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ORIGINAL RESEARCH article

Front. Phys., 07 July 2020
Sec. Statistical and Computational Physics
This article is part of the Research Topic Complex Dynamical Systems with Analytical and Numerical Wave Solutions View all 6 articles

Different Types of Progressive Wave Solutions via the 2D-Chiral Nonlinear Schrödinger Equation

  • 1Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
  • 2Department of Mathematics, Faculty of Applied Science, Umm Alqura University, Makkah, Saudi Arabia
  • 3Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara, Turkey
  • 4Institute of Space Sciences, Măgurele, Romania
  • 5Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
  • 6Department of Mathematics, Mirpur University of Science and Technology (MUST), Mirpur, Pakistan
  • 7Department of Mathematics, Art-Science Faculty, Kastamonu University, Kastamonu, Turkey
  • 8Punjab University College of Information Technology, University of the Punjab, Lahore, Pakistan
  • 9Department of Mathematics, COMSATS University Islamabad, Lahore, Pakistan

A versatile integration tool, namely the protracted (or extended) Fan sub-equation (PFS-E) technique, is devoted to retrieving a variety of solutions for different models in many fields of the sciences. This essay presents the dynamics of progressive wave solutions via the 2D-chiral nonlinear Schrödinger (2D-CNLS) equation. The solutions acquired comprise dark optical solitons, periodic solitons, singular dark (bright) solitons, and singular periodic solutions. By comparing the results gained in this work with other literature, it can be noticed that the PFS-E method is an useful technique for finding solutions to other similar problems. Furthermore, some new types of solutions are revealed that will help us better understand the dynamic behaviors of the 2D-CNLS model.

1. Introduction

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 [15]. 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 [826].

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 [2932]:

iΦt+μ1(Φxx+Φyy)          +i(μ2(ΦΦx*+Φ*Φx)+μ3(ΦΦy*+Φ*Φy))Φ=0,    (1)

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 [2932]. 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.

2. Mathematical Analysis

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

Φ(x,y,t)=eiψΩ(ξ),    (2)

while the amplitude

ξ=α1x+β1y-λt,    (3)

and

ψ=α0x+β0y+λ0t+η0.    (4)

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

δ1Ω-δ2Ω3-δ3Ω=0,    (5)

where

δ1=μ1(α02+β02)+λ0,δ2=2(α0μ2+β0μ3),δ3=μ1(α12+β12),    (6)

and

(2μ1(α0α1+β0β1)-λ)Ωξ=0.    (7)

From (7), we have

λ=2μ1(α0α1+β0β1).    (8)

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

Ω=a0+a1ϕ(ξ),    (9)

where ϕ is given by the following auxiliary equation,

(dϕ(ξ)dξ)2=ζ0+ζ1ϕ(ξ)+ζ2ϕ2(ξ)+ζ3ϕ3(ξ)+ζ4ϕ4(ξ),    (10)

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

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

a03(-δ2)+a0δ1-12a1δ3ζ1=0,-3a1a02δ2+a1δ1-a1δ3ζ2=0,          -3a0a12δ2-32a1δ3ζ3=0,                    -a13δ2-2a1δ3ζ4=0.    (11)

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

ζ0=ζ0,ζ1=-2(a03δ1+3a0δ3)3a1δ2,ζ2=-a02δ1+δ3δ2,ζ3=-2a0a1δ13δ2,ζ4=-a12δ16δ2,    (12)

which give

a0=δ1-δ3ζ23δ2,a1=2-δ3ζ4δ2.    (13)

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

Φ1I(ξ)=[a0+ a1(-ϑ12-4ϑ2ϑ3tanh(12ξϑ12-4ϑ2ϑ3)+ϑ12ϑ2)]×eiψ.    (14)

A variety of bright-dark optical soliton is gained as

Φ3I(ξ)=[a0a12ϑ2(ϑ124ϑ2ϑ3                    (isech(ξϑ124ϑ2ϑ3)                   +tanh(ξϑ124ϑ2ϑ3))+ϑ1)]×eiψ.    (15)

A set of singular dark optical solitons is obtained as

Φ5I(ξ)=[a0a12ϑ2(ϑ124ϑ2ϑ3                     (tanh(14ξϑ124ϑ2ϑ3)                     +coth(14ξϑ124ϑ2ϑ3))+ϑ1)]×eiψ.    (16)

The family of solitons is obtained as

Φ10I(ξ)=[a0+a1(2cosh(ξϑ124ϑ2ϑ3)                      (ϑ124ϑ2ϑ3sinh(ξϑ124ϑ2ϑ3)                 (ϑ1cosh(ξϑ124ϑ2ϑ3)                  ±iϑ124ϑ2ϑ3))1)]×eiψ.    (17)

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

Φ13I(ξ)=[a0+a1(4ϑ2ϑ3ϑ12 tan(12ξ4ϑ2ϑ3ϑ12)ϑ12ϑ2)]×eiψ,    (18)
Φ20I(ξ)=[a0+a1(2ϑ3cos(12ξ4ϑ2ϑ3ϑ12)4ϑ2ϑ3ϑ12sin(12ξ4ϑ2ϑ3ϑ12)+ϑ1cos(12ξ4ϑ2ϑ3ϑ12))]×eiψ,    (19)
Φ24I(ξ)=[a0+a1((4rsin(14ξ4ϑ2ϑ3ϑ12)cos(14ξ4ϑ2ϑ3ϑ12))(24ϑ2ϑ3ϑ12cos2(14ξ4ϑ2ϑ3ϑ12)2ϑ1sin(14ξ4ϑ2ϑ3ϑ12)cos(14ξ4ϑ2ϑ3ϑ12)4ϑ2ϑ3ϑ12)1)]×eiψ.    (20)

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

Φ1II(ξ)=[a0+a1(6ϑ2ϑ3 tanh(12ξ6ϑ2ϑ3)+2ϑ2ϑ32ϑ2)]×eiψ.    (21)

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

Φ5II(ξ)=[a0+a1(6qr(tanh(14ξ6qr)+coth(14ξ6qr))+22qr4q)]×eiψ,    (22)

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.

Φ1III(ξ)=[a0+a1(λ1sech(ξ)λ2sech(ξ)+λ3)]×eiψ.    (23)

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

Φ2III(ξ)=[a0+a1(λ1csch(ξ)λ2csch(ξ)+λ3)]×eiψ.    (24)

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

Φ1III(ξ)=[a0+a1(λ1sech(ξ)λ3)]×eiψ.    (25)
Φ2III(ξ)=[a0+a1(λ1csch(ξ)λ3)]×eiψ.    (26)

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

Φ3III(ξ)=[a0+a1(λ1sech2(ξ)λ2tanh(ξ)+λ3+λ4sech2(ξ))]×eiψ.    (27)

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

Φ4III(ξ)=[a0+a1(λ1csch2(ξ)λ2coth(ξ)+λ3+λ4csch2(ξ))]×eiψ.    (28)

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

Φ4III(ξ)=[a0+a1(λ1csch2(ξ)λ2coth(ξ)+λ3+λ2csch2(ξ))]×eiψ.    (29)

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

Φ6III(ξ)=[a0+a1(λ1(sinh(λ1ξ)+cosh(λ1ξ))(sinh(λ1ξ)+cosh(λ1ξ)+λ2)λ3)]×eiψ.    (30)

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

Φ8III(ξ)=[a0+a1(λ1csc(ξ)λ2csc(ξ)+λ3)]×eiψ.    (31)

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

Φ9III(ξ)=[a0+a1(λ1sec2(ξ)λ2tan(ξ)+λ3+λ4sec2(ξ))]×eiψ.    (32)

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

Φ3IV(ξ)=[a0+a1(cnξ)]×eiψ,    (33)

leads to the bright optical soliton when m → 1,

Φ3IV(ξ)=[a0+a1sech(ξ)]×eiψ,    (34)

and the singular periodic solutions when m → 0,

Φ3IV(ξ)=[a0+a1cos(ξ)]×eiψ,    (35)

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

Φ13IV(ξ)=[a0+a1(nsξ±csξ)]×eiψ,    (36)

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

Φ13IV(ξ)=[a0+a1(coth(ξ)+csch(ξ))]×eiψ,     (37)

and singular periodic solutions when m → 0,

Φ13IV(ξ)=[a0+a1(cot(ξ)+csc(ξ))]×eiψ.    (38)

3. Results and Discussion

In this part, the physical aspects of the solutions obtained are discussed by means of graphical 3D representations. Figures 14 show different categories of background for soliton solutions classified into dark or singular soliton solutions.

FIGURE 1
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Figure 1. |Φ(x,y,t)|1I: 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.

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.

FIGURE 2
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Figure 2. |Φ(x,y,t)|5II: 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 3
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Figure 3. |Φ(x,y,t)|1III: 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.

FIGURE 4
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Figure 4. |Φ(x,y,t)|13IV for m → 1: 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 24 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.

4. Conclusions

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.

Author Contributions

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

Conflict of Interest

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.

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Keywords: 2D-CNLS equation, PFS-E algorithm, solitons, analytical solutions, waves structures

Citation: Osman MS, Baleanu D, Tariq KU-H, Kaplan M, Younis M and Rizvi STR (2020) Different Types of Progressive Wave Solutions via the 2D-Chiral Nonlinear Schrödinger Equation. Front. Phys. 8:215. doi: 10.3389/fphy.2020.00215

Received: 14 December 2019; Accepted: 20 May 2020;
Published: 07 July 2020.

Edited by:

Grienggrai Rajchakit, Maejo University, Thailand

Reviewed by:

Prasanta Panigrahi, Indian Institute of Science Education and Research Kolkata, India
Anouar Ben Mabrouk, University of Kairouan, Tunisia

Copyright © 2020 Osman, Baleanu, Tariq, Kaplan, Younis and Rizvi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: M. S. Osman, mofatzi@sci.cu.edu.eg; Dumitru Baleanu, dumitru@cankaya.edu.tr

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