b 4 , β 0 = β 0 , β 1 = β 1 , θ 2 = 1 2 ( b 1 κ 1 λ 1 ( − log 2 ( d ) ) − 2 b 1 κ 2 λ 2 ) ,

From Eqs. (35), (42), and (46), we obtain

(47) g ( x , y , t ) = − b 1 b 3 κ 1 log ( d ) 2 b 2 b 4 β 0 − β 1 d τ β 0 + β 1 d τ × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(48) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − b 1 b 3 κ 1 log ( d ) 2 b 2 b 4 β 0 − β 1 d τ β 0 + β 1 d τ 2 .

5.2 Solutions to the extended ( G ′ ∕ G ) -expansion method

For m = 1 , Eq. (17) becomes:

(49) G ( τ ) = α − 1 G ′ ( τ ) G ( τ ) − 1 + α 0 + α 1 G ′ ( τ ) G ( τ ) ,

where α − 1 , α 0 , and α 1 are undetermined parameters.

Substituting Eq. (49) with Eq. (18) into Eq. (41) and solving the system for α − 1 , α 0 , α 1 , and other parameters, we obtain the following solution sets:

Set 1:

(50) α − 1 = − 2 a b 1 b 3 κ 1 b 2 b 4 d , α 0 = − b b 1 b 3 κ 1 2 b 2 b 4 d , α 1 = 0 , θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) + b 2 ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Case 1:

From Eqs. (19), (35), (49), and (50), we obtain

(51) g ( x , y , t ) = − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d − 1 × exp ( ι ( κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ) ) ,

(52) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d − 1 2 ,

where θ 2 is given in Eq. (50). As an explanation, the dynamic properties of Eq. (52) are demonstrated in Figure 2.

Figure 2 Solution (52) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = γ = a = b = c = 1 {b}_{1}={b}_{2}={b}_{3}={b}_{4}={\theta }_{1}={\kappa }_{1}={\lambda }_{1}=\gamma =a=b=c=1 , y = 0 y=0 , C 1 = d = 2 {C}_{1}=d=2 , (a) α = 0.6 \alpha =0.6 , (b) α = 0.7 \alpha =0.7 , and (c) α = 0.8 \alpha =0.8 .
Figure 2

Solution (52) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = γ = a = b = c = 1 , y = 0 , C 1 = d = 2 , (a) α = 0.6 , (b) α = 0.7 , and (c) α = 0.8 .

Case 2:

From Eqs. (20), (35), (49), and (50), and we obtain

(53) g ( x , y , t ) = − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d − 1 × exp ( ι ( κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ) ) ,

(54) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a ( b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d ) − 1 2 ,

where θ 2 is given in Eq. (50). As an explanation, the dynamic properties of Eq. (56) are demonstrated in Figure 3.

Figure 3 Solution (56) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = γ = a = b = C 2 = 1 {b}_{1}={b}_{2}={b}_{3}={b}_{4}={\theta }_{1}={\kappa }_{1}={\lambda }_{1}=\gamma =a=b={C}_{2}=1 , y = 0 y=0 , c = 3 c=3 , C 1 = d = 2 {C}_{1}=d=2 , (a) α = 0.1 \alpha =0.1 , (b) α = 0.2 \alpha =0.2 , and (c) α = 0.3 \alpha =0.3 .
Figure 3

Solution (56) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = γ = a = b = C 2 = 1 , y = 0 , c = 3 , C 1 = d = 2 , (a) α = 0.1 , (b) α = 0.2 , and (c) α = 0.3 .

Case 3:

From Eqs. (22), (35), (49), and (50), we obtain

(55) g ( x , y , t ) = − a 2 b 1 b 3 κ 1 b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d − 1 × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(56) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − 2 a b 1 b 3 κ 1 b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d − 1 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Case 4:

From Eqs. (23), (35), (49), and (50), we obtain

(57) g ( x , y , t ) = − a 2 b 1 b 3 κ 1 b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d − 1 × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(58) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − 2 a b 1 b 3 κ 1 b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d − 1 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Set 2:

(59) α − 1 = 0 , α 0 = − b b 1 b 3 κ 1 2 b 2 b 4 d , α 1 = − 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d , θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) + b 2 ) + 2 d 2 κ 2 λ 2 ) 2 d 2 ,

Case 1:

From Eqs. (19), (35), (49), and (59), and we obtain

(60) g ( x , y , t ) = − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(61) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d 2 ,

where θ 2 is given in Eq. (59).

Case 2:

From Eqs. (20), (35), (49), and (59), we obtain

(62) g ( x , y , t ) = − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(63) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d 2 ,

where θ 2 is given in Eq. (59).

Case 3:

From Eqs. (22), (35), (49), and (59), we obtain

(64) g ( x , y , t ) = − 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(65) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Case 4:

From Eqs. (23), (35), (49), and (59), we obtain

(66) g ( x , y , t ) = − 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(67) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 − 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Set 3:

(68) α − 1 = 2 a b 1 b 3 κ 1 b 2 b 4 d , α 0 = b b 1 b 3 κ 1 2 b 2 b 4 d , α 1 = 0 , θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) + b 2 ) + 2 d 2 κ 2 λ 2 ) 2 d 2 ,

Case 1:

From Eqs. (19), (35), (49), and (68), we obtain

(69) g ( x , y , t ) = b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d − 1 × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(70) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d − 1 2 ,

where θ 2 is given in Eq. (68).

Case 2:

From Eqs. (20), (35), (49), and (68), we obtain

(71) g ( x , y , t ) = b 1 b 3 κ 1 2 b 2 b 4 d ( b + 2 a b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d − 1 ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(72) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 b 1 b 3 κ 1 2 b 2 b 4 d b + 2 a b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d − 1 2 ,

where θ 2 is given in Eq. (68).

Case 3:

From Eqs. (22), (35), (49), and (68), we obtain

(73) g ( x , y , t ) = a 2 b 1 b 3 κ 1 b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d − 1 × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(74) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 2 a b 1 b 3 κ 1 b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d − 1 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Case 4:

From Eqs. (23), (35), (49), and (68), we obtain

(75) g ( x , y , t ) = a 2 b 1 b 3 κ 1 b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d − 1 × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(76) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 2 a b 1 b 3 κ 1 b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d − 1 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Set 4:

(77) α − 1 = 0 , α 0 = b b 1 b 3 κ 1 2 b 2 b 4 d , α 1 = 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d , θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) + b 2 ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

Case 1: From Eqs. (19), (35), (49), and (77), we obtain

(78) g ( x , y , t ) = b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(79) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + − 4 a c + 4 a d + b 2 2 ( d − c ) C 1 sinh τ − 4 a c + 4 a d + b 2 2 d + C 2 cosh τ − 4 a c + 4 a d + b 2 2 d C 1 cosh τ − 4 a c + 4 a d + b 2 2 d + C 2 sinh τ − 4 a c + 4 a d + b 2 2 d 2 ,

where θ 2 is given in Eq. (77).

Case 2:

From Eqs. (20), (35), (49), and (77), we obtain

(80) g ( x , y , t ) = b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(81) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 b 1 b 3 κ 1 2 b 2 b 4 d b + 2 ( c − d ) b 2 ( d − c ) + 4 a c − 4 a d − b 2 2 ( d − c ) C 2 cos τ 4 a c − 4 a d − b 2 2 d − C 1 sin τ 4 a c − 4 a d − b 2 2 d C 1 cos τ 4 a c − 4 a d − b 2 2 d + C 2 sin τ 4 a c − 4 a d − b 2 2 d 2 ,

where θ 2 is given in Eq. (77).

Case 3:

From Eqs. (22), (35), (49), and (77), we obtain

(82) g ( x , y , t ) = 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(83) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a d − a c ( d − c ) C 1 sinh τ a d − a c d + C 2 cosh τ a d − a c d C 1 cosh τ a d − a c d + C 2 sinh τ a d − a c d 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 ,

Case 4:

From Eqs. (23), (35), (49), and (77), we obtain

(84) g ( x , y , t ) = 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a c − a d d − c ( C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(85) h ( x , y , t ) = − λ 1 b 4 κ 1 b 3 2 b 1 b 3 κ 1 ( c − d ) b 2 b 4 d a c − a d d − c C 2 cos τ a c − a d d − C 1 sin τ a c − a d d C 1 cos τ a c − a d d + C 2 sin τ a c − a d d 2 ,

where θ 2 = − b 1 ( κ 1 λ 1 ( 4 a ( d − c ) ) + 2 d 2 κ 2 λ 2 ) 2 d 2 .

5.3 Exact soliton solutions to the extended ShGEEM

For m = 1 , Eqs. (27), (32), and (33) become:

(86) G ( τ ) = α 0 ± ι β 1 sech ( τ ) ± α 1 tanh ( τ ) ,

(87) G ( τ ) = α 0 ± β 1 csch ( τ ) ± α 1 coth ( τ ) ,

(88) G ( τ ) = α 0 + β 1 sinh ( p ) + α 1 cosh ( p ) ,

where α 0 , α 1 , and β 1 are the unknowns. Inserting Eq. (88) into Eq. (41), we achieve the algebraic equations containing α 0 , α 1 , β 1 , and other parameters. Now with the help of software, we obtain the following sets:

Set 1:

(89) α 0 = 0 , α 1 = − 2 b 1 b 3 κ 1 b 2 b 4 , β 1 = 0 , θ 2 = − b 1 ( 2 κ 1 λ 1 + κ 2 λ 2 ) .

From Eqs. (35), (86), (88), and (89), we obtain

(90) g 1 ( x , y , t ) = ∓ 2 b 1 b 3 κ 1 b 2 b 4 tanh ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(91) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ 2 b 1 b 3 κ 1 b 2 b 4 tanh ( τ ) 2 .

As an explanation, the dynamic properties of Eq. (91) are demonstrated in Figure 4.

Figure 4 Solution (91) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = κ 2 = λ 2 = γ = 1 {b}_{1}={b}_{2}={b}_{3}={b}_{4}={\theta }_{1}={\kappa }_{1}={\lambda }_{1}={\kappa }_{2}={\lambda }_{2}=\gamma =1 , y = 0 y=0 , (a) α = 0.4 \alpha =0.4 , (b) α = 0.5 \alpha =0.5 , and (c) α = 0.6 \alpha =0.6 .
Figure 4

Solution (91) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = κ 2 = λ 2 = γ = 1 , y = 0 , (a) α = 0.4 , (b) α = 0.5 , and (c) α = 0.6 .

From Eqs. (35), (87), (88), and (89), we obtain

(92) g 2 ( x , y , t ) = ∓ 2 b 1 b 3 κ 1 b 2 b 4 coth ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(93) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ 2 b 1 b 3 κ 1 b 2 b 4 coth ( τ ) 2 .

Set 2:

(94) α 0 = 0 , α 1 = 2 b 1 b 3 κ 1 b 2 b 4 , β 1 = 0 , θ 2 = − b 1 ( 2 κ 1 λ 1 + κ 2 λ 2 ) .

From Eqs. (35), (86), (88), and (94), we obtain

(95) g 1 ( x , y , t ) = ± 2 b 1 b 3 κ 1 b 2 b 4 tanh ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(96) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ± 2 b 1 b 3 κ 1 b 2 b 4 tanh ( τ ) 2 .

From Eqs. (35), (87), (88), and (94), we obtain

(97) g 2 ( x , y , t ) = ± 2 b 1 b 3 κ 1 b 2 b 4 coth ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(98) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ± 2 b 1 b 3 κ 1 b 2 b 4 coth ( τ ) 2 .

Set 3:

(99) α 0 = 0 , α 1 = − b 1 b 3 κ 1 2 b 2 b 4 , β 1 = − b 1 b 3 κ 1 2 b 2 b 4 , θ 2 = − 1 2 b 1 ( κ 1 λ 1 + 2 κ 2 λ 2 ) .

From Eqs. (35), (86), (88), and (99) we obtain

(100) g 1 ( x , y , t ) = ∓ b 1 b 3 κ 1 2 b 2 b 4 ( ι sech ( τ ) + tanh ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(101) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ b 1 b 3 κ 1 2 b 2 b 4 ( ι sech ( τ ) + tanh ( τ ) ) 2 .

As an explanation, the dynamic properties of Eq. (101) are demonstrated in Figure 5. From Eqs. (35), (87), (88), and (99), we obtain

(102) g 2 ( x , y , t ) = ∓ b 1 b 3 κ 1 2 b 2 b 4 ( coth ( τ ) + csch ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(103) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ b 1 b 3 κ 1 2 b 2 b 4 ( coth ( τ ) + csch ( τ ) ) 2 .

Figure 5 Solution (101) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = κ 2 = λ 2 = γ = 1 {b}_{1}={b}_{2}={b}_{3}={b}_{4}={\theta }_{1}={\kappa }_{1}={\lambda }_{1}={\kappa }_{2}={\lambda }_{2}=\gamma =1 , y = 0 y=0 , ι = 1 \iota =1 , (a) α = 0.6 \alpha =0.6 , (b) α = 0.7 \alpha =0.7 , (c) α = 0.8 \alpha =0.8 .
Figure 5

Solution (101) with b 1 = b 2 = b 3 = b 4 = θ 1 = κ 1 = λ 1 = κ 2 = λ 2 = γ = 1 , y = 0 , ι = 1 , (a) α = 0.6 , (b) α = 0.7 , (c) α = 0.8 .

Set 4:

(104) α 0 = 0 , α 1 = b 1 b 3 κ 1 2 b 2 b 4 , β 1 = − b 1 b 3 κ 1 2 b 2 b 4 , θ 2 = − 1 2 b 1 ( κ 1 λ 1 + 2 κ 2 λ 2 ) .

From Eqs. (104), (88), (35), and (86), we obtain

(105) g 1 ( x , y , t ) = ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ∓ ι sech ( τ ) ± tanh ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(106) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ∓ ι sech ( τ ) ± tanh ( τ ) ) 2 .

From Eqs. (104), (88), (35), and (87), we obtain

(107) g 2 ( x , y , t ) = ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ∓ csch ( τ ) ± coth ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(108) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ∓ csch ( τ ) ± coth ( τ ) ) 2 .

Set 5:

(109) α 0 = 0 , α 1 = − b 1 b 3 κ 1 2 b 2 b 4 , β 1 = b 1 b 3 κ 1 2 b 2 b 4 , θ 2 = − 1 2 b 1 ( κ 1 λ 1 + 2 κ 2 λ 2 ) .

From Eqs. (109), (88), (35), and (86), we obtain

(110) g 1 ( x , y , t ) = ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ± ι sech ( τ ) ∓ tanh ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(111) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ± ι sech ( τ ) ∓ tanh ( τ ) ) 2 .

From Eqs. (109), (88), (35) and (87), we obtain

(112) g 2 ( x , y , t ) = ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ± csch ( τ ) ∓ coth ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(113) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ( b 1 b 3 κ 1 ) 2 b 2 b 4 ( ± csch ( τ ) ∓ coth ( τ ) ) 2 ,

Set 6:

(114) α 0 = 0 , α 1 = b 1 b 3 κ 1 2 b 2 b 4 , β 1 = b 1 b 3 κ 1 2 b 2 b 4 , θ 2 = − 1 2 b 1 ( κ 1 λ 1 + 2 κ 2 λ 2 ) .

From Eqs. (114), (88), (35), and (86), we obtain

(115) g 1 ( x , y , t ) = ± b 1 b 3 κ 1 2 b 2 b 4 ( ι sech ( τ ) + tanh ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(116) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ± b 1 b 3 κ 1 2 b 2 b 4 ( ι sech ( τ ) + tanh ( τ ) ) 2 .

From Eqs. (114), (88), (35), and (87), we obtain

(117) g 2 ( x , y , t ) = ± b 1 b 3 κ 1 2 b 2 b 4 ( coth ( τ ) + csch ( τ ) ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(118) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ± b 1 b 3 κ 1 2 b 2 b 4 ( coth ( τ ) + csch ( τ ) ) 2 .

Set 7:

(119) α 0 = 0 , α 1 = 0 , β 1 = − 2 b 1 b 3 κ 1 b 2 b 4 , θ 2 = b 1 ( κ 1 λ 1 − κ 2 λ 2 ) .

From Eqs. (119), (88), (35), and (86), we obtain

(120) g 1 ( x , y , t ) = ∓ ι 2 b 1 b 3 κ 1 b 2 b 4 sech ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(121) h 1 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ ι 2 b 1 b 3 κ 1 b 2 b 4 sech ( τ ) 2 .

From Eqs. (119), (88), (35), and (87), we obtain

(122) g 2 ( x , y , t ) = ∓ 2 b 1 b 3 κ 1 b 2 b 4 csch ( τ ) × exp ι κ 2 x + λ 2 y + θ 2 Γ ( γ + 1 ) α t α ,

(123) h 2 ( x , y , t ) = − λ 1 b 4 κ 1 b 3 ∓ 2 b 1 b 3 κ 1 b 2 b 4 csch ( τ ) 2 ,

where τ = κ 1 x + λ 1 y − b 1 ( κ 1 λ 2 + κ 2 λ 1 ) Γ ( γ + 1 ) α t α .

6 Conclusion

In this paper, we have gained the new optical solitons of the integrable generalized (2+1)-dimensional NLS system along with novel definition of derivative by applying the modified integration methods, Exp a function, extended ( G ′ ∕ G ) -expansion, and extended ShGEEM. The achieved solitons are including dark, bright and combo optical solitons. These solutions are also verified by using the MATHEMATICA software [69,70]. The obtained solutions are newer than the existing solutions of this model in the literature. The important thing of this paper is that truncated M-fractional derivative is first time used for this model. These optical solitons suggest that these methods are effective, straight forward, and reliable compared to other methods.

  1. Funding information: Project supported by National Natural Science Foundation of China (Grant No: 12161048), Doctoral Research Foundation of Jiangxi University of Chinese Medicine (Grant No: 2021WBZR007) and Jiangxi University of Chinese Medicine Science and Technology Innovation Team Development Program (Grant No: CXTD22015).

  2. Author contributions: All authors contributed to writing-original draft, methodology, software, formal analysis, funding acquisition. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] Amiri P, Rezapour S, Shahzad N. Fixed points of generalized alpha-psi-contractions. Racsam Rev R Acad A. 2014;108:519–26. 10.1007/s13398-013-0123-9Search in Google Scholar

[2] Baleanu D, Hedayati V, Rezapour S, Al QMM. On two fractional differential inclusions. SpringerPlus. 2016;5:882. 10.1186/s40064-016-2564-zSearch in Google Scholar PubMed PubMed Central

[3] Baleanu D, Mohammadi H, Rezapour S. On a nonlinear fractional differential equation on partially ordered metric spaces. Adv Differ Equ. 2013;2013:83. 10.1186/1687-1847-2013-83Search in Google Scholar

[4] Alam M, Zada A, Popa IL, Kheiryan A, Rezapour S, Kaabar MKA. A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers-Ulam stability. Bound Value Probl. 2021;2021:73. 10.1186/s13661-021-01549-ySearch in Google Scholar

[5] Haghi RH, Rezapour S. Fixed points of multifunctions on regular cone metric spaces. Expo Math. 2010;28(1):71–7. 10.1016/j.exmath.2009.04.001Search in Google Scholar

[6] Mohammadi H, Kumar S, Rezapour S, Etemad S. A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Soliton Fract. 2021;144:110668. 10.1016/j.chaos.2021.110668Search in Google Scholar

[7] Matar MM, Abbas MI, Alzabut J, Kaabar MKA, Etemad S, Rezapour S. Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivatives. Adv Differ Equ. 2021;2021:68. 10.1186/s13662-021-03228-9Search in Google Scholar

[8] Rezapour S, Samei ME. On the existence of solutions for a multi-singular pointwise defined fractional q-integro-differential equation. Bound Value Probl. 2020;2020:38. 10.1186/s13661-020-01342-3Search in Google Scholar

[9] Baleanu D, Rezapour S, Saberpour Z. On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation. Bound Value Probl. 2019;2019:79. 10.1186/s13661-019-1194-0Search in Google Scholar

[10] Baleanu D, Etemad S, Pourrazi S, Rezapour S. On the new fractional hybrid boundary value problems with three-point integral hybrid conditions. Adv Differ Equ. 2019;2019:473. 10.1186/s13662-019-2407-7Search in Google Scholar

[11] Aydogan MS, Baleanu D, Mousalou A, Rezapour S. On high order fractional integro-differential equations including the Caputo-Fabrizio derivative. Bound Value Probl. 2018;2018:90. 10.1186/s13661-018-1008-9Search in Google Scholar

[12] Baleanu D, Mohammadi H, Rezapour S. Analysis of the model of HIV-1 infection of CD4. CD4+ T-cell with a new approach of fractional derivative. Adv Differ Equ. 2020;2020:71. 10.1186/s13662-020-02544-wSearch in Google Scholar

[13] Wazwaz AM. A study on linear and nonlinear Schrödinger equations by the variational iteration method. Chaos Soliton Fract. 2008;37(4):1136–42. 10.1016/j.chaos.2006.10.009Search in Google Scholar

[14] Manafian J. Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan(ϕ∕2) -expansion method. Optik. 2016;127(10):4222–45. 10.1016/j.ijleo.2016.01.078Search in Google Scholar

[15] Zhou Q, Biswas A. Optical soliton in parity-time-symmetric mixed linear and nonlinear lattice with non-Kerr law nonlinearity. Superlatice Microst. 2017;109:588–98. 10.1016/j.spmi.2017.05.049Search in Google Scholar

[16] Asim Z, Raheel M, Ahmed B. Expolring the dark and singular soliton solutions of Biswas-Arshed model with full nonlinear form. Optik. 2020;204:164133. 10.1016/j.ijleo.2019.164133Search in Google Scholar

[17] Biswas A, Alqahtani RT. Chirp-free bright optical solitons for perturbed Gerdjikov-Ivanov equation by semi-inverse variational principle. Optik. 2017;147:72–6. 10.1016/j.ijleo.2017.08.019Search in Google Scholar

[18] Bekir A. Application of the extended tanh method for coupled nonlinear evolution equation. Commun Nonlinear Sci. 2008;13:1742–51. 10.1016/j.cnsns.2007.05.001Search in Google Scholar

[19] Yasar E, Yılırım Y, Yasar E. New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method. Results Phys. 2018;9:1666–72. 10.1016/j.rinp.2018.04.058Search in Google Scholar

[20] Haci MB, Osman MS, Hamood R, Muhammad R, Muhammad T, Shagufta A. On pulse propagation of soliton wave solutions related to the perturbed Chen-Lee-Liu equation in an optical fiber. Opt Quant Electron. 2021;53:556. 10.1007/s11082-021-03190-6Search in Google Scholar

[21] Kudryashov NA. General solution of the traveling wave reduction for the perturbed Chen-Lee-Liu equation. Optik. 2019;186:339–49. 10.1016/j.ijleo.2019.04.127Search in Google Scholar

[22] Esen H, Ozdemir N, Secer A, Bayram M. On solitary wave solutions for the perturbed Chen-Lee-Liu equation via an analytical approach. Optik. 2021;245:167641. 10.1016/j.ijleo.2021.167641Search in Google Scholar

[23] Yépez-Martínez H, Rezazadeh H, Inc M, Ali AM. New solutions to the fractional perturbed Chen-Lee-Liu equation with a new local fractional derivative, Wave Random Complex. 2021. 10.1080/17455030.2021.1930280. Search in Google Scholar

[24] Alrashed R, Alshaery AA, Alkhateeb S. Optical solitons via the collective variable method for the classical and perturbed Chen-Lee-Liu equations. Open Phys. 2021;19(1):559–67. 10.1515/phys-2021-0065Search in Google Scholar

[25] Ghanbari B, Osman MS, Baleanu D. Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative. Mod Phys Lett A. 2019;34(20):1950155. 10.1142/S0217732319501554Search in Google Scholar

[26] Osman MS, Ghanbari B. New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach. Optik. 2018;175:328–33. 10.1016/j.ijleo.2018.08.007Search in Google Scholar

[27] Neirameh A, Eslami M. New solitary wave solutions for fractional Jaulent-Miodek hierarchy equation. Mod Phys Lett B. 2022;36(7):2150612. 10.1142/S0217984921506120Search in Google Scholar

[28] Rezazadeh H, Osman MS, Eslami M, Ekici M, Sonmezoglu A, Asma M, et al. Mitigating Internet bottleneck with fractional temporal evolution of optical solitons having quadratic-cubic nonlinearity. Optik. 2018;164:84–92. 10.1016/j.ijleo.2018.03.006Search in Google Scholar

[29] Osman MS. One-soliton shaping and inelastic collision between double solitons in the fifth-order variable-coefficient Sawada-Kotera equation. Nonlinear Dyn. 2019;96(2):1491–6. 10.1007/s11071-019-04866-1Search in Google Scholar

[30] Ali KK, Wazwaz AM, Osman MS. Optical soliton solutions to the generalized nonautonomous nonlinear Schrödinger equations in optical fibers via the sine-Gordon expansion method. Optik. 2020;208:164132. 10.1016/j.ijleo.2019.164132Search in Google Scholar

[31] Aktar MS, Akbar MA, Osman MS. Spatio-temporal dynamic solitary wave solutions and diffusion effects to the nonlinear diffusive predator-prey system and the diffusion-reaction equations. Chaos Soliton Fract. 2022;160:112212. 10.1016/j.chaos.2022.112212Search in Google Scholar

[32] Inan B, Osman MS, Ak T, Baleanu D. Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations. Math Method Appl Sci. 2020;43(5):2588–600. 10.1002/mma.6067Search in Google Scholar

[33] Adel M, Baleanu D, Sadiya U, Arefin MA, Uddin MH, Elamin MA, et al. Inelastic soliton wave solutions with different geometrical structures to fractional order nonlinear evolution equations. Results Phys. 2022;38:105661. 10.1016/j.rinp.2022.105661Search in Google Scholar

[34] Kumar S, Dhiman SK, Baleanu D, Osman MS, Wazwaz AM. Lie symmetries, closed-form solutions, and various dynamical profiles of solitons for the variable coefficient (2+1)-dimensional KP equations. Symmetry. 2022;14(3):597. 10.3390/sym14030597Search in Google Scholar

[35] Ali KK, Yilmazer R, Osman MS. Dynamic behavior of the (3+1)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff equation. Opt Quant Electron. 2022;54(3):160. 10.1007/s11082-022-03528-8Search in Google Scholar

[36] Tarla S, Ali KK, Yilmazer R, Osman MS. The dynamic behaviors of the Radhakrishnan-Kundu-Lakshmanan equation by Jacobi elliptic function expansion technique. Opt Quant Electron. 2022;54(5):292. 10.1007/s11082-022-03710-ySearch in Google Scholar

[37] Rashid S, Kubra KT, Sultana S, Agarwal P, Osman MS. An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method. J Comput Appl Math. 2022;413:114378. 10.1016/j.cam.2022.114378Search in Google Scholar

[38] Ismael HF, Okumuş I, Aktürk T, Bulut H, Osman MS. Analyzing study for the 3D potential Yu-Toda-Sasa-Fukuyama equation in the two-layer liquid medium. J Ocean Eng Sci. 2022. 10.1016/j.joes.2022.03.017. Search in Google Scholar

[39] Seadawy AR, Cheemaa N, Biswas A. Optical dromions and domain walls in (2+1)-dimensional coupled system. Optik. 2021;227:165669. 10.1016/j.ijleo.2020.165669Search in Google Scholar

[40] El-Ganaini S, Al-Amr MO. New abundant solitary wave structures for a variety of some nonlinear models of surface wave propagation with their geometric interpretations. Math Method Appl Sci. 2022. 10.1002/mma.8232. Search in Google Scholar

[41] Al-Amr MO, Rezazadeh H, Ali KK, Korkmazki A. N1-soliton solution for Schrödinger equation with competing weakly nonlocal and parabolic law nonlinearities. Commun Theor Phys. 2020;72(6):065503. 10.1088/1572-9494/ab8a12Search in Google Scholar

[42] Rasheed NM, Al-Amr MO, Az-Zo’bi EA, Tashtoush MA, Akinyemi L. Stable optical solitons for the Higher-order Non-Kerr NLSE via the modified simple equation method. Mathematics. 2021;9(16):1986. 10.3390/math9161986Search in Google Scholar

[43] Eslami M, Rezazadeh H. The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo. 2016;53(3):475–85. 10.1007/s10092-015-0158-8Search in Google Scholar

[44] Rezazadeh H, Kumar D, Neirameh A, Eslami M, Mirzazadeh M. Applications of three methods for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model with Kerr law nonlinearity. Pramana. 2020;94(1):39. 10.1007/s12043-019-1881-5Search in Google Scholar

[45] Zafar A, Raheel M, Mirzazadeh M, Eslami M. Different soliton solutions to the modified equal-width wave equation with Beta-time fractional derivative via two different methods. Rev Mex Fis. 2022;68(1):010701. 10.31349/RevMexFis.68.010701Search in Google Scholar

[46] Sahoo S, Ray SS, Abdou MA. New exact solutions for time-fractional Kaup-Kupershmidt equation using improved (G′∕G)-expansion and extended (G′∕G) -expansion methods. Alex Eng J. 2020;59(5):3105–10. 10.1016/j.aej.2020.06.043Search in Google Scholar

[47] Ugurlu Y. Exp-function method for the some nonlinear partial differentialequations. Math Aeter. 2013;3(1):57–70. Search in Google Scholar

[48] Kumar S, Chauhan RP, Osman MS, Mohiuddine SA. A study on fractional HIV-AIDs transmission model with awareness effect. Math Method Appl Sci. 2021. 10.1002/mma.7838. Search in Google Scholar

[49] Seadawy AR, Ali A, Albarakati WA. Analytical wave solutions of the (2+1)-dimensional first integro-differential Kadomtsev-Petviashivili hierarchy equation by using modified mathematical methods. Results Phys. 2019;15:102775. 10.1016/j.rinp.2019.102775Search in Google Scholar

[50] Seadawy AR, Kumar D, Chakrabarty AK. Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method. Eur Phys J Plus. 2018;133(5):182. 10.1140/epjp/i2018-12027-9Search in Google Scholar

[51] Alharbi AR, Almatrafi MB, Seadawy AR. Construction of the numerical and analytical wave solutions of the Joseph-Egri dynamical equation for the long waves in nonlinear dispersive systems. Int J Mod Phys B. 2020;34(30):2050289. 10.1142/S0217979220502896Search in Google Scholar

[52] Akinyemi L, Şcenol M, Rezazadeh H, Ahmad H, Wang H. Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system. Results Phys. 2021;25:104177. 10.1016/j.rinp.2021.104177Search in Google Scholar

[53] Hosseini K, Sadri K, Mirzazadeh M, Salahshour S. An integrable (2. 1)-dimensional nonlinear Schrödinger system and its optical soliton solutions. Optik. 2021;229:166247. 10.1016/j.ijleo.2020.166247Search in Google Scholar

[54] Hosseini K, Ayati Z, Ansari R. New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method. J Mod Optic. 2018;65(7):847–51. 10.1080/09500340.2017.1407002Search in Google Scholar

[55] Asim Z, Ahmed B, Muhammad R, Hadi R. Investigation for optical soliton solutions of two nonlinear Schrödinger equations via two concrete finite series methods. Int J Appl Comput Math. 2020;6(3):1–13. 10.1007/s40819-020-00818-1Search in Google Scholar

[56] Ali AT, Hassan ER. General Expa-function method for nonlinear evolution equations. Appl Math Comput. 2010;217(2):451–9. 10.1016/j.amc.2010.06.025Search in Google Scholar

[57] Hosseini K, Zabihi A, Samadani F, Ansari R. New explicit exact solutions of the unstable nonlinear Schrödinger’s equation using the expa and hyperbolic function methods. Opt Quant Electron. 2018;50(2):82. 10.1007/s11082-018-1350-2Search in Google Scholar

[58] Zhou Q, Ekici M, Sonmezoglu A, Mirzazadeh M. Optical solitons with Biswas–Milovic equation by extended (G′∕G)-expansion method. Optik. 2016;127(16):6277–90. 10.1016/j.ijleo.2016.04.119Search in Google Scholar

[59] Zuo JM. Application of the extended (G′∕G) -expansion method to solve the Pochhammer–Chree equations. Appl Math Comput. 2010;217(1):376–83. 10.1016/j.amc.2010.05.072Search in Google Scholar

[60] Aslan I. The Ablowitz–Ladik lattic system by means of the extended (G′∕G)-expansion method. Appl Math Comput. 2010;216(9):2778–82. 10.1016/j.amc.2010.03.124Search in Google Scholar

[61] Gulnur Y, Carlo C, Haci MB, Gao W. On the complex simulations with dark-bright to the Hirota-Maccari system. J Comput Nonlinear Dynam. 2021;16(6):061005. 10.1115/1.4050677Search in Google Scholar

[62] Asim Z, Ahmed B, Raheel M, Waseem R. Optical soliton solutions to Biswas-Arshed model with truncated M-fractional derivative. Optik. 2020;222:165355. 10.1016/j.ijleo.2020.165355Search in Google Scholar

[63] Yang XL, Tang JS. Travelling wave solutions for Konopelchenko-Dubrovsky equation using an extended sinh-Gordon equation expansion method. Commun Theor Phys. 2018;50:10471051. Search in Google Scholar

[64] Baskonus HM, Sulaiman TA, Bulut H. Bright, dark optical and other solitons to the generalized higher-order NLSE in optical fibers. Opt Quant Electron. 2018;50:253. 10.1007/s11082-018-1522-0Search in Google Scholar

[65] Tukur AS, Gulnur Y, Hasan B. M-fractional solitons and periodic wave solutions to the Hirota-Maccari system. Mod Phys Lett B. 2019;33(5):1950052. 10.1142/S0217984919500520Search in Google Scholar

[66] Sousa JVC, Capelas OE. A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties. Int J Anal Appl. 2018;16(1):83–96. Search in Google Scholar

[67] Zayed EME, Al-Nowehy AG. Generalized kudryashov method and general expa function method for solving a high order nonlinear Schrödinger equation. J Space Explor. 2017;6(2):120. Search in Google Scholar

[68] Zafar A. The expa function method and the conformable time-fractional KdV equations. Nonlinear Eng. 2019;8:728–32. 10.1515/nleng-2018-0094Search in Google Scholar

[69] Zhou Q, Houria T, Xu J, Zeng Z, Liu W, Anjan B. Perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos Soliton Fract. 2011;160(13):112198. 10.1016/j.chaos.2022.112198Search in Google Scholar

[70] Zhou Q, Wang T, Anjan B, Liu W. Nonlinear control of logic structure of all-optical logic devices using soliton interactions. Nonlinear Dyn. 2022;107(2):1–8. 10.1007/s11071-021-07027-5Search in Google Scholar
Received: 2022-05-17
Revised: 2022-07-06
Accepted: 2022-07-17
Published Online: 2022-09-08

© 2022 Wen-Hui Zhu et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Test influence of screen thickness on double-N six-light-screen sky screen target
  3. Analysis on the speed properties of the shock wave in light curtain
  4. Abundant accurate analytical and semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation
  5. Measured distribution of cloud chamber tracks from radioactive decay: A new empirical approach to investigating the quantum measurement problem
  6. Nuclear radiation detection based on the convolutional neural network under public surveillance scenarios
  7. Effect of process parameters on density and mechanical behaviour of a selective laser melted 17-4PH stainless steel alloy
  8. Performance evaluation of self-mixing interferometer with the ceramic type piezoelectric accelerometers
  9. Effect of geometry error on the non-Newtonian flow in the ceramic microchannel molded by SLA
  10. Numerical investigation of ozone decomposition by self-excited oscillation cavitation jet
  11. Modeling electrostatic potential in FDSOI MOSFETS: An approach based on homotopy perturbations
  12. Modeling analysis of microenvironment of 3D cell mechanics based on machine vision
  13. Numerical solution for two-dimensional partial differential equations using SM’s method
  14. Multiple velocity composition in the standard synchronization
  15. Electroosmotic flow for Eyring fluid with Navier slip boundary condition under high zeta potential in a parallel microchannel
  16. Soliton solutions of Calogero–Degasperis–Fokas dynamical equation via modified mathematical methods
  17. Performance evaluation of a high-performance offshore cementing wastes accelerating agent
  18. Sapphire irradiation by phosphorus as an approach to improve its optical properties
  19. A physical model for calculating cementing quality based on the XGboost algorithm
  20. Experimental investigation and numerical analysis of stress concentration distribution at the typical slots for stiffeners
  21. An analytical model for solute transport from blood to tissue
  22. Finite-size effects in one-dimensional Bose–Einstein condensation of photons
  23. Drying kinetics of Pleurotus eryngii slices during hot air drying
  24. Computer-aided measurement technology for Cu2ZnSnS4 thin-film solar cell characteristics
  25. QCD phase diagram in a finite volume in the PNJL model
  26. Study on abundant analytical solutions of the new coupled Konno–Oono equation in the magnetic field
  27. Experimental analysis of a laser beam propagating in angular turbulence
  28. Numerical investigation of heat transfer in the nanofluids under the impact of length and radius of carbon nanotubes
  29. Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation
  30. Optical properties and thermal stability of the H+-implanted Dy3+/Tm3+-codoped GeS2–Ga2S3–PbI2 chalcohalide glass waveguide
  31. Nonlinear dynamics for different nonautonomous wave structure solutions
  32. Numerical analysis of bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: New iterative method
  33. Modeling extreme value data with an upside down bathtub-shaped failure rate model
  34. Abundant optical soliton structures to the Fokas system arising in monomode optical fibers
  35. Analysis of the partially ionized kerosene oil-based ternary nanofluid flow over a convectively heated rotating surface
  36. Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts
  37. Magnetofluid unsteady electroosmotic flow of Jeffrey fluid at high zeta potential in parallel microchannels
  38. Effect of plasma-activated water on microbial quality and physicochemical properties of fresh beef
  39. The finite element modeling of the impacting process of hard particles on pump components
  40. Analysis of respiratory mechanics models with different kernels
  41. Extended warranty decision model of failure dependence wind turbine system based on cost-effectiveness analysis
  42. Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation
  43. First-principle calculation of electronic structure and optical properties of (P, Ga, P–Ga) doped graphene
  44. Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula
  45. Optimization method for detection a flying bullet
  46. Angle error control model of laser profilometer contact measurement
  47. Numerical study on flue gas–liquid flow with side-entering mixing
  48. Travelling waves solutions of the KP equation in weakly dispersive media
  49. Characterization of damage morphology of structural SiO2 film induced by nanosecond pulsed laser
  50. A study of generalized hypergeometric Matrix functions via two-parameter Mittag–Leffler matrix function
  51. Study of the length and influencing factors of air plasma ignition time
  52. Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches
  53. The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions
  54. Generalized notion of integral inequalities of variables
  55. The seasonal variation in the polarization (Ex/Ey) of the characteristic wave in ionosphere plasma
  56. Impact of COVID 19 on the demand for an inventory model under preservation technology and advance payment facility
  57. Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach
  58. Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods
  59. Irreversibility analysis in time-dependent Darcy–Forchheimer flow of viscous fluid with diffusion-thermo and thermo-diffusion effects
  60. Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media
  61. NTIM solution of the fractional order parabolic partial differential equations
  62. Jointly Rayleigh lifetime products in the presence of competing risks model
  63. Abundant exact solutions of higher-order dispersion variable coefficient KdV equation
  64. Laser cutting tobacco slice experiment: Effects of cutting power and cutting speed
  65. Performance evaluation of common-aperture visible and long-wave infrared imaging system based on a comprehensive resolution
  66. Diesel engine small-sample transfer learning fault diagnosis algorithm based on STFT time–frequency image and hyperparameter autonomous optimization deep convolutional network improved by PSO–GWO–BPNN surrogate model
  67. Analyses of electrokinetic energy conversion for periodic electromagnetohydrodynamic (EMHD) nanofluid through the rectangular microchannel under the Hall effects
  68. Propagation properties of cosh-Airy beams in an inhomogeneous medium with Gaussian PT-symmetric potentials
  69. Dynamics investigation on a Kadomtsev–Petviashvili equation with variable coefficients
  70. Study on fine characterization and reconstruction modeling of porous media based on spatially-resolved nuclear magnetic resonance technology
  71. Optimal block replacement policy for two-dimensional products considering imperfect maintenance with improved Salp swarm algorithm
  72. A hybrid forecasting model based on the group method of data handling and wavelet decomposition for monthly rivers streamflow data sets
  73. Hybrid pencil beam model based on photon characteristic line algorithm for lung radiotherapy in small fields
  74. Surface waves on a coated incompressible elastic half-space
  75. Radiation dose measurement on bone scintigraphy and planning clinical management
  76. Lie symmetry analysis for generalized short pulse equation
  77. Spectroscopic characteristics and dissociation of nitrogen trifluoride under external electric fields: Theoretical study
  78. Cross electromagnetic nanofluid flow examination with infinite shear rate viscosity and melting heat through Skan-Falkner wedge
  79. Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives
  80. Weak nonlinear analysis of nanofluid convection with g-jitter using the Ginzburg--Landau model
  81. Strip waveguides in Yb3+-doped silicate glass formed by combination of He+ ion implantation and precise ultrashort pulse laser ablation
  82. Best selected forecasting models for COVID-19 pandemic
  83. Research on attenuation motion test at oblique incidence based on double-N six-light-screen system
  84. Review Articles
  85. Progress in epitaxial growth of stanene
  86. Review and validation of photovoltaic solar simulation tools/software based on case study
  87. Brief Report
  88. The Debye–Scherrer technique – rapid detection for applications
  89. Rapid Communication
  90. Radial oscillations of an electron in a Coulomb attracting field
  91. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part II
  92. The exact solutions of the stochastic fractional-space Allen–Cahn equation
  93. Propagation of some new traveling wave patterns of the double dispersive equation
  94. A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations
  95. An orthotropic thermo-viscoelastic infinite medium with a cylindrical cavity of temperature dependent properties via MGT thermoelasticity
  96. Modeling of hepatitis B epidemic model with fractional operator
  97. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part III
  98. Investigation of effective thermal conductivity of SiC foam ceramics with various pore densities
  99. Nonlocal magneto-thermoelastic infinite half-space due to a periodically varying heat flow under Caputo–Fabrizio fractional derivative heat equation
  100. The flow and heat transfer characteristics of DPF porous media with different structures based on LBM
  101. Homotopy analysis method with application to thin-film flow of couple stress fluid through a vertical cylinder
  102. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part II
  103. Asymptotic analysis of hepatitis B epidemic model using Caputo Fabrizio fractional operator
  104. Influence of chemical reaction on MHD Newtonian fluid flow on vertical plate in porous medium in conjunction with thermal radiation
  105. Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation
  106. Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling
  107. On theoretical analysis of nonlinear fractional order partial Benney equations under nonsingular kernel
  108. The solutions of nonlinear fractional partial differential equations by using a novel technique
  109. Modelling and graphing the Wi-Fi wave field using the shape function
  110. Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative
  111. Impact of the convergent geometric profile on boundary layer separation in the supersonic over-expanded nozzle
  112. Variable stepsize construction of a two-step optimized hybrid block method with relative stability
  113. Thermal transport with nanoparticles of fractional Oldroyd-B fluid under the effects of magnetic field, radiations, and viscous dissipation: Entropy generation; via finite difference method
  114. Special Issue on Advanced Energy Materials - Part I
  115. Voltage regulation and power-saving method of asynchronous motor based on fuzzy control theory
  116. The structure design of mobile charging piles
  117. Analysis and modeling of pitaya slices in a heat pump drying system
  118. Design of pulse laser high-precision ranging algorithm under low signal-to-noise ratio
  119. Special Issue on Geological Modeling and Geospatial Data Analysis
  120. Determination of luminescent characteristics of organometallic complex in land and coal mining
  121. InSAR terrain mapping error sources based on satellite interferometry
https://doi.org/10.1515/phys-2022-0191
Keywords for this article
integrable generalized NLS system; Exp a function method; extended (G′/G)-expansion method; extended sinh-Gordon equation expansion method; optical solitons
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Test influence of screen thickness on double-N six-light-screen sky screen target
  3. Analysis on the speed properties of the shock wave in light curtain
  4. Abundant accurate analytical and semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation
  5. Measured distribution of cloud chamber tracks from radioactive decay: A new empirical approach to investigating the quantum measurement problem
  6. Nuclear radiation detection based on the convolutional neural network under public surveillance scenarios
  7. Effect of process parameters on density and mechanical behaviour of a selective laser melted 17-4PH stainless steel alloy
  8. Performance evaluation of self-mixing interferometer with the ceramic type piezoelectric accelerometers
  9. Effect of geometry error on the non-Newtonian flow in the ceramic microchannel molded by SLA
  10. Numerical investigation of ozone decomposition by self-excited oscillation cavitation jet
  11. Modeling electrostatic potential in FDSOI MOSFETS: An approach based on homotopy perturbations
  12. Modeling analysis of microenvironment of 3D cell mechanics based on machine vision
  13. Numerical solution for two-dimensional partial differential equations using SM’s method
  14. Multiple velocity composition in the standard synchronization
  15. Electroosmotic flow for Eyring fluid with Navier slip boundary condition under high zeta potential in a parallel microchannel
  16. Soliton solutions of Calogero–Degasperis–Fokas dynamical equation via modified mathematical methods
  17. Performance evaluation of a high-performance offshore cementing wastes accelerating agent
  18. Sapphire irradiation by phosphorus as an approach to improve its optical properties
  19. A physical model for calculating cementing quality based on the XGboost algorithm
  20. Experimental investigation and numerical analysis of stress concentration distribution at the typical slots for stiffeners
  21. An analytical model for solute transport from blood to tissue
  22. Finite-size effects in one-dimensional Bose–Einstein condensation of photons
  23. Drying kinetics of Pleurotus eryngii slices during hot air drying
  24. Computer-aided measurement technology for Cu2ZnSnS4 thin-film solar cell characteristics
  25. QCD phase diagram in a finite volume in the PNJL model
  26. Study on abundant analytical solutions of the new coupled Konno–Oono equation in the magnetic field
  27. Experimental analysis of a laser beam propagating in angular turbulence
  28. Numerical investigation of heat transfer in the nanofluids under the impact of length and radius of carbon nanotubes
  29. Multiple rogue wave solutions of a generalized (3+1)-dimensional variable-coefficient Kadomtsev--Petviashvili equation
  30. Optical properties and thermal stability of the H+-implanted Dy3+/Tm3+-codoped GeS2–Ga2S3–PbI2 chalcohalide glass waveguide
  31. Nonlinear dynamics for different nonautonomous wave structure solutions
  32. Numerical analysis of bioconvection-MHD flow of Williamson nanofluid with gyrotactic microbes and thermal radiation: New iterative method
  33. Modeling extreme value data with an upside down bathtub-shaped failure rate model
  34. Abundant optical soliton structures to the Fokas system arising in monomode optical fibers
  35. Analysis of the partially ionized kerosene oil-based ternary nanofluid flow over a convectively heated rotating surface
  36. Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts
  37. Magnetofluid unsteady electroosmotic flow of Jeffrey fluid at high zeta potential in parallel microchannels
  38. Effect of plasma-activated water on microbial quality and physicochemical properties of fresh beef
  39. The finite element modeling of the impacting process of hard particles on pump components
  40. Analysis of respiratory mechanics models with different kernels
  41. Extended warranty decision model of failure dependence wind turbine system based on cost-effectiveness analysis
  42. Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation
  43. First-principle calculation of electronic structure and optical properties of (P, Ga, P–Ga) doped graphene
  44. Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula
  45. Optimization method for detection a flying bullet
  46. Angle error control model of laser profilometer contact measurement
  47. Numerical study on flue gas–liquid flow with side-entering mixing
  48. Travelling waves solutions of the KP equation in weakly dispersive media
  49. Characterization of damage morphology of structural SiO2 film induced by nanosecond pulsed laser
  50. A study of generalized hypergeometric Matrix functions via two-parameter Mittag–Leffler matrix function
  51. Study of the length and influencing factors of air plasma ignition time
  52. Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches
  53. The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions
  54. Generalized notion of integral inequalities of variables
  55. The seasonal variation in the polarization (Ex/Ey) of the characteristic wave in ionosphere plasma
  56. Impact of COVID 19 on the demand for an inventory model under preservation technology and advance payment facility
  57. Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach
  58. Exploring the new optical solitons to the time-fractional integrable generalized (2+1)-dimensional nonlinear Schrödinger system via three different methods
  59. Irreversibility analysis in time-dependent Darcy–Forchheimer flow of viscous fluid with diffusion-thermo and thermo-diffusion effects
  60. Double diffusion in a combined cavity occupied by a nanofluid and heterogeneous porous media
  61. NTIM solution of the fractional order parabolic partial differential equations
  62. Jointly Rayleigh lifetime products in the presence of competing risks model
  63. Abundant exact solutions of higher-order dispersion variable coefficient KdV equation
  64. Laser cutting tobacco slice experiment: Effects of cutting power and cutting speed
  65. Performance evaluation of common-aperture visible and long-wave infrared imaging system based on a comprehensive resolution
  66. Diesel engine small-sample transfer learning fault diagnosis algorithm based on STFT time–frequency image and hyperparameter autonomous optimization deep convolutional network improved by PSO–GWO–BPNN surrogate model
  67. Analyses of electrokinetic energy conversion for periodic electromagnetohydrodynamic (EMHD) nanofluid through the rectangular microchannel under the Hall effects
  68. Propagation properties of cosh-Airy beams in an inhomogeneous medium with Gaussian PT-symmetric potentials
  69. Dynamics investigation on a Kadomtsev–Petviashvili equation with variable coefficients
  70. Study on fine characterization and reconstruction modeling of porous media based on spatially-resolved nuclear magnetic resonance technology
  71. Optimal block replacement policy for two-dimensional products considering imperfect maintenance with improved Salp swarm algorithm
  72. A hybrid forecasting model based on the group method of data handling and wavelet decomposition for monthly rivers streamflow data sets
  73. Hybrid pencil beam model based on photon characteristic line algorithm for lung radiotherapy in small fields
  74. Surface waves on a coated incompressible elastic half-space
  75. Radiation dose measurement on bone scintigraphy and planning clinical management
  76. Lie symmetry analysis for generalized short pulse equation
  77. Spectroscopic characteristics and dissociation of nitrogen trifluoride under external electric fields: Theoretical study
  78. Cross electromagnetic nanofluid flow examination with infinite shear rate viscosity and melting heat through Skan-Falkner wedge
  79. Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives
  80. Weak nonlinear analysis of nanofluid convection with g-jitter using the Ginzburg--Landau model
  81. Strip waveguides in Yb3+-doped silicate glass formed by combination of He+ ion implantation and precise ultrashort pulse laser ablation
  82. Best selected forecasting models for COVID-19 pandemic
  83. Research on attenuation motion test at oblique incidence based on double-N six-light-screen system
  84. Review Articles
  85. Progress in epitaxial growth of stanene
  86. Review and validation of photovoltaic solar simulation tools/software based on case study
  87. Brief Report
  88. The Debye–Scherrer technique – rapid detection for applications
  89. Rapid Communication
  90. Radial oscillations of an electron in a Coulomb attracting field
  91. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part II
  92. The exact solutions of the stochastic fractional-space Allen–Cahn equation
  93. Propagation of some new traveling wave patterns of the double dispersive equation
  94. A new modified technique to study the dynamics of fractional hyperbolic-telegraph equations
  95. An orthotropic thermo-viscoelastic infinite medium with a cylindrical cavity of temperature dependent properties via MGT thermoelasticity
  96. Modeling of hepatitis B epidemic model with fractional operator
  97. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part III
  98. Investigation of effective thermal conductivity of SiC foam ceramics with various pore densities
  99. Nonlocal magneto-thermoelastic infinite half-space due to a periodically varying heat flow under Caputo–Fabrizio fractional derivative heat equation
  100. The flow and heat transfer characteristics of DPF porous media with different structures based on LBM
  101. Homotopy analysis method with application to thin-film flow of couple stress fluid through a vertical cylinder
  102. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part II
  103. Asymptotic analysis of hepatitis B epidemic model using Caputo Fabrizio fractional operator
  104. Influence of chemical reaction on MHD Newtonian fluid flow on vertical plate in porous medium in conjunction with thermal radiation
  105. Structure of analytical ion-acoustic solitary wave solutions for the dynamical system of nonlinear wave propagation
  106. Evaluation of ESBL resistance dynamics in Escherichia coli isolates by mathematical modeling
  107. On theoretical analysis of nonlinear fractional order partial Benney equations under nonsingular kernel
  108. The solutions of nonlinear fractional partial differential equations by using a novel technique
  109. Modelling and graphing the Wi-Fi wave field using the shape function
  110. Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative
  111. Impact of the convergent geometric profile on boundary layer separation in the supersonic over-expanded nozzle
  112. Variable stepsize construction of a two-step optimized hybrid block method with relative stability
  113. Thermal transport with nanoparticles of fractional Oldroyd-B fluid under the effects of magnetic field, radiations, and viscous dissipation: Entropy generation; via finite difference method
  114. Special Issue on Advanced Energy Materials - Part I
  115. Voltage regulation and power-saving method of asynchronous motor based on fuzzy control theory
  116. The structure design of mobile charging piles
  117. Analysis and modeling of pitaya slices in a heat pump drying system
  118. Design of pulse laser high-precision ranging algorithm under low signal-to-noise ratio
  119. Special Issue on Geological Modeling and Geospatial Data Analysis
  120. Determination of luminescent characteristics of organometallic complex in land and coal mining
  121. InSAR terrain mapping error sources based on satellite interferometry
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 15.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2022-0191/html
Scroll to top button