The attainment of analytical solutions for different models described by NLPDEs plays a major role in applied mathematics, fluid mechanics, fluid dynamics, plasma and solid-state physics, nonlinear optics, and chemistry. Among these solutions, the optical solitons, which have significant applications in modern communication systems, and have attracted particular attention from physicists as well as mathematicians [1–5]. Optical solitons can propagate over extremely large distances without shape change when a balance between the linear dispersion and nonlinear effects is achieved. There are many types of solitons, including bright, dark, anti-dark, and singular solitons, amongst many others. Bright solitons exist on a zero-intensity background, while dark solitons arise as an intensity dip in an infinitely extended constant background. Moreover, dark solitons are less influenced by the perturbations, which means that dark solitons could be more preferable than bright ones in optical communication systems. The anti-dark solitons have profiles similar to those of the bright ones but exist on a nonzero background like the dark ones [6, 7]. Many effective methods have been presented to solve these equations [8–26].

The PFS-E method [27, 28] is a direct and concise method to solve nonlinear evolution equations. It is employed to find and study the wave solutions of the 2D-CNLS equation. The predominating equation is described by [29–32]:

where Φ = Φ(x, y, t) refers to the complex-valued function, μ₁ is the second-order dispersion coefficient, and μ₂, μ₃ are the self-steepening coefficients. The 2D-CNLS equation has been established by a one-dimensional reduction of the structure that defines the fractional quantum Hall effect (it is a quantum-mechanical version of the Hall effect existing in 2D electron systems related to strong magnetic fields and low temperatures). An extraordinary characteristic of Equation (1) is the nonlinearity of the current density, which informs the new execution for the SPM and self-focusing through the current [29–32]. This equation cannot pass the Painlevè test of integrability and is not invariant under the Galilean transformation [32].

Bulut et al. [30] discussed Equation (1) in 1D and 2D and found bright and dark soliton solutions via the extended sinh-Gordon equation method. Nishino et al. [33] solved Equation (1) in 1D only and introduced two categories of wave solutions like bright and dark soliton trains. Very recently, Raza and Javid [32] investigated the singular and dark solitons for the 2D-CNLS equation by two different approaches, namely the extended direct algebraic and trial equation methods. To the best of our knowledge, no studies have found optical wave solutions for (1) via the extended Fan sub-equation method.

The paper is organized as follows. Different solutions for the 2D-CNLS equation are evaluated in section 2. The physical interpretation of the solutions is discussed in section 3. The main deductions are presented in section 4.

In this section, we use the PFS-E technique to find more forms of exact solutions for Equation (1) by considering a more general transformation stated in [34, 35]. The PFS-E method includes an algebraic strategy to find different analytical solutions for NLPDEs that can be expressed as a polynomial in the variable that satisfies the general Riccati equation. The most significant achievement of this approach is that it offers all the solutions that can be found by the use of other methods such as processes using the Riccati equation, an elliptic equation of the first kind, an auxiliary ordinary equation, or the generalized Riccati equation as mapping equation.

Let the wave profile be defined as

while the amplitude

and

Inserting (2) into (1) and separating its real and imaginary parts, we get

where

and

From (7), we have

From the homogeneous balance condition on (5), the general solution can be written as

where ϕ is given by the following auxiliary equation,

where ζ_i(i = 0, 1, 2, 3, 4) are free parameters.

Plugging (9) and (10) into (5) and setting the coefficients of ϕ^j, j = 0, 1, ⋯ , 4 identical zero, we get

Here, suitable values are selected for ζ_i, (i = 0, 1, 2, 3, 4).

which give

Different cases [34, 35] can be introduced to obtain the following solutions.

Case I.

If ζ0=ϑ32,ζ1=2ϑ1ϑ3,ζ2=2ϑ2ϑ3+ϑ12,ζ3=2ϑ1ϑ2,ζ4=ϑ22, we get the solution of (1) in the form ΦηI,(η=1,3,5,10,13,20,24). A variety of significant solitons are obtained below.

Type I: when ϑ12-4ϑ2ϑ3>0, ϑ₁ϑ₂ ≠ 0, ϑ₂ϑ₃ ≠ 0. A set of dark optical solitons is acquired as

A variety of bright-dark optical soliton is gained as

A set of singular dark optical solitons is obtained as

The family of solitons is obtained as

Type II: when ϑ12-4ϑ2ϑ3<0, ϑ₁ϑ₂ ≠ 0, ϑ₂ϑ₃ ≠ 0. The following collections of periodic solitons are given by

Case II.

If ζ0=ϑ32,ζ1=2ϑ1ϑ3,ζ2=0,ζ3=2ϑ1ϑ2,ζ4=ϑ22, we get the solution of (1) in the form ΦηII,(η=1,5). A collection of dark optical solitons is given by

Also, a different shape of singular dark optical soliton is acquired

Case III.

If ζ₀ = ζ₁ = 0, we get the solution of (1) in the form ΦηIII,(η=1,2,3,4,6,8,9).

Type I: ζ2=1,ζ3=-2λ3λ1,ζ4=λ32-λ22λ12, where λ₁, λ₂, λ₃ are arbitrary constants.

Type II: ζ2=1,ζ3=-2λ3λ1,ζ4=λ32+λ22λ12, where λ₁, λ₂, λ₃ are arbitrary constants.

In particular, if we take λ₂ = 0 in the above Equations (23), (24), we get

Type III: ζ2=4,ζ3=-4(2λ2+λ4)λ1,ζ4=4λ22+4λ4λ2+λ32λ12, where λ₁, λ₂, λ₃, λ₄ are arbitrary constants.

Type IV: ζ2=4,ζ3=4(λ4-2λ2)λ1,ζ4=4λ22-4λ4λ2+λ32λ12, where λ₁, λ₂, λ₃, λ₄ are arbitrary constants.

In particular, if we consider λ₂ = λ₄, we have another solution as

Type V: ζ2=-1,ζ3=2λ3λ1,ζ4=λ32-λ22λ12, where λ₁, λ₂, λ₃ are arbitrary constants.

Type VI: ζ2=4,ζ3=-2λ3λ1,ζ4=λ32-λ22λ12, where λ₁, λ₂, λ₃ are arbitrary constants.

Type VII: ζ2=-4,ζ3=4(2λ2+λ4)λ1,ζ4=-4λ22+4λ4λ2-λ32λ12, where λ₁, λ₂, λ₃, λ₄ are arbitrary constants.

Case IV.

If ζ₁ = ζ₃ = 0, we get the solution of (1) in the form ΦηIV,(η=3,13).

For ζ0=14,ζ2=1-2m22,ζ4=14, the solution of (1) of the form

leads to the bright optical soliton when m → 1,

and the singular periodic solutions when m → 0,

For ζ0=14,ζ2=1-2m22,ζ4=14, the solution of (1) of the form

leads to a collection of singular dark solutions when m → 1,

and singular periodic solutions when m → 0,

In this part, the physical aspects of the solutions obtained are discussed by means of graphical 3D representations. Figures 1–4 show different categories of background for soliton solutions classified into dark or singular soliton solutions.

Figure 1 illustrates |Φ(x,y,t)|1I established in Case I (Type I) for ϑ₁ = 1, ϑ₂ = −1, ϑ₃ = 1, μ₁ = 1, μ₂ = 1, μ₃ = 1, α₀ = 1, β₀ = 2, λ₀ = 3, α₁ = 4, β₁ = 5, λ = 6, η₀ = 1, t = 0.01.

Moreover, Figure 2 illustrates |Φ(x,y,t)|5II found in Case II for ϑ₁ = 1, ϑ₂ = −1, ϑ₃ = 1, μ₁ = 1, μ₂ = 1, μ₃ = 1, α₀ = 1, β₀ = 2, λ₀ = 3, α₁ = 4, β₁ = 5, λ = 6, η₀ = 1, t = 0.1.

Similarly, Figure 3 illustrates |Φ(x,y,t)|1III found in Case III (Type I) for λ₁ = 1, λ₂ = 2, λ₃ = 3, μ₁ = 1, μ₂ = 1, μ₃ = 1, α₀ = 1, β₀ = 2, λ₀ = 3, α₁ = 4, β₁ = 5, λ = 6, η₀ = 1, t = 1. Similarly, Figure 4 expresses |Φ(x,y,t)|13IV observed in Case IV (m → 1) for μ₁=1, μ₂=1, μ₃ = 2, α₀ = 1, β₀ = 2, λ₀ = 3, α₁ = 0.1, β₁ = 1, λ = 0.5, η₀ = 1, t = 0.1.

Figure 1 represents the absolute value of the complex wave solution given by Equation (14). We observe that this solution is a dark (or kink) soliton wave propagating along the y-axis. The kink wave is an essential aspect of numerous physical phenomena containing self-reinforcing, impulsive systems, and reaction-diffusion-advection. It is clear that there is a transmission of the dark soliton with invariant amplitude (without any gain or loss) in the homogeneous medium of motion. Due to the homogeneous (constant) coefficients of Equation (1), we cannot provide a possible way to control the propagation of the dark solitons in this medium. Figures 2–4 represent the absolute value of the complex wave solutions given by Equations (22), (23), and (37), respectively. We observe that these solutions are singular solitons that can be linked to a solitary wave when its center becomes an imaginary position. Furthermore, It is clear that their intensity gets Stronger, and consequently, they are not stable. These solutions have a cusp, which may lead to the formation of Rogue waves.

Herein, a large set of new analytical solutions with different wave structures of the 2D-CNLS equation has been reproduced with the aid of the PFS-E technique. As a positive result, a wide variety of unprecedented exact solutions were gained in an easy manner. Our study presents whether the suggested approach is trustworthy in treatment NLPDEs to promote a variety of exact solutions. Finally, we have plotted some 3D graphs of these solutions and have shown that these graphs can be controlled by adjusting the parameters. According to our knowledge, the obtained solutions are likely to provide a useful supplement to the existing literature on nonlinear optics.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.