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BY 4.0 license Open Access Published by De Gruyter Open Access March 17, 2021

On the optical solutions to nonlinear Schrödinger equation with second-order spatiotemporal dispersion

Hadi Rezazadeh, Waleed Adel, Mostafa Eslami, Kalim U. Tariq, Seyed Mehdi Mirhosseini-Alizamini, Ahmet Bekir and Yu-Ming Chu
From the journal Open Physics

Abstract

In this article, the sine-Gordon expansion method is employed to find some new traveling wave solutions to the nonlinear Schrödinger equation with the coefficients of both group velocity dispersion and second-order spatiotemporal dispersion. The nonlinear model is reduced to an ordinary differential equation by introducing an intelligible wave transformation. A set of new exact solutions are observed corresponding to various parameters. These novel soliton solutions are depicted in figures, revealing the new physical behavior of the acquired solutions. The method proves its ability to provide good new approximate solutions with some applications in science. Moreover, the associated solution of the presented method can be extended to solve more complex models.

1 Introduction

The nonlinear Schrödinger equation (NLSE) is one of the most powerful generic family of models, fascinating great attention of both mathematicians and physicists because of their potential applications in the recent era of the optical theory. A lot of natural complex phenomena can be described by this model of the nonlinear type. A good understanding of the solutions, configurations, interdependence, and supplementary features may contribute to a more study of more complex models in several areas of science and engineering. For example, electromagnetic theory, condensed matter physics, acoustics, cosmology, and plasma physics are some of the areas that benefit from studying this type of equation. With these above-mentioned applications, the need to further study the NLSE is of interest and this was the motivation to investigate more about the behavior of this model. The study with an effective method which may provide accurate results with physical meaning is an ongoing research for such model and similar ones. In the sense of fractional calculus, an extended model of the NLSE can be proposed and studied to take into account the effect of the fractional term. Fractional calculus has a great amount of work for solving models with applications and continues to prove the ability to provide more realistic models. For example, optimal control of diabetes [1], blood ethanol concentration system modeling [2], and dengue fever modeling [3] are some of the real-life applications of models with fractional derivatives. We are interested in the future to simulate the fractional NLSE. For more details regarding other areas of application, one may see refs. [4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26,27, 28,29,30].

Many researchers were interested in nonlinear models due to their complexity. Analytical solutions can elucidate the physical behavior of a natural system more accurately corresponding to a particular process. New, innovative, and accurate techniques are being developed to find a new solution to nonlinear equations, which may contribute in recent areas of science and technology. Recently, many numerical and analytical approaches are being developed such as the auxiliary equation method [31], Cole-Hopf transformation, exp-function method [32], sine-cosine method [33], Darboux transformation [34], Hirota method [35], Lie group analysis [36], modified simple equation method [37], similarity reduced method, tanh method, inverse scattering scheme [38], Bäcklund transform method [39], homogeneous balance scheme [40], sine-cosine method, tanh-coth method, extended FAN sub-equation method [41], auxiliary equation method [42], and many more.

One of these important and effective methods that may provide good solutions with important physical behaviors is the sine-Gordon expansion method. The method has been used numerous times for solving different science and engineering models of physical importance. For example, Baskonus in ref. [43] applied this method for investigating the behavior of a Davey–Stewartson equation with power-law nonlinearity, which has some applications in fluid dynamics. Also, Yel et al. [44] adopted the same method for solving the new coupled Konno–Oono equation acquiring new solitons like solutions. In ref. [45], the method is used to find new dark-bright solitons for the shallow water wave model. Other related models that have been solved using this method including Fokas–Lenells equation [46], nonautonomous NLSEs equations [47], conformable time-fractional equations in RLW-class [48], 2D complex Ginzburg–Landau equation [49], time-fractional Fitzhugh–Nagumo equation [50], and references therein. It is worth mentioning that this study is the first to be dealing with finding the solution to the Schrödinger equation with the coefficients of both group velocity dispersion and second-order spatiotemporal dispersion using this method.

In the present article, we use the sine-Gordon expansion method to derive exact traveling wave solutions for the NLSE with its coefficients of both group velocity and spatiotemporal dispersion. The model can take the following form:

(1) i q x + α q t + β 2 q t 2 + γ 2 q x 2 + q 2 q = 0 ,

where q ( x , t ) , α , β , and γ are defined in refs. [51,52,53].

This article is organized as follows. In Section 2, we describe the sine-Gordon expansion method. The application of the method is presented in Section 3. The conclusions are drawn in Section 4.

2 sine-Gordon expansion method

The main steps of the sine-Gordon expansion method are described below to determine an exact solution for the partial differential equation. The sine-Gordon equation can take the following form [48,54]:

(2) u x x u t t = m 2 sin ( u ) ,

where u = u ( x , t ) and m is a constant. Next, equation (2) can be reduced into a nonlinear ordinary differential equation with the aid of a traveling wave transform u ( x , t ) = U ( ξ ) , ξ = x ν t into the following:

(3) U = m 2 1 ν 2 sin ( U ) ,

where ν is the wave velocity in the aforementioned wave transform. Then, by multiplying both sides of equation (3) with the term U and integrating one, we reach the following:

(4) U 2 2 = m 2 1 ν 2 sin 2 U 2 + C ,

where C is an integration constant. Assuming that C = 0 , U 2 = H ( ξ ) , and m 2 1 ν 2 = a 2 in equation (4), we obtain

(5) H = a sin ( H ) ,

and by replacing the coefficient a = 1 into equation (5), we acquire the following equation:

(6) H = sin ( H ) .

As can be seen, equation (6) can be considered as the known sine-Gordon equation with a simplified form. Now, to solve equation (6), we adapt the separation of variables method and with some simplifications, one can find the following relations:

(7) sin ( H ( ξ ) ) = sech ( ξ ) , cos ( H ( ξ ) ) = tanh ( ξ ) ,

(8) sin ( H ( ξ ) ) = i csch ( ξ ) , cos ( H ( ξ ) ) = coth ( ξ ) .

Now, consider a nonlinear partial differential equation as follows:

(9) P ( u , u x , u t , u x x , u x t , u t t , ) = 0 ,

by using the transformation u ( x , t ) = U ( ξ ) with ξ = x ν t , equation (8) can be converted into the following form:

(10) G ( U , U , U , ) = 0 .

The trial solution to equation (9) is assumed to be of the following form:

(11) U ( H ) = j = 1 N cos j 1 ( ξ ) [ B j sin ( H ) + A j cos ( H ) ] + A 0 .

Based on equations (7) and (8), the solution of equations (11) can be written as follows:

(12) U ( ξ ) = j = 1 N tanh j 1 ( ξ ) [ B j sech ( ξ ) + A j tanh ( ξ ) ] + A 0

and

(13) U ( ξ ) = j = 1 N cos j 1 ( ξ ) [ i B j csch ( ξ ) + A j coth ( ξ ) ] + A 0 ,

where N is an integer value that can be calculated by balancing the terms of the highest derivative with the nonlinear terms. Inserting equation (11) into (10) and some algebra, yields a polynomial equation in sin j ( H ) cos j ( H ) . Then, by setting the coefficients of sin j ( H ) cos j ( H ) to zero will result in a set of over-determined algebraic equations in A j , B j , and ν . Next, the algebraic system is tried to be solved for the coefficients A j , B j , and ν . For the last step, A j , B j values are substituted into equations (12) and (13), which will result in the new solution to equation (9) in the form of a traveling wave.

3 Application of the method

To begin, we take the travelling wave transformation as:

(14) q ( x , t ) = U ( ξ ) e i ϕ , ξ = x ν t , ϕ = κ x + ω t + θ 0 ,

where

(15) q x = ( U i κ U ) e i ϕ , q t = ( ν U + i ω U ) e i ϕ ,

(16) q x x = ( U 2 i κ U κ 2 U ) e i ϕ , q t t = ( ν 2 U 2 i ω ν U ω 2 U ) e i ϕ .

Substituting equation (14) into equation (1), we have

(17) i ( 1 α ν ) U ( α ω κ ) U + ( β ν 2 + γ ) U 2 i ( ω ν β + γ κ ) U ( β ω 2 + γ κ 2 ) U + U 3 = 0 .

Imaginary part:

(18) 1 α ν 2 ( ω ν β + γ κ ) = 0 ν = 1 2 γ κ α + 2 ω β .

Real part:

(19) ( β ν 2 + γ ) U + ( κ α ω β ω 2 + γ κ 2 ) U + U 3 = 0 .

By applying equation (18) in equation (19), we get

(20) β 1 2 γ κ α + 2 ω β 2 + γ U + κ α ω β ω 2 + γ κ 2 U + U 3 = 0 .

Thus, we obtain

(21) β ( 1 2 γ κ ) 2 + γ ( α + 2 ω β ) 2 U + ( α + 2 ω β ) 2 ( κ α ω β ω 2 + γ κ 2 ) U + U 3 = 0 .

With the aid of the homogenous principle, and by balancing the two terms U and U 3 will yield N = 1 .

With N = 1 , equations (11), (12), and (13) take the form

(22) U ( H ) = B 1 sin ( H ) + A 1 cos ( H ) + A 0 ,

(23) U ( ξ ) = B 1 sech ( ξ ) + A 1 tanh ( ξ ) + A 0 ,

and

(24) U ( ξ ) = i B 1 csch ( ξ ) + A 1 coth ( ξ ) + A 0 .

Then, by substituting the form of equation (22) along with its second derivative into (21), a polynomial in powers of a hyperbolic function form will result. By setting the summation of the coefficients of the trigonometric identities with the same power to zero, we find a group of algebraic equations. This set of equations is simplified and the parameter values can be found. For each case, the solution of equation (1) can be found by substituting the values of the parameters into equations (23) and (24) and then, into equation (14).

Case I:

A 0 = 0 , A 1 = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ , B 1 = 0 , κ = 1 2 γ 1 ± 8 γ ω 2 β + γ ω α + 2 γ 2 1 4 β ω + 1 2 α 2 β 2 ω 2 + β ( α ω + 2 γ ) + 1 4 α 2 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ 1 1 2 .

From (14), we deduce the following exact solutions:

q 1 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ tanh ( x ν t ) × exp i 1 2 γ 1 ± 8 γ ω 2 β + γ ω α + 2 γ 2 1 4 β ω + 1 2 α 2 × β 2 ω 2 + β ( α ω + 2 γ ) + 1 4 α 2 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ 1 1 2 x + ω t + θ 0 ,

and

q 2 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ coth ( x ν t ) × exp i 1 2 γ 1 ± 8 γ ω 2 β + γ ω α + 2 γ 2 1 4 β ω + 1 2 α 2 × β 2 ω 2 + β ( α ω + 2 γ ) + 1 4 α 2 4 β 2 ω 2 + 4 α β ω + α 2 + 8 β γ 1 1 2 x + ω t + θ 0 .

Case II:

A 0 = 0 , A 1 = 0 , B 1 = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 4 α β ω α 2 + 4 β γ , B 1 = 0 , κ = 1 2 γ 1 ± 8 γ ω 2 β + γ ω α γ 2 1 4 β ω + 1 2 α 2 β 2 ω 2 + β ( α ω γ ) + 1 4 α 2 4 β 2 ω 2 4 α β ω α 2 + 4 β γ .

From (14), we deduce the following exact solutions:

q 3 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 4 α β ω α 2 + 4 β γ sech ( x ν t ) × exp i 1 2 γ 1 ± 8 γ ω 2 β + γ ω α γ 2 1 4 β ω + 1 2 α 2 × β 2 ω 2 + β ( α ω γ ) + 1 4 α 2 4 β 2 ω 2 4 α β ω α 2 + 4 β γ 1 1 2 x + ω t + θ 0 ,

and

q 4 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 4 α β ω α 2 + 4 β γ csch ( x ν t ) × exp i 1 2 γ 1 ± 8 γ ω 2 β + γ ω α γ 2 1 4 β ω + 1 2 α 2 × β 2 ω 2 + β ( α ω γ ) + 1 4 α 2 4 β 2 ω 2 4 α β ω α 2 + 4 β γ 1 1 2 x + ω t + θ 0 .

Case III:

A 0 = 0 , A 1 = ± 1 2 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ , B 1 = ± 1 2 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ , κ = 1 2 γ 1 ± ( 4 γ ω 2 β + 4 γ ω α + 2 γ 2 1 ) ( 2 β ω + α ) 2 ( 4 β 2 ω 2 + 2 β ( 2 α ω + γ ) + α 2 ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ .

From (14), we deduce the following exact solutions:

q 5 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ sech ( x ν t ) + i tanh ( x ν t ) × exp i 1 2 γ 1 ± ( 4 γ ω 2 β + 4 γ ω α + 2 γ 2 1 ) ( 2 β ω + α ) 2 × ( 4 β 2 ω 2 + 2 β ( 2 α ω + γ ) + α 2 ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ 1 1 2 x + ω t + θ 0 ,

and

q 6 ( x , t ) = ± 2 ( α 2 γ + β ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ csch ( x ν t ) + coth ( x ν t ) × exp i 1 2 γ 1 ± ( 4 γ ω 2 β + 4 γ ω α + 2 γ 2 1 ) ( 2 β ω + α ) 2 × ( 4 β 2 ω 2 + 2 β ( 2 α ω + γ ) + α 2 ) 4 β 2 ω 2 + 4 α β ω + α 2 + 2 β γ 1 1 2 x + ω t + θ 0 .

4 Graphical representation of solutions

In this section, the solitons solution for the main equation for different cases and different values of the parameters is being investigated and represented throughout the following figures with the help of Mathematica 11.0.

5 Conclusions

In this study, the sine-Gordon expansion method was employed to integrate the NLSE with the coefficients of group velocity dispersion and second-order spatiotemporal dispersion. Some new traveling wave solutions are found while changing the values of the parameters. The new form of solutions possesses some novel traveling wave behaviors. A graphical representation of these solutions is provided in Figures 1, 2, 3, 4, 5, and 6. The proposed method is shown to provide a solution with important physical representation which may help in dealing with similar complex nonlinear models with applications in contemporary science and other related areas. The method proves to be a reliable method for solving such models with high accuracy. This work, thus, provides a lot of encouragement for subsequent research in this area, and the results of that research will be reported in near future.

Figure 1 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 1
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{1}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           3
                        
                        {\varsigma }_{2}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           1
                        
                        {\varsigma }_{3}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           3
                        
                        {\vartheta }_{1}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           1
                        
                        {\vartheta }_{2}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           1
                        
                        {\vartheta }_{3}=1
                     
                  , 
                     
                        
                        
                           α
                           =
                           3
                        
                        \alpha =3
                     
                  , 
                     
                        
                        
                           β
                           =
                           2
                        
                        \beta =2
                     
                  , 
                     
                        
                        
                           γ
                           =
                           4
                        
                        \gamma =4
                     
                  , 
                     
                        
                        
                           μ
                           =
                           3
                        
                        \mu =3
                     
                  , 
                     
                        
                        
                           ν
                           =
                           2
                        
                        \nu =2
                     
                  , 
                     
                        
                        
                           ω
                           =
                           3
                        
                        \omega =3
                     
                  .

Figure 1

Graphical representation of solution q 1 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 3 , ς 3 = 1 , ϑ 1 = 3 , ϑ 2 = 1 , ϑ 3 = 1 , α = 3 , β = 2 , γ = 4 , μ = 3 , ν = 2 , ω = 3 .

Figure 2 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 3
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{3}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           3
                        
                        {\varsigma }_{2}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           1
                        
                        {\varsigma }_{3}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           5
                        
                        {\vartheta }_{1}=5
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           3
                        
                        {\vartheta }_{2}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           2
                        
                        {\vartheta }_{3}=2
                     
                  , 
                     
                        
                        
                           α
                           =
                           −
                           1
                        
                        \alpha =-1
                     
                  , 
                     
                        
                        
                           β
                           =
                           2
                        
                        \beta =2
                     
                  , 
                     
                        
                        
                           γ
                           =
                           −
                           2
                        
                        \gamma =-2
                     
                  , 
                     
                        
                        
                           μ
                           =
                           2
                        
                        \mu =2
                     
                  , 
                     
                        
                        
                           ν
                           =
                           −
                           2
                        
                        \nu =-2
                     
                  , 
                     
                        
                        
                           ω
                           =
                           3
                        
                        \omega =3
                     
                  .

Figure 2

Graphical representation of solution q 3 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 3 , ς 3 = 1 , ϑ 1 = 5 , ϑ 2 = 3 , ϑ 3 = 2 , α = 1 , β = 2 , γ = 2 , μ = 2 , ν = 2 , ω = 3 .

Figure 3 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 3
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{3}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           4
                        
                        {\varsigma }_{2}=4
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           1
                        
                        {\varsigma }_{3}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           1
                        
                        {\vartheta }_{1}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           2
                        
                        {\vartheta }_{2}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           3
                        
                        {\vartheta }_{3}=3
                     
                  , 
                     
                        
                        
                           α
                           =
                           3
                        
                        \alpha =3
                     
                  , 
                     
                        
                        
                           β
                           =
                           2
                        
                        \beta =2
                     
                  , 
                     
                        
                        
                           γ
                           =
                           −
                           2
                        
                        \gamma =-2
                     
                  , 
                     
                        
                        
                           μ
                           =
                           2
                        
                        \mu =2
                     
                  , 
                     
                        
                        
                           ν
                           =
                           3
                        
                        \nu =3
                     
                  , 
                     
                        
                        
                           ω
                           =
                           2
                        
                        \omega =2
                     
                  .

Figure 3

Graphical representation of solution q 3 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 4 , ς 3 = 1 , ϑ 1 = 1 , ϑ 2 = 2 , ϑ 3 = 3 , α = 3 , β = 2 , γ = 2 , μ = 2 , ν = 3 , ω = 2 .

Figure 4 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 4
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{4}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           3
                        
                        {\varsigma }_{2}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           5
                        
                        {\varsigma }_{3}=5
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           4
                        
                        {\vartheta }_{1}=4
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           5
                        
                        {\vartheta }_{2}=5
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           3
                        
                        {\vartheta }_{3}=3
                     
                  , 
                     
                        
                        
                           α
                           =
                           4
                        
                        \alpha =4
                     
                  , 
                     
                        
                        
                           β
                           =
                           3
                        
                        \beta =3
                     
                  , 
                     
                        
                        
                           γ
                           =
                           −
                           4
                        
                        \gamma =-4
                     
                  , 
                     
                        
                        
                           μ
                           =
                           2
                        
                        \mu =2
                     
                  , 
                     
                        
                        
                           ν
                           =
                           2
                        
                        \nu =2
                     
                  , 
                     
                        
                        
                           ω
                           =
                           3
                        
                        \omega =3
                     
                  .

Figure 4

Graphical representation of solution q 4 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 3 , ς 3 = 5 , ϑ 1 = 4 , ϑ 2 = 5 , ϑ 3 = 3 , α = 4 , β = 3 , γ = 4 , μ = 2 , ν = 2 , ω = 3 .

Figure 5 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 5
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{5}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           −
                           3
                        
                        {\varsigma }_{2}=-3
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           4
                        
                        {\varsigma }_{3}=4
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           0
                        
                        {\vartheta }_{1}=0
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           −
                           1
                        
                        {\vartheta }_{2}=-1
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           1
                        
                        {\vartheta }_{3}=1
                     
                  , 
                     
                        
                        
                           α
                           =
                           −
                           4
                        
                        \alpha =-4
                     
                  , 
                     
                        
                        
                           β
                           =
                           2
                        
                        \beta =2
                     
                  , 
                     
                        
                        
                           γ
                           =
                           −
                           2
                        
                        \gamma =-2
                     
                  , 
                     
                        
                        
                           μ
                           =
                           2
                        
                        \mu =2
                     
                  , 
                     
                        
                        
                           ν
                           =
                           1
                        
                        \nu =1
                     
                  , 
                     
                        
                        
                           ω
                           =
                           −
                           3
                        
                        \omega =-3
                     
                  , 
                     
                        
                        
                           
                              
                                 λ
                              
                              
                                 1
                              
                           
                           =
                           −
                           1
                        
                        {\lambda }_{1}=-1
                     
                  , 
                     
                        
                        
                           
                              
                                 λ
                              
                              
                                 2
                              
                           
                           =
                           1
                        
                        {\lambda }_{2}=1
                     
                  .

Figure 5

Graphical representation of solution q 5 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 3 , ς 3 = 4 , ϑ 1 = 0 , ϑ 2 = 1 , ϑ 3 = 1 , α = 4 , β = 2 , γ = 2 , μ = 2 , ν = 1 , ω = 3 , λ 1 = 1 , λ 2 = 1 .

Figure 6 
               Graphical representation of solution 
                     
                        
                        
                           
                              
                                 q
                              
                              
                                 6
                              
                           
                           
                              (
                              
                                 x
                                 ,
                                 t
                              
                              )
                           
                        
                        {q}_{6}\left(x,t)
                     
                   with the parameter values as: 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 1
                              
                           
                           =
                           2
                        
                        {\varsigma }_{1}=2
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 2
                              
                           
                           =
                           1
                        
                        {\varsigma }_{2}=1
                     
                  , 
                     
                        
                        
                           
                              
                                 ς
                              
                              
                                 3
                              
                           
                           =
                           3
                        
                        {\varsigma }_{3}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 1
                              
                           
                           =
                           0
                        
                        {\vartheta }_{1}=0
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 2
                              
                           
                           =
                           3
                        
                        {\vartheta }_{2}=3
                     
                  , 
                     
                        
                        
                           
                              
                                 ϑ
                              
                              
                                 3
                              
                           
                           =
                           −
                           2
                        
                        {\vartheta }_{3}=-2
                     
                  , 
                     
                        
                        
                           α
                           =
                           4
                        
                        \alpha =4
                     
                  , 
                     
                        
                        
                           β
                           =
                           2
                        
                        \beta =2
                     
                  , 
                     
                        
                        
                           γ
                           =
                           2
                        
                        \gamma =2
                     
                  , 
                     
                        
                        
                           μ
                           =
                           2
                        
                        \mu =2
                     
                  , 
                     
                        
                        
                           ν
                           =
                           4
                        
                        \nu =4
                     
                  , 
                     
                        
                        
                           ω
                           =
                           2
                        
                        \omega =2
                     
                  .

Figure 6

Graphical representation of solution q 6 ( x , t ) with the parameter values as: ς 1 = 2 , ς 2 = 1 , ς 3 = 3 , ϑ 1 = 0 , ϑ 2 = 3 , ϑ 3 = 2 , α = 4 , β = 2 , γ = 2 , μ = 2 , ν = 4 , ω = 2 .

Acknowledgements

The authors would like to express their sincere thanks to the support of National Natural Science Foundation of China.

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

  2. Funding information: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

References

[1] Jajarmi A, Ghanbari B, Baleanu D. A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence. Chaos: Interdisciplinary J Nonlin Sci. 2019 Sep 9;29(9):093111. Search in Google Scholar

[2] Qureshi S, Yusuf A, Shaikh AA, Inc M, Baleanu D. Fractional modeling of blood ethanol concentration system with real data application. Chaos: Interdisciplinary J Nonlin Sci. 2019 Jan 31;29(1):013143. Search in Google Scholar

[3] Jajarmi A, Arshad S, Baleanu D. A new fractional modelling and control strategy for the outbreak of dengue fever. Physica A: Stat Mech Appl. 2019 Dec 1;535:122524. Search in Google Scholar

[4] Qureshi S, Rangaig NA, Baleanu D. New numerical aspects of Caputo-Fabrizio fractional derivative operator. Mathematics. 2019 Apr;7(4):374. Search in Google Scholar

[5] Munawar M, Jhangeer A, Pervaiz A, Ibraheem F. New general extended direct algebraic approach for optical solitons of Biswas-Arshed equation through birefringent fibers. Optik. 2021 Feb 1;228:165790.Search in Google Scholar

[6] Jhangeer A, Hussain A, Junaid-U-Rehman M, Baleanu D, Riaz MB. Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation. Chaos, Solitons & Fractals. 2021 Feb 1;143:110578.Search in Google Scholar

[7] Hussain A, Jhangeer A, Abbas N, Khan I, Sherif ES. Optical solitons of fractional complex Ginzburg–Landau equation with conformable, beta, and M-truncated derivatives: a comparative study. Adv Differ Equ. 2020 Dec;2020(1):1–9. Search in Google Scholar

[8] Hosseini K, Osman MS, Mirzazadeh M, Rabiei F. Investigation of different wave structures to the generalized third-order nonlinear Scrödinger equation. Optik. 2020 Mar 1;206:164259.Search in Google Scholar

[9] Hosseini K, Mirzazadeh M, Vahidi J, Asghari R. Optical wave structures to the Fokas-Lenells equation. Optik. 2020 Apr 1;207:164450.Search in Google Scholar

[10] Rezazadeh H. New solitons solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity. Optik. 2018 Aug 1;167:218–27. Search in Google Scholar

[11] Hosseini K, Mirzazadeh M, Ilie M, Gómez-Aguilar JF. Biswas-Arshed equation with the beta time derivative: optical solitons and other solutions. Optik. 2020 Sep 1;217:164801.Search in Google Scholar

[12] Yokus A, Durur H, Ahmad H, Thounthong P, Zhang YF. Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G∕G,1∕G)-expansion and (1∕G)-expansion techniques. Results Phys. 2020 Dec 1;19:103409.Search in Google Scholar

[13] Yokus A, Durur H, Ahmad H. Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system. Facta Universitatis, Series: Mathematics and Informatics. 2020 May 28;35(2):523–31. Search in Google Scholar

[14] Yokus A, Durur H, Ahmad H, Yao SW. Construction of different types analytic solutions for the Zhiber-Shabat equation. Math. 2020 Jun;8(6):908. Search in Google Scholar

[15] Khater MM, Lu D, Hamed YS. Computational simulation for the (1+1)-dimensional Ito equation arising quantum mechanics and nonlinear optics. Results Phys. 2020 Dec 1;19:103572.Search in Google Scholar

[16] Abdel-Aty AH, Khater MM, Baleanu D, Khalil EM, Bouslimi J, Omri M. Abundant distinct types of solutions for the nervous biological fractional FitzHugh-Nagumo equation via three different sorts of schemes. Adv Differ Equ. 2020 Dec;2020(1):1–7. Search in Google Scholar

[17] Chu Y, Khater MM, Hamed YS. Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model. AIP Advances. 2021 Jan 1;11(1):015223.Search in Google Scholar

[18] Srivastava HM, Baleanu D, Machado JA, Osman MS, Rezazadeh H, Arshed S, Günerhan H. Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method. Physica Scripta. 2020 Jun 1;95(7):075217.Search in Google Scholar

[19] Ghanbari B, Nisar KS, Aldhaifallah M. Abundant solitary wave solutions to an extended nonlinear Schrödinger’s equation with conformable derivative using an efficient integration method. Adv Differ Equ. 2020 Dec;2020(1):1–25. Search in Google Scholar

[20] Ghanbari B, Yusuf A, Baleanu D. The new exact solitary wave solutions and stability analysis for the (2+1)-dimensional Zakharov-Kuznetsov equation. Adv Differ Equ. 2019 Dec;2019(1):1–5. Search in Google Scholar

[21] Munusamy K, Ravichandran C, Nisar KS, Ghanbari B. Existence of solutions for some functional integrodifferential equations with nonlocal conditions. Math Meth Appl Sci. 2020 Nov 30;43(17):10319–31. Search in Google Scholar

[22] Inc M, Khan MN, Ahmad I, Yao SW, Ahmad H, Thounthong P. Analysing time-fractional exotic options via efficient local meshless method. Results Phys. 2020 Dec 1;19:103385.Search in Google Scholar

[23] Rezazadeh H, Inc M, Baleanu D. New solitary wave solutions for variants of (3+1)-dimensional Wazwaz–Benjamin–Bona–Mahony equations. Front Phys. 2020 Sep 4;8:332.Search in Google Scholar

[24] Khater MM, Inc M, Attia RA, Lu D, Almohsen B. Abundant new computational wave solutions of the GM-DP-CH equation via two modified recent computational schemes. J Taibah Univ Sci. 2020 Jan 1;14(1):1554–62. Search in Google Scholar

[25] Rahman G, Nisar KS, Ghanbari B, Abdeljawad T. On generalized fractional integral inequalities for the monotone weighted Chebyshev functionals. Adv Differ Equ. 2020 Dec;2020(1):1–9. Search in Google Scholar

[26] Akinyemi L, Senol M, Huseen SN. Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma. Adv Differ Equ. 2021 Dec;2021(1):1–27. Search in Google Scholar

[27] Akinyemi L. A fractional analysis of Noyes-Field model for the nonlinear Belousov–Zhabotinsky reaction. Comput Appl Math. 2020 Sep;39:1–34. Search in Google Scholar

[28] Akinyemi L, Senol M, Iyiola OS. Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method. Math Comput Simul. 2021 April 1;182:211–33. Search in Google Scholar

[29] Senol M, Iyiola OS, Kasmaei HD, Akinyemi L. Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent-Miodek system with energy-dependent Schrödinger potential. Adv Differ Equ. 2019 Dec;2019(1):1–21. Search in Google Scholar

[30] Khater MM. On the dynamics of strong Langmuir turbulence through the five recent numerical schemes in the plasma physics. Numer Methods Partial Differ Equ. 2020 Dec 4. Search in Google Scholar

[31] Tariq KU, Seadawy AR. On the soliton solutions to the modified Benjamin-Bona-Mahony and coupled Drinfelad-Sokolov-Wilson models and its applications. J King Saud Univ Sci. 2020 Jan 1;32(1):156–62. Search in Google Scholar

[32] Bhrawy AH, Biswas A, Javidi M, Ma WX, Pınar Z, Yıldırım A. New solutions for (1+1)-dimensional and (2+1)-dimensional Kaup-Kupershmidt equations. Results Math. 2013 Feb;63(1):675–86. Search in Google Scholar

[33] Mirzazadeh M, Eslami M, Zerrad E, Mahmood MF, Biswas A, Belic M. Optical solitons in nonlinear directional couplers by sine-cosine function method and Bernoulli’s equation approach. Nonlin Dyn. 2015 Sep;81(4):1933–49. Search in Google Scholar

[34] Qiao Z. Darboux transformation and explicit solutions for two integrable equations. J Math Anal Appl. 2011 Aug 15;380(2):794–806. Search in Google Scholar

[35] Wazwaz AM, El-Tantawy SA. Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlin Dyn. 2017 Jun;88(4):3017–21. Search in Google Scholar

[36] Yang S, Hua C. Lie symmetry reductions and exact solutions of a coupled KdV-Burgers equation. Appl Math Comput. 2014 May 15;234:579–83. Search in Google Scholar

[37] Younis M. A new approach for the exact solutions of nonlinear equations of fractional order via modified simple equation method. Appl Math. 2014 Jul 7;5(13):1927–32.Search in Google Scholar

[38] Ablowitz MJ, Ablowitz MA, Clarkson PA, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. London: Cambridge University Press; 1991 Dec 12. Search in Google Scholar

[39] Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett. 1971 Nov 1;27(18):1192. Search in Google Scholar

[40] Wang M. Exact solutions for a compound KdV-Burgers equation. Phys Lett A. 1996 Apr 29;213(5–6):279–87. Search in Google Scholar

[41] Tariq KU, Seadawy AR, Younis M, Rizvi ST. Dispersive traveling wave solutions to the space-time fractional equal-width dynamical equation and its applications. Opt Quant Electron. 2018 Mar;50(3):1–6. Search in Google Scholar

[42] Tariq KU, Seadawy AR. Bistable Bright-Dark solitary wave solutions of the (3+1)-dimensional Breaking soliton, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations and their applications. Results Phys. 2017 Jan 1;7:1143–9. Search in Google Scholar

[43] Baskonus HM. New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamics. Nonlin Dyn. 2016 Oct;86(1):177–83. Search in Google Scholar

[44] Yel G, Baskonus HM, Bulut H. Novel archetypes of new coupled Konno-Oono equation by using sine-Gordon expansion method. Opt Quant Electron. 2017 Sep;49(9):1–10. Search in Google Scholar

[45] Yel G, Baskonus HM, Gao W. New dark-bright soliton in the shallow water wave model. Aims Math. 2020 Apr 20;5(4):4027–44. Search in Google Scholar

[46] Ismael HF, Bulut H, Baskonus HM. Optical soliton solutions to the Fokas-Lenells equation via sine-Gordon expansion method and (m+(G′∕G))-expansion method. Pramana. 2020 Dec;94(1):1–9. Search in Google Scholar

[47] 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 Apr 1;208:164132.Search in Google Scholar

[48] Korkmaz A, Hepson OE, Hosseini K, Rezazadeh H, Eslami M. Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. J King Saud Univ Sci. 2020 Jan 1;32(1):567–74. Search in Google Scholar

[49] Leta TD, El Achab A, Liu W, Ding J. Application of bifurcation method and rational sine-Gordon expansion method for solving 2D complex Ginzburg–Landau equation. Int J Mod Phys B. 2020 Apr 10;34(9):2050079. Search in Google Scholar

[50] Tasbozan O, Kurt A. The new travelling wave solutions of time fractional Fitzhugh–Nagumo equation with Sine-Gordon expansion method. Adıyaman Universitesi Fen Bilimleri Dergisi. 2020 June 26;10(1):256–63. Search in Google Scholar

[51] Tariq KU, Younis M. Bright, dark and other optical solitons with second order spatiotemporal dispersion. Optik. 2017 Aug 1;142:446–50. Search in Google Scholar

[52] Christian JM, McDonald GS, Hodgkinson TF, Chamorro-Posada P. Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves. Phys Rev A. 2012 Aug 21;86(2):023838. Search in Google Scholar

[53] Christian JM, McDonald GS, Hodgkinson TF, Chamorro-Posada P. Wave envelopes with second-order spatiotemporal dispersion. II. Modulational instabilities and dark Kerr solitons. Phys Rev A. 2012 Aug 21;86(2):023839. Search in Google Scholar

[54] Rezazadeh H, Mirhosseini-Alizamini SM, Neirameh A, Souleymanou A, Korkmaz A, Bekir A. Fractional Sine-Gordon equation approach to the coupled Higgs system defined in time-fractional form. Iran J Sci Technol Trans A: Sci. 2019 Dec;43(6):2965–73. Search in Google Scholar

Received: 2020-11-22
Revised: 2021-01-30
Accepted: 2021-02-16
Published Online: 2021-03-17

© 2021 Hadi Rezazadeh et al., published by De Gruyter

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

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