2.1.The auxiliary equation method

The following steps describe the auxiliary equation method ^26,27. Considering a given nonlinear partial differential equation (NPDE) with independent variables (x,t) and dependent variables a(x,t)

Step 1: The traveling-wave solution of (10) is used in the following form

where v is the traveling-wave speed. Then (10) was converted into a nonlinear ordinary differential equation as follows

Step 2: Suppose that the exact solutions of (12) can be expressed

and g(ξ) satisfies the following auxiliary equation

with gξ=(∂g/∂ξ), C_i(i = (0,1,2,3,4)), A₀, A_i (i = 1,2,…n) are real constants to be determined later.

Step 3: Under the terms of the method, it is assumed that the solution of Eq.(12) can be written in the following form

where A0,A1,A3,A4, and A _n are real constants to be determined later. To calculate the value of n, we balance the highest-order nonlinear terms in (12), and then the value of n can be determined.

Step 4: Substituting (16), (15) and (14) into (12) provides a polynomial of g(ξ). Next, collecting all the coefficients g(ξ)i, (i=0,1,2,........n) forms a system of algebraic equations. Solving this system, we describe the variable coefficients of A₀, A _i , i = (1,2,……,n), so the solution to (12) can be obtained in terms of g(ξ).

Step 5: To obtain the exact solutions to (10), the following solutions of (14) or (15) are used.

Case 1: for C0=C1=C3=0, C2>0, C4<0,

Case 2: for C0= (C22/4C4), C1=C3=0, C2<0, C4>0,

Case 3: for C0=C1=0 C2>0, C4>0,

Case 4: for C0=C1=0, C2>0, C32-4C2C4>0,

Case 5: for C0=C1=0, C2>0,

where C ₀, C ₁, C ₂, C ₃ and C ₄ are arbitrary constants. Therefore, using Eq.(17-21) and (16), the exact solutions to (10) can be obtained.

2.2.The modified auxiliary equation method

Investigation of exact traveling wave solutions of certain nonlinear partial differential equations by using the modified auxiliary equation method has been carried out recently ^24-26.

Consider a nonlinear evolution partial differential equation as in Eq.(10) where a=a(x,t) is an unknown function of independent variables x and t.

The main steps of the method to obtain exact solutions of (10) can be given as follows.

Step 1: Suppose that the formal solution of the ODE in (12) can be expressed as

where Ai,Bi,K are arbitrary real constants and f(ξ) satisfy the following auxiliary equation:

where α,β,σ are arbitrary constants and K>0,K≠1. The solutions of (23) are given by

Case 1: for β2-4ασ<0 and σ≠0,

Case 2: for β2-4ασ>0 and σ≠0,

Case 3: for β2-4ασ=0 and σ≠0,

The explicit exact solutions of (10) can be obtained by inserting the values 0, A₀, A_i, B_i (j = 1,2,3,…..n).

2.3.Sine-Gordon expansion approach

To integrate the ordinary differential equation (ODE) (12), we consider the sine-Gordon equation given by

The solutions of (27) are represented by

The series can be utilized to derive the solution of (12),

The parameter n can be obtained using the balancing principle. Making all the necessary computations, the solutions of the NPDE under consideration can be obtained.

3.1.The Auxiliary equation method

Now, to use the homogeneous balanced principle between ϕ'' and ϕ3 in (9), it is obtained n = 1. Subsequently, it is introduced into (13) the value of n, and taking into account (14-15), a system of equation in terms of g(ξ)i is obtained. Thus, we set all the coefficients of each individual term g(ξ)i to zero, which gives the results below.

Set 1:

Case 1: For C0=C1=C3=0, and C2>0, C4<0 using (31), the bright soliton to the governing model (2) is

Case 2: For C0=(C22/4C4), C1=C3=0, and C2<0, C4>0, using (31), the dark soliton is found to be

Set 2:

Case 3: For C0=C1=0 C2>0 C4>0 using

Eq.(34), one can find

Case 4: For C₀ = C₁ = 0, and C2>0, C32-4C2C4>0, using (34), we obtain

Case 5: For C₀ = C₁ = 0, and C2>0, using (34),

3.2.The modified auxiliary equation method

Now, the MAE method will be utilized to get the general solution of Eq.(2). In this perspective, we obtain a system of algebraic equations which solves to

Set 1:

Set 2:

Using the values of the parameters in Set 1 given by (38), we get the solitary wave solutions of (2) in the following formulas:

When β2-4ασ<0 and σ≠0, we obtain trigonometric function solutions:

or

When β2-4ασ>0 and σ≠0 we obtain dark soliton solutions:

or bright soliton solutions

When β2-4ασ=0 and σ≠0 we obtain rational function solutions:

Using the values of the parameters in Set 2 given by Eq.(39), we get the solitary wave solutions of Eq.(2) in the following formulas: When β2-4ασ<0 and σ≠0, we obtain trigonometric functions solutions:

or

When β2-4ασ>0 and σ≠0 we obtain dark soliton solutions:

or bright soliton solutions

When β2-4ασ=0 and σ≠0 we obtain rational function solutions:

Figures 1 and 2 show (a) the spatiotemporal evolution in 3D, (b) contour plot, and (c) the evolution at a different time in 2D of the bright and dark soliton solutions in optical fibers for |a1,1|2 and |a1,2|2, respectively. It is observed that the evolution plot of bright and dark solitons (c) for -100≤x≤100, at t = 0, t = 10, t = 15, t = 20 shift from left to right caused by the group velocity dispersion (GVD) term (β2), which is fully annulled by the nonlinear phase shift modulation caused by (SPM), coming from a pulse that spreads undisturbed down the fiber.

Figure 1 The plot of the bright soliton of |a _1;1 |² of the solution Eq. (32) at A₁ = 1, C₂ = 0:019, C₄ = -0:3, v = 7. 

Figure 2 The plot of dark soliton of |a _1;1 |² of the solution Eq. (33) at A₁ = 1, C₀ = 520:62, C₂ = -58:91, C₄ = 1:66; v = 0:3. 

3.3.The Sine-Gordon expansion approach

Plugging the predicted solution (27) and its necessary derivatives into (9), we get an over-determined system containing the combination of cos(ω) and sin(ω). Setting the same powers of sini(ω)cosj(ω) equal to zero, a system of algebraic equations is deduced. Solving this system, we report the following result:

Set 1:

Figure 3 The plot of dark soliton |a _1;1 |² of the solution Eq. (33) at A₁ = 1:5, C₀ = 0:5, C₄ = 0:5, v = 4:5, a) [C₂ = -0:25;C₂ = -0:5;C₂ = -0:75]; b) [C₂ = -1:25;C₂ = -1:50;C₂ = -1:75] respectively. 

Then, the solutions of (2) corresponding to (50) are

where (51) and (52) represent dark optical and singular soliton solutions, respectively.

Furthermore, (33) is read like (51) under the constraint conditions C₂ = -1, and C₄ = 1/2 (3). On the other side when C2≠-1, (33) and (51) (dark-solitons) have the same form, but differ in width and amplitude during long distance communication taking advantage of its stability under the influence of the material losses (see 4). So, it is important to conserve the obtained two results.

Set 2:

Then, the solutions of (2) corresponding to (53) are

where (54) and (55) represent bright optical and singular soliton solutions, respectively.

Figure 4 The plot of dark soliton |a _1;2 |² of the solution Eq. (33) at [A₁ = 1:5, C₀ = 51:62, C₂ = -2:31, C₄ = 0:26, c = 1:2, v = 0:15, γ = 0:35, β₂ = 1, k₁ = 1:65], [A₁ = 1:5, C₀ = 38:43, C₂ = -2:002, C₄ = 0:026, c = 1:2, v = 0:15, γ = 0:35, β₂ = 1, k₂ = 2:65], [A₁ = 1:5, C₀ = 27:18, C₂ = -1:68, C₄ = 0:026, c = 1:2, v = 0:15, γ = 0:35, β₂ = 1, k₁ = 3:65] respectively. 

Remark: By integrating from (4) with along to (7), we have

where ξ0 is an integration constant. However when k₁ = 0, which corresponds to the conventional two-core fiber design ²¹, the chirp obtained depends only with the GVD and the speed of the soliton.