Optical solitons in birefringent fibers with quadratic-cubic nonlinearity using three integration architectures

Gómez-Aguilar, J. F.; Osman, M. S.; Raza, Nauman; Zubair, Asad; Arshed, Saima; Ghoneim, Mohamed E.; Mahmoud, Emad E.; Abdel-Aty, Abdel-Haleem

doi:10.1063/5.0038038

In this work, the nonlinear Schrödinger’s equation is studied for birefringent fibers incorporating four-wave mixing. The improved $\tan (\frac{ϕ (ξ)}{2})$ -expansion, first integral, and $\frac{G^{'}}{G^{2}}$ -expansion methods are used to extract a novel class of optical solitons in the quadratic-cubic nonlinear medium. The extracted solutions are dark, periodic, singular, and dark-singular, along with other soliton solutions. These solutions are listed with their respective existence criteria. The recommended computational methods here are uncomplicated, outspoken, and consistent and minimize the computational work size, which give it a wide range of applicability. A detailed comparison with the results that already exist is also presented.

I. INTRODUCTION

Optical solitons are a valuable accretion in the field of fiber optic communications.^1–3 The nonlinear Schrödinger’s equation (NLSE) is the governing model that describes the propagation of optical solitons with different forms of nonlinear media.^4–14 The nonlinear media based on the Kerr law have been extensively studied through various research papers.^12–15 Nowadays, a growing interest to study optical solitons in the non-Kerr law medium can be observed. There are various forms of nonlinearities that are studied in the context of the non-Kerr law, i.e., polynomial law, parabolic law, power law, dual-power law, saturable law, triple power law, among others.^16–20 For more than a couple of decades, the study of optical solitons has been carried out with quadratic-cubic (QC) nonlinearity. This form of nonlinearity first appeared during 1994.²¹ Later, interest was rekindled with this model during 2011.²² There are several results with a variety of mathematical methods that are reported.^23–33 These include the traveling wave hypothesis, semi-inverse variational principle, method of undetermined coefficients, conservation laws, the unified method and its generalized technique, and various other aspects.^34–48

In this study, in the presence of birefringent fibers and the four-wave mixing effect, we consider the NLSE with quadratic-cubic nonlinearity (4WM). It is likely that birefringence exists in an optical waveguide. By splitting pulses into two, this phenomenon steers toward differential party latency. Birefringence implies the mutual reaction to this pause.

Refractive indices caused by orthogonally polarized materials vary significantly in the case of fiber birefringence in optical modes. Nonlinear structures are significantly affected by this refractive index, which plays a significant role in the field of fiber optics. If Bragg gratings are rendered within the center of polarization-maintaining fibers, their effects should be included. It is worthwhile extending the coupled-mode principle to account for fiber birefringence.^49–51 In this case, a series of four coupled equations comprising forward and backward propagating waves explain the evolution of two orthogonally polarized elements, which makes the topic very complicated. A major class of nonlinear phenomena of functional implementation, such as optical logic gates, is given by this ramification. To extract soliton solutions to the models, multiple integration schemes are available.^{17,18,52–54} The first integral approach is an exceptionally desirable and stable method for scrutinizing reliable evolution equation solutions since it copes fairly well with both integrable and non-integrable equations. Feng⁵⁵ introduced this method by solving the Burgers KdV equation. This approach is fundamental to the ring theory of commutative algebra and is used all over the world.^56,57

The $\frac{G^{'}}{G^{2}}$ -expansion method⁵⁸ is a novel approach for calculating the soliton solutions of single and combined nonlinear equations that exist in different fields of physics, fluid mechanics, problems of wave propagation, population dynamics, etc. Owing to its easy thinking and suitability, it has become very popular. In the form of hyperbolic, trigonometric, and rational functions, the solutions obtained using the approach proposed can be expressed.

The present model, however, has been already studied in the past with the extended trial function method,⁵⁹ extended Jacobi’s elliptic function expansion method,⁶⁰ (G′/G)-expansion method,⁶¹ and F-expansion method.⁶² The current paper will apply the improved $\tan (\frac{ϕ (ξ)}{2})$ -expansion method,⁶³ first integral method,⁵⁵ and $\frac{G^{'}}{G^{2}}$ -expansion method⁵⁸ to extract a novel class of soliton solutions to the model. Dark, singular, periodic, dark-singular, and some other solitons are the part of this novel class. After a quick intro to the model, the details are enumerated in the rest of the paper.

A. Governing model

The governing equation for the propagation of solitons through optical fibers with QC nonlinearity is given by the following dimensionless form of the NLSE:³⁸

i ψ_{t} + α ψ_{x x} + (β_{1} | ψ | + β_{2} | ψ |^{2}) ψ = 0 .

(1)

In Eq. (1), the independent variables are t and x that represent temporal and spatial variables, respectively. The dependent variable ψ(x, t) gives the complex valued wave profile. The coefficient of the real-valued constant α is group velocity dispersion (GVD). The two nonlinear terms are with coefficients β₁ and β₂, which are both real-valued constants. This leads to the formulation of the NLSE with QC nonlinearity. With birefringence, Eq. (1) splits into two components as follows:⁵⁹

\begin{align} i r_{t} + α_{1} r_{x x} + β_{1} r \sqrt{| r |^{2} + | s |^{2} + r s^{*} + r^{*} s} \\ + (μ_{1} | r |^{2} + γ_{1} | s |^{2}) r + τ_{1} s^{2} r^{*} = 0, \end{align}

(2)

\begin{align} i s_{t} + α_{2} s_{x x} + β_{2} s \sqrt{| s |^{2} + | r |^{2} + s r^{*} + s^{*} r} \\ + (μ_{2} | s |^{2} + γ_{2} | r |^{2}) s + τ_{2} r^{2} s^{*} = 0 . \end{align}

(3)

In Eqs. (2) and (3), α_j, for j = 1or2, gives GVD for the two components. The coefficients of β_j, the first two terms in the radical, account for self-phase modulation (SPM) and cross-phase modulation (XPM) terms, while the third and fourth terms stem from 4WM due to quadratic nonlinearity. Next, for the cubic nonlinear term, μ_j and ν_j are from SPM and XPM, respectively, while τ_j gives the effect of 4WM. With this complete picture to the model, the rest of the paper explores the mathematical scheme to enlist the solution spectrum.

II. MATHEMATICAL PRELIMINARIES

In order to solve Eqs. (2) and (3), we first adopt the following transformations:

r (x, t) = U_{1} (ξ) e^{i ϕ},

(4)

s (x, t) = U_{2} (ξ) e^{i ϕ},

(5)

where

ξ = x - v t,

(6)

while the phase component ϕ is given as

ϕ = - κ x + ω t + θ .

(7)

Here, v is the velocity of the soliton, κ refers to the frequency of the solitons in each of the two components while θ is the phase constant and ω is the soliton wave number. Substituting Eqs. (4) and (5) into Eqs. (2) and (3) and splitting into real and imaginary parts, respectively, yield

α_{l} U_{l}^{″} - (α_{l} κ^{2} + ω) U_{l} + β_{l} U_{l}^{2} + β_{l} U_{l} U_{\bar{l}} + μ_{l} U_{l}^{3} + (γ_{l} + τ_{l}) U_{l} U_{\bar{l}}^{2} = 0,

(8)

(2 α_{l} κ + v) U_{l}^{'} = 0,

(9)

for l = 1, 2 and $\bar{l} = 3 - l$ ⁠. From Eq. (9), the soliton speed is

v = - 2 α_{l} κ .

(10)

On comparing both soliton speed expressions, we get

α_{l} = α_{2} = α .

(11)

Thus, the speed of the solitons becomes

v = - 2 α κ .

(12)

Therefore, Eqs. (2) and (3) can be rewritten as

\begin{matrix} i r_{t} + α r_{x x} + β_{1} r \sqrt{| r |^{2} + | s |^{2} + r s^{*} + r^{*} s} \\ + (μ_{1} | r |^{2} + γ_{1} | s |^{2}) r + τ_{1} s^{2} r^{*} = 0, \\ i s_{t} + α s_{x x} + β_{1} s \sqrt{| s |^{2} + | r |^{2} + s r^{*} + s^{*} r} \\ + (μ_{1} | s |^{2} + γ_{1} | r |^{2}) s + τ_{1} r^{2} s^{*} = 0 . \end{matrix}

(13)

In this case, the real part (8) changes to

α U_{l}^{″} - (α κ^{2} + ω) U_{l} + β_{l} U_{l}^{2} + β_{l} U_{l} U_{\bar{l}} + μ_{l} U_{l}^{3} + (γ_{l} + τ_{l}) U_{l} U_{\bar{l}}^{2} = 0 .

(14)

Next, using the balancing principle, a relationship is acquired, which is given by

U_{l} = U_{\bar{l}} .

(15)

As a consequence, Eq. (16) turns into

α U_{l}^{″} - (α κ^{2} + ω) U_{l} + 2 β_{l} U_{l}^{2} + (μ_{l} + γ_{l} + τ_{l}) U_{l}^{3} = 0 .

(16)

III. EXTRACTION OF SOLITONS BY THE IMPROVED $t a n (\frac{ϕ (ξ)}{2})$ -EXPANSION METHOD

This section deals with the integration of Eqs. (2) and (3) by the aid of a mathematical tool called the improved $\tan (\frac{ϕ (ξ)}{2})$ -expansion method.⁶³ Suppose that Eq. (16) has a solution in terms of the $t a n (\frac{ϕ (ξ)}{2})$ function,

U (ξ) = \sum_{k = 0}^{m} A_{k} {[p + \tan (\frac{ϕ (ξ)}{2})]}^{k} + \sum_{k = 1}^{m} B_{k} {[p + \tan (\frac{ϕ (ξ)}{2})]}^{- k},

(17)

where A_k and B_k are constants to be determined, such that A_m ≠ 0, B_m ≠ 0, and ϕ(ξ) satisfies the following differential equation:

ϕ^{'} (ξ) = E \sin (ϕ (ξ)) + F \cos (ϕ (ξ)) + G .

(18)

Balancing U″ and U³ in Eq. (16), yields m = 1. Hence, Eq. (17) along with p = 0 takes the following form:

U (ξ) = A_{0} + A_{1} [\tan (\frac{ϕ (ξ)}{2})] + B_{1} {[\tan (\frac{ϕ (ξ)}{2})]}^{- 1} .

(19)

Here, the objective is to find the values of A₀, A₁, and B₁. In order to find these values, Eq. (19) is substituted into Eq. (16) and all the coefficients of ${(tan (\frac{ϕ}{2}))}^{n}$ are compared, where n = −3, −2, −1, 0, 1, 2, and 3 with zero providing the following set of algebraic equations:

μ_{l} {A_{1}}^{3} + τ_{l} {A_{1}}^{3} - α F A_{1} G + γ_{l} {A_{1}}^{3} + \frac{1}{2} α A_{1} G^{2} + \frac{1}{2} F^{2} α A_{1} = 0,

(20)

\begin{align} 3 γ_{l} A_{0} {A_{1}}^{2} + \frac{3}{2} α G A_{1} E + 3 μ_{l} A_{0} {A_{1}}^{2} + 3 τ_{l} A_{0} {A_{1}}^{2} \\ + 2 β_{l} {A_{1}}^{2} - \frac{3}{2} α F A_{1} E = 0, \end{align}

(21)

\begin{align} 4 β_{l} A_{0} A_{1} - \frac{1}{2} F^{2} α A_{1} + 3 γ_{l} {A_{0}}^{2} A_{1} + 3 τ_{l} {A_{0}}^{2} A_{1} + 3 μ_{l} {A_{0}}^{2} A_{1} \\ + \frac{1}{2} α A_{1} G^{2} + 3 τ_{l} {A_{1}}^{2} B_{1} + α E^{2} A_{1} + 3 μ_{l} {A_{1}}^{2} B_{1} \\ - ω A_{1} - α κ^{2} A_{1} + 3 γ_{l} {A_{1}}^{2} B_{1} = 0, \end{align}

(22)

\begin{align} 6 γ_{l} A_{0} A_{1} B_{1} - \frac{1}{2} α E B_{1} F + \frac{1}{2} α E B_{1} G + \frac{1}{2} α F A_{1} E + \frac{1}{2} α G A_{1} E \\ + μ_{l} {A_{0}}^{3} + 4 β_{l} A_{1} B_{1} + γ_{l} {A_{0}}^{3} + 2 β_{l} {A_{0}}^{2} + τ_{l} {A_{0}}^{3} - α κ^{2} A_{0} \\ + 6 μ_{l} A_{0} A_{1} B_{1} + 6 τ_{l} A_{0} A_{1} B_{1} - ω A_{0} = 0, \end{align}

(23)

\begin{align} 4 β_{l} A_{0} B_{1} - ω B_{1} + α E^{2} B_{1} + \frac{1}{2} α B_{1} G^{2} - \frac{1}{2} α B_{1} F^{2} + 3 γ_{l} A_{1} {B_{1}}^{2} \\ + 3 γ_{l} {A_{0}}^{2} B_{1} + 3 τ_{l} {A_{0}}^{2} B_{1} + 3 μ_{l} {A_{0}}^{2} B_{1} + 3 μ_{l} A_{1} {B_{1}}^{2} \\ - α κ^{2} B_{1} + 3 τ_{l} A_{1} {B_{1}}^{2} = 0, \end{align}

(24)

\begin{align} \frac{3}{2} α E B_{1} G + 3 τ_{l} A_{0} {B_{1}}^{2} + 2 β_{l} {B_{1}}^{2} + 3 γ_{l} A_{0} {B_{1}}^{2} \\ + \frac{3}{2} α E B_{1} F + 3 μ_{l} A_{0} {B_{1}}^{2} = 0, \end{align}

(25)

\frac{1}{2} α B_{1} F^{2} + \frac{1}{2} α B_{1} G^{2} + γ_{l} {B_{1}}^{3} + τ_{l} {B_{1}}^{3} + μ_{l} {B_{1}}^{3} + α F B_{1} G = 0 .

(26)

On solving the above-mentioned system of algebraic equations with the help of Maple, we get the subsequent cases for values of A₀, A₁, B₁, α, and ω.

Case 1:

\begin{gathered} A_{0} = - \frac{4}{3} \frac{β_{l}}{μ_{l} + γ_{l} + τ_{l}}, A_{1} = \frac{2}{3} \frac{(- G + F) β_{l}}{E (μ_{l} + γ_{l} + τ_{l})}, \\ B_{1} = - \frac{2}{3} \frac{(G + F) β_{l}}{E (μ_{l} + γ_{l} + τ_{l})}, α = - \frac{8}{9} \frac{{β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}}, \\ ω = - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} . \end{gathered}

(27)

Substituting the above-mentioned values in Eq. (19) and using the relation in Eq. (18) yield the following soliton solutions for systems (2) and (3).

When E² + F² − G² = −M² < 0 and F − G ≠ 0, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 + \frac{M}{E} \tan (\frac{M}{2} (x + 2 α κ t + C))) \\ (+ \frac{(F + G) (F - G)}{E^{2} - M E \tan (\frac{M}{2} (x + 2 α κ t + C))}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{(M^{2} + κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(28)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 + \frac{M}{E} \tan (\frac{M}{2} (x + 2 α κ t + C))) \\ (+ \frac{(F + G) (F - G)}{E^{2} - M E \tan (\frac{M}{2} (x + 2 α κ t + C))}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{(M^{2} + κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(29)

When E² + F² − G² = R² > 0 and F − G ≠ 0, then the subsequent dark-singular soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} - γ_{1} + τ_{1})} [1 + \frac{R}{E} \tanh (\frac{R}{2} (x + 2 α κ t + C))) \\ (+ \frac{(F + G) (F - G)}{E^{2} + R E \tanh (\frac{R}{2} (x + 2 α κ t + C))}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(R^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(30)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 - \frac{R}{E} \tanh (\frac{R}{2} (x + 2 α κ t + C))) \\ (+ \frac{(F + G) (F - G)}{E^{2} + E R \tanh (\frac{R}{2} (x + 2 α κ t + C))}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(R^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(31)

When E² + F² − G² > 0, F ≠ 0, and G = 0, then the subsequent dark-singular soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 - \frac{\sqrt{F^{2} + E^{2}}}{E}) \\ (\times \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))) \\ (+ \frac{F^{2}}{E} (\frac{1}{E + \sqrt{F^{2} + E^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(32)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 - \frac{\sqrt{F^{2} + E^{2}}}{E}) \\ (\times \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))) \\ (+ \frac{F^{2}}{E} (\frac{1}{E + \sqrt{F^{2} + E^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(33)

When E² + F² − G² < 0, G ≠ 0, and F = 0, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 + \frac{\sqrt{G^{2} - E^{2}}}{E}) \\ (\times \tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C)) + \frac{G^{2}}{E}) \\ (\times (\frac{1}{E - \sqrt{G^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(34)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 + \frac{\sqrt{G^{2} - E^{2}}}{E}) \\ \times \tan ((\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C)) +) \\ (\times \frac{G^{2}}{E} (\frac{1}{E - \sqrt{G^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(35)

When E² + F² = G², then the following rational soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [2 - \frac{E (x + 2 α κ t + C) + 2}{E (x + 2 α κ t + C)}) \\ (- \frac{E (x + 2 α κ t + C)}{E (x + 2 α κ t + C) + 2}] e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(36)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [2 - \frac{E (x + 2 α κ t + C) + 2}{E (x + 2 α κ t + C)}) \\ (- \frac{E (x + 2 α κ t + C)}{E (x + 2 α κ t + C) + 2}] e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(37)

When E = F = G = kE, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 + \frac{1}{e^{k E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) k^{2} E^{2}} t + θ)}, \end{align}

(38)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 + \frac{1}{e^{k E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) k^{2} E^{2}} t + θ)} . \end{align}

(39)

When E = G = kE and F = −kE, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 + \frac{e^{k E (x + 2 α κ t + C)}}{1 - e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) k^{2} E^{2}} t + θ)}, \end{align}

(40)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 + \frac{e^{k E (x + 2 α κ t + C)}}{1 - e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) k^{2} E^{2}} t + θ)} . \end{align}

(41)

When G = E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \\ (\times [2 +, \frac{(F - E)}{E}, (\frac{(E + F) e^{F (x + 2 α κ t + C)} - 1}{(E - F) e^{F (x + 2 α κ t + C)} - 1}))) \\ - (\frac{(F + E)}{E} (\frac{(E - F) e^{F (x + 2 α κ t + C)} - 1}{(E + F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(42)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \\ (\times [2 +, \frac{(F - E)}{E}, (\frac{(E + F) e^{F (x + 2 α κ t + C)} - 1}{(E - F) e^{F (x + 2 α κ t + C)} - 1}))) \\ (- \frac{(F + E)}{E} (\frac{(E - F) e^{F (x + 2 α κ t + C)} - 1}{(E + F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(43)

When G = −E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \\ (\times [2 -, \frac{(F - E)}{E}, (\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} + F - E}))) \\ (+ \frac{(F + E)}{E} (\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} + F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(44)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \\ (\times [2 -, \frac{(F - E)}{E}, (\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} + F - E}))) \\ (+ \frac{(F + E)}{E} (\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} + F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(45)

When F = −G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 - \frac{G e^{E (x + 2 α κ t + C)}}{e^{k E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(46)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 - \frac{G e^{E (x + 2 α κ t + C)}}{e^{k E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(47)

When F = 0 and E = G, then the subsequent rational soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [2 - \frac{G (x + 2 α κ t + C) + 2}{G (x + 2 α κ t + C)}) \\ (- \frac{G (x + 2 α κ t + C)}{G (x + 2 α κ t + C) + 2}] e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) G^{2}} t + θ)}, \end{align}

(48)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [2 - \frac{G (x + 2 α κ t + C) + 2}{G (x + 2 α κ t + C)}) \\ (- \frac{G (x + 2 α κ t + C)}{G (x + 2 α κ t + C) + 2}] e^{i (- κ x + \frac{8}{9} \frac{κ^{2} {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) G^{2}} t + θ)} . \end{align}

(49)

When F = G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [1 + \frac{G}{e^{E (x + 2 α κ t + C)} - F}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)}, \end{align}

(50)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [1 + \frac{G}{e^{E (x + 2 α κ t + C)} - F}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{(μ_{l} + γ_{l} + τ_{l}) E^{2}} t + θ)} . \end{align}

(51)

Case 2:

\begin{gathered} A_{0} = - \frac{2}{3} \frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}}) β_{l}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})}, \\ B_{1} = \frac{2}{3} \frac{(G + F) β_{l}}{\sqrt{F^{2} + E^{2} - G^{2}} (μ_{l} + γ_{l} + τ_{l})}, \\ A_{1} = 0, α = - \frac{8}{9} \frac{{β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})}, \\ ω = - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} . \end{gathered}

(52)

Substituting the above-mentioned values in Eq. (19) and using the relation in Eq. (18) yield the following soliton solutions for systems (2) and (3).

When E² + F² − G² < 0 and F − G ≠ 0, then the following singular periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})} - \frac{(F^{2} - G^{2})}{\sqrt{F^{2} + E^{2} - G^{2}}}) \\ (\times (\frac{1}{E - \sqrt{G^{2} - F^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - F^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(53)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})} - \frac{(F^{2} - G^{2})}{\sqrt{F^{2} + E^{2} - G^{2}}}) \\ (\times (\frac{1}{E - \sqrt{G^{2} - F^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - F^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(54)

When E² + F² − G² > 0 and F − G ≠ 0, then the subsequent singular soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})} - \frac{(F^{2} - G^{2})}{\sqrt{F^{2} + E^{2} - G^{2}}}) \\ (\times (\frac{1}{E + \sqrt{F^{2} + E^{2} - G^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2} - G^{2}}}{2} (x + 2 α κ t + C))})] e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(55)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})} - \frac{(F^{2} - G^{2})}{\sqrt{F^{2} + E^{2} - G^{2}}}) \\ (\times (\frac{1}{E + \sqrt{F^{2} + E^{2} - G^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2} - G^{2}}}{2} (x + 2 α κ t + C))})] e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(56)

When E² + F² − G² > 0, F ≠ 0, and G = 0, then the subsequent singular soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - E \sqrt{F^{2} + E^{2}})}{(F^{2} + E^{2})} - \frac{(F^{2})}{\sqrt{F^{2} + E^{2}}} (\frac{1}{E + \sqrt{F^{2} + E^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(57)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - E \sqrt{F^{2} + E^{2}})}{(F^{2} + E^{2})} - \frac{(F^{2})}{\sqrt{F^{2} + E^{2}}} (\frac{1}{E + \sqrt{F^{2} + E^{2}} \tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(58)

When E² + F² − G² < 0, G ≠ 0, and F = 0, then the following singular periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(E^{2} - G^{2} - E \sqrt{E^{2} - G^{2}})}{(E^{2} - G^{2})} + \frac{(G^{2})}{\sqrt{E^{2} - G^{2}}} (\frac{1}{E - \sqrt{G^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(59)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(E^{2} - G^{2} - E \sqrt{E^{2} - G^{2}})}{(E^{2} - G^{2})} + \frac{(G^{2})}{\sqrt{E^{2} - G^{2}}} (\frac{1}{E - \sqrt{G^{2} - E^{2}} \tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(60)

When E² + F² − G² > 0, F − G ≠ 0, and E = 0, then the subsequent singular soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \\ \times [1 - (\frac{1}{\tanh (\frac{\sqrt{F^{2} - G^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(61)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \\ \times [1 - (\frac{1}{\tanh (\frac{\sqrt{F^{2} - G^{2}}}{2} (x + 2 α κ t + C))})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(62)

When E = F = G = kE, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{1}{e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(k^{2} E^{2} - κ^{2}) {β_{l}}^{2}}{(K^{2} E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(63)

\begin{align} s (x, t) & = \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{1}{e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(k^{2} E^{2} - κ^{2}) {β_{l}}^{2}}{(K^{2} E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(64)

When G = E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - E}{F}) \\ + (\frac{F + E}{F}) ((\frac{(E - F) e^{F (x + 2 α κ t + C)} - 1}{(E + F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(65)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - E}{F}) \\ + (\frac{F + E}{F}) ((\frac{(E - F) e^{F (x + 2 α κ t + C)} - 1}{(E + F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(66)

When E = G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - G}{F}) \\ - (\frac{F + G}{F}) ((\frac{(F - G) e^{F (x + 2 α κ t + C)} - 1}{(F + G) e^{F (x + 2 α κ t + C)} + 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(67)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - G}{F}) \\ - (\frac{F + G}{F}) ((\frac{(F - G) e^{F (x + 2 α κ t + C)} - 1}{(F + G) e^{F (x + 2 α κ t + C)} + 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(68)

When G = −E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - E}{F}) \\ - (\frac{F - E}{F}) ((\frac{e^{F (x + 2 α κ t + C)} - F - E}{e^{F (x + 2 α κ t + C)} + F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(69)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - E}{F} - (\frac{F - E}{F})) \\ \times ((\frac{e^{F (x + 2 α κ t + C)} - F - E}{e^{F (x + 2 α κ t + C)} + F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(70)

When F = G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{G}{e^{E (x + 2 α κ t + C)} - F}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(71)

\begin{align} s (x, t) & = \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{G}{e^{E (x + 2 α κ t + C)} - F}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(72)

Case 3:

\begin{gathered} A_{0} = - \frac{2}{3} \frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}}) β_{l}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})}, \\ A_{1} = - \frac{2}{3} \frac{(F - G) β_{l}}{\sqrt{F^{2} + E^{2} - G^{2}} (μ_{l} + γ_{l} + τ_{l})}, \\ B_{1} = 0, α = - \frac{8}{9} \frac{{β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})}, \\ ω = - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} . \end{gathered}

(73)

Substituting the above-mentioned values in Eq. (19) and using the relation in Eq. (18) yield the following soliton solutions for systems (2) and (3).

When E² + F² − G² < 0 and F − G ≠ 0, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2} - G^{2}}} (E - \sqrt{G^{2} - F^{2} - E^{2}})) \\ (\times (\tan (\frac{\sqrt{G^{2} - F^{2} - E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(74)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2} - G^{2}}} (E - \sqrt{G^{2} - F^{2} - E^{2}})) \\ (\times (\tan (\frac{\sqrt{G^{2} - F^{2} - E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(75)

When E² + F² − G² > 0 and F − G ≠ 0, then the subsequent dark soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2} - G^{2}}} (E + \sqrt{F^{2} + E^{2} - G^{2}})) \\ (\times (\tanh (\frac{\sqrt{F^{2} + E^{2} - G^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(76)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - G^{2} - E \sqrt{F^{2} + E^{2} - G^{2}})}{(F^{2} + E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2} - G^{2}}} (E + \sqrt{F^{2} + E^{2} - G^{2}})) \\ (\times (\tanh (\frac{\sqrt{F^{2} + E^{2} - G^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(77)

When E² + F² − G² > 0, F ≠ 0, and G = 0, then the subsequent dark soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(F^{2} + E^{2} - E \sqrt{F^{2} + E^{2}})}{(F^{2} + E^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2}}} (E + \sqrt{F^{2} + E^{2}})) \\ (\times (\tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(78)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(F^{2} + E^{2} - E \sqrt{F^{2} + E^{2}})}{(F^{2} + E^{2})}) \\ (+ \frac{1}{\sqrt{F^{2} + E^{2}}} (E + \sqrt{F^{2} + E^{2}})) \\ (\times (\tanh (\frac{\sqrt{F^{2} + E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} + F^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} + E^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(79)

When E² + F² − G² < 0, G ≠ 0, and F = 0, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{(E^{2} - G^{2} - E \sqrt{E^{2} - G^{2}})}{(E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{E^{2} - G^{2}}} (E - \sqrt{G^{2} - E^{2}})) \\ (\times (\tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(80)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{(E^{2} - G^{2} - E \sqrt{E^{2} - G^{2}})}{(E^{2} - G^{2})}) \\ (+ \frac{1}{\sqrt{E^{2} - G^{2}}} (E - \sqrt{G^{2} - E^{2}})) \\ (\times (\tan (\frac{\sqrt{G^{2} - E^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(E^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(81)

When E² + F² − G² > 0, F − G ≠ 0, and E = 0, then the subsequent dark soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \\ \times [1 + (\tanh (\frac{\sqrt{F^{2} - G^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(82)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \\ \times [1 + (\tanh (\frac{\sqrt{F^{2} - G^{2}}}{2} (x + 2 α κ t + C)))] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - G^{2} - κ^{2}) {β_{l}}^{2}}{(F^{2} - G^{2}) (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(83)

When E = G = kE and F = −kE, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{e^{k E (x + 2 α κ t + C)}}{1 - e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(k^{2} E^{2} - κ^{2}) {β_{l}}^{2}}{K^{2} E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(84)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{e^{k E (x + 2 α κ t + C)}}{1 - e^{k E (x + 2 α κ t + C)}}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(k^{2} E^{2} - κ^{2}) {β_{l}}^{2}}{K^{2} E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(85)

When G = E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - E}{F} - (\frac{F - E}{F})) \\ \times ((\frac{(E + F) e^{F (x + 2 α κ t + C)} - 1}{(E - F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(86)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - E}{F} - (\frac{F - E}{F})) \\ (\times (\frac{(E + F) e^{F (x + 2 α κ t + C)} - 1}{(E - F) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(87)

When E = G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - G}{F} + (\frac{F - G}{F})) \\ \times ((\frac{(F + G) e^{F (x + 2 α κ t + C)} + 1}{(F - G) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(88)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - G}{F} + (\frac{F - G}{F})) \\ \times ((\frac{(F + G) e^{F (x + 2 α κ t + C)} + 1}{(F - G) e^{F (x + 2 α κ t + C)} - 1})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(89)

When G = −E, then the following periodic soliton solutions are obtained:

\begin{align} r (x, t) & = - \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{F - E}{F} + (\frac{E + F}{F})) \\ \times ((\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} - F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(90)

\begin{align} s (x, t) & = - \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{F - E}{F} + (\frac{E + F}{F})) \\ \times ((\frac{e^{F (x + 2 α κ t + C)} + F - E}{e^{F (x + 2 α κ t + C)} - F - E})] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(F^{2} - κ^{2}) {β_{l}}^{2}}{F^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(91)

When F = −G, then the subsequent periodic soliton solutions are obtained,

\begin{align} r (x, t) & = - \frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} [\frac{G e^{E (x + 2 α κ t + C)}}{G e^{E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)}, \end{align}

(92)

\begin{align} s (x, t) & = - \frac{4 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} [\frac{G e^{E (x + 2 α κ t + C)}}{G e^{E (x + 2 α κ t + C)} - 1}] \\ \times e^{i (- κ x - \frac{8}{9} \frac{(E^{2} - κ^{2}) {β_{l}}^{2}}{E^{2} (μ_{l} + γ_{l} + τ_{l})} t + θ)} . \end{align}

(93)

It is important to mention that all the above-mentioned solutions are valid for

μ_{1} + γ_{1} + τ_{1} > 0,

(94)

μ_{2} + γ_{2} + τ_{2} > 0 .

(95)

A. The first integral method

In order to solve Eq. (16) by using the first integral method we first introduce the following transformations:^55–57

U (ξ) = X (ξ), Y (ξ) = X^{'} (ξ) .

(96)

Equation (16) takes the following form:

α Y_{l}^{'} - (α κ^{2} + ω) X_{l} + 2 β_{l} X_{l}^{2} + (μ_{l} + γ_{l} + τ_{l}) X_{l}^{3} = 0 .

(97)

Equation (97) can be rewritten as

Y_{l}^{'} (ξ) = \frac{α κ^{2} + ω}{α} X_{l} - \frac{2 β_{l}}{α} X_{l}^{2} - \frac{μ_{l} + γ_{l} + τ_{l}}{α} X_{l}^{3} .

(98)

According to the first integral method, it is assumed that X_l(ξ) and Y_l(ξ) are non-trivial solutions of Eq. (98) and the polynomial $Q (X_{l}, Y_{l}) = \sum_{j = 0}^{m} a_{j} (X_{l}) Y_{l}^{j} (ξ)$ is an irreducible polynomial in the complex domain C[X_l, Y_l] such that

Q (X_{l} (ξ), Y_{l} (ξ)) = \sum_{j = 0}^{m} a_{j} (X_{l}) Y_{l}^{j} = 0,

(99)

where a_j(X_l)(j = 0, 1, 2, 3, …, m) are polynomials in X_l and a_m(X_l) ≠ 0. By the division theorem, there exists a polynomial [g(X_l) + h(X_l)Y_l] in the complex domain C[X_l, Y_l] such that

\frac{d Q}{d ξ} = \frac{\partial Q}{\partial X_{l}} \frac{d X_{l}}{d ξ} + \frac{\partial Q}{\partial Y_{l}} \frac{d Y_{l}}{d ξ} = [g (X_{l}) + h (X_{l}) Y_{l} (ξ)] \sum_{j = 0}^{m} a_{j} (X_{l}) Y_{l}^{j} .

(100)

For m = 1, Eq. (99) becomes

Q (X_{l}, Y_{l}) = a_{0} (X_{l}) + a_{1} (X_{l}) Y_{l} = 0 .

(101)

Equation (100) becomes

\begin{align} \frac{d a_{0} (X_{l})}{d X_{l}} Y_{l} + \frac{d a_{1} (X_{l})}{d X_{l}} Y_{l}^{2} + a_{1} (X_{l}) \\ \times [\frac{ω + a κ^{2}}{α} X_{l} - \frac{2 β_{l}}{α} X_{l}^{2} - \frac{μ_{l} + γ_{l} + τ_{l}}{α} X_{l}^{3}] \\ = a_{0} (X_{l}) g (X_{l}) + [a_{1} (X_{l}) g (X_{l}) + a_{0} (X_{l}) h (X_{l})] Y_{l} \\ + a_{1} (X_{l}) h (X_{l}) Y_{l}^{2} . \end{align}

(102)

Equating the coefficients of $Y_{l}^{j} (j = 0, 1, 2)$ on both sides of Eq. (102) gives

\begin{align} Y_{l}^{0} : a_{0} (X_{l}) g (X_{l}) = a_{1} (X_{l}) [\frac{ω + a κ^{2}}{α} X_{l} - \frac{2 β_{l}}{α} X_{l}^{2}) \\ (- \frac{μ_{l} + γ_{l} + τ_{l}}{α} X_{l}^{3}], \end{align}

(103)

Y_{l}^{1} : \frac{d a_{0} (X_{l})}{d X_{l}} = a_{1} (X_{l}) g (X_{l}) + a_{0} (X_{l}) h (X_{l}),

(104)

Y_{l}^{2} : \frac{d a_{1} (X_{l})}{d X_{l}} = a_{1} (X_{l}) h (X_{l}) .

(105)

Since a_j(X_l)(j = 0, 1) are assumed to be polynomials, if we choose h(X_l) = 0, Eq. (105) yields a₁(X_l) = constant. For simplicity, take a₁(X_l) = 1. Balancing the degrees of a₀(X_l) and g(X_l) gives a degree of g(X_l) = 1. Suppose that

g (X_{l}) = A X_{l} + B,

(106)

where A ≠ 0. From Eq. (104), we obtain

a_{0} (X_{l}) = \frac{A X_{l}^{2}}{2} + B X_{l} + C,

(107)

where C is a constant of integration. When substituting a₀(X_l), a₁(X_l) and g(X_l) in Eq. (103) and equating the coefficients of $X_{l}^{j} (j = 0, 1, 2, 3)$ on both sides, the system of nonlinear algebraic equations is obtained. After solving the system, the following values of constants are obtained:

A = \pm \frac{\sqrt{2} \sqrt{μ_{l} + γ_{l} + τ_{l}}}{\sqrt{- α}}, B = \frac{2 \sqrt{2} β_{l}}{3 \sqrt{- α} \sqrt{μ_{l} + γ_{l} + τ_{l}}}, C = 0 .

(108)

Using the above-mentioned values in Eq. (101), we obtain

Y (ξ) = \mp \frac{2 \sqrt{2} β_{l}}{3 \sqrt{- α} \sqrt{μ_{l} + γ_{l} + τ_{l}}} X_{l} \mp \frac{\sqrt{μ_{l} + γ_{l} + τ_{l}}}{\sqrt{- 2 α}} X_{l}^{2},

(109)

In order to solve Eq. (109), we refer to the Bernoulli equation,

v^{'} (ξ) = l_{1} v (ξ) + l_{2} v^{β} (ξ),

(110)

where l₁, l₂, and β are real integers and l₁l₂ ≠ 0 and β ≠ 1. Its general solution is of the form

v (ξ) = {[\frac{- l_{1} l_{2}}{ϵ_{0} \exp (l_{1} (1 - β) ξ) + 1}]}^{\frac{1}{β - 1}}

or

\begin{align} v (ξ) & = {[\frac{- l_{1}}{2 l_{2}} [1 + \tanh (\frac{l_{1} (1 - β)}{2} ξ - \frac{\ln ϵ_{0}}{2})]]}^{\frac{1}{β - 1}}, if ϵ_{0} > 0 \\ = {[\frac{- l_{1}}{2 l_{2}} [1 + \coth (\frac{l_{1} (1 - β)}{2} ξ - \frac{\ln (- ϵ_{0})}{2})]]}^{\frac{1}{β - 1}}, if ϵ_{0} < 0 \\ = {(- l_{1} l_{2})}^{\frac{1}{β - 1}}, if ϵ_{0} = 0 . \end{align}

(111)

If we choose $l_{1} = \mp \frac{2 \sqrt{2} β_{l}}{3 \sqrt{- α} \sqrt{μ_{l} + γ_{l} + τ_{l}}}$ ⁠, $l_{2} = \mp \frac{\sqrt{μ_{l} + γ_{l} + τ_{l}}}{\sqrt{- 2 α}}$ ⁠, and β = 2 in Eq. (111), then we get our required solutions.

If ϵ₀ > 0, the dark soliton solution is obtained as

\begin{align} r (x, t) & = [\frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\times [1 \pm \tanh (\frac{\sqrt{2} β_{1}}{3 \sqrt{- α} \sqrt{μ_{1} + γ_{1} + τ_{1}}} ξ - \frac{\ln ϵ_{0}}{2})]] \\ \times e^{i (- κ x + ω t + θ)}, \end{align}

(112)

\begin{align} s (x, t) & = [\frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\times [1 \pm \tanh (\frac{\sqrt{2} β_{2}}{3 \sqrt{- α} \sqrt{μ_{2} + γ_{2} + τ_{2}}} ξ - \frac{\ln ϵ_{0}}{2})]] \\ \times e^{i (- κ x + ω t + θ)} . \end{align}

(113)

If ϵ₀ < 0, we obtain the singular soliton solution,

\begin{align} r (x, t) & = [\frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\times [1 \pm \coth (\frac{\sqrt{2} β_{1}}{3 \sqrt{- α} \sqrt{μ_{1} + γ_{1} + τ_{1}}} ξ - \frac{\ln (- ϵ_{0})}{2})]] \\ \times e^{i (- κ x + ω t + θ)}, \end{align}

(114)

\begin{align} s (x, t) & = [\frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\times [1 \pm \coth (\frac{\sqrt{2} β_{2}}{3 \sqrt{- α} \sqrt{μ_{2} + γ_{2} + τ_{2}}} ξ - \frac{\ln (- ϵ_{0})}{2})]] \\ \times e^{i (- κ x + ω t + θ)} . \end{align}

(115)

If ϵ₀ = 0, we obtain the plane wave solution,

r (x, t) = [\frac{4 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}] e^{i (- κ x + ω t + θ)},

(116)

s (x, t) = [\frac{4 β_{2}}{3 (μ_{2} + γ_{l} + τ_{2})}] e^{i (- κ x + ω t + θ)} .

(117)

The constraint conditions for the existence of soliton solutions obtained by the first integral method are α < 0, μ₁ + γ₁ + τ₁ > 0, and μ₁ + γ₁ + τ₁ > 0.

B. $\frac{G^{'}}{G^{2}}$ -Expansion method

In this section, the $\frac{G^{'}}{G^{2}}$ -expansion method⁵⁸ is employed to solve Eq. (16). According to this method, the traveling wave solution can be expressed as

U (ξ) = a_{0} + \sum_{n = 1}^{N} (a_{n} {(\frac{G^{'}}{G^{2}})}^{n} + b_{n} {(\frac{G^{'}}{G^{2}})}^{- n}),

(118)

where G = G(ξ) satisfies

{(\frac{G^{'}}{G^{2}})}^{'} = ϵ + δ {(\frac{G^{'}}{G^{2}})}^{2},

(119)

in which δ ≠ 0 and ϵ ≠ 1 are integers. The unknown constants a₀, a_n, and b_n (n = 1, 2, 3, …, N) are to be determined. Balancing the terms U″_l and $U_{l}^{3}$ in Eq. (16) by using the homogeneous principle yields N = 1. As a result, Eq. (118) takes the form

U (ξ) = a_{0} + a_{1} (\frac{G^{'}}{G^{2}}) + b_{1} {(\frac{G^{'}}{G^{2}})}^{- 1} .

(120)

Now putting Eq. (120) into Eq. (16) and then comparing the coefficients of same powers of ${(\frac{G^{'}}{G^{2}})}^{j}$ ⁠, (j = 0, ±1, ±2, ±3, ±4), to 0 provide a system of algebraic equations. The following set of solutions are retrieved on solving the system of algebraic equations.

Set 1:

a_{0} = - \frac{2 β_{l}}{3 (μ_{l} + γ_{l} + τ_{l})}, a_{1} = 0, b_{1} = \pm \frac{\sqrt{2} \sqrt{- α} ϵ}{\sqrt{μ_{l} + γ_{l} + τ_{l}}},

δ = \frac{2 β_{l}^{2}}{9 α (μ_{l} + γ_{l} + τ_{l}) ϵ}, ω = - α κ^{2} - \frac{8 β_{l}^{2}}{9 (μ_{l} + γ_{l} + τ_{l})} .

Set 2:

a_{0} = - \frac{2 β_{l}}{3 (μ_{l} + γ_{l} + τ_{l})}, a_{1} = \pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{l} + γ_{l} + τ_{l}}}, b_{1} = 0,

ϵ = \frac{2 β_{l}^{2}}{9 α (μ_{l} + γ_{l} + τ_{l}) δ}, ω = - α κ^{2} - \frac{8 β_{l}^{2}}{9 (μ_{l} + γ_{l} + τ_{l})} .

Set 3:

\begin{gathered} a_{0} = - \frac{2 β_{l}}{3 (μ_{l} + γ_{l} + τ_{l})}, a_{1} = \pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{l} + γ_{l} + τ_{l}}}, \\ b_{1} = \pm \frac{β_{l}^{2}}{9 \sqrt{2} \sqrt{- α} \sqrt{{(μ_{l} + γ_{l} + τ_{l})}^{3}} δ}, \end{gathered}

ϵ = \frac{β_{l}^{2}}{18 α (μ_{l} + γ_{l} + τ_{l}) δ}, ω = - α κ^{2} - \frac{8 β_{l}^{2}}{9 (μ_{l} + γ_{l} + τ_{l})} .

According to set 1, the subsequent solutions are retrieved.

If ϵδ > 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{1} + γ_{1} + τ_{1}} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}], \end{align}

(121)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{2} + γ_{2} + τ_{2}} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}] . \end{align}

(122)

If ϵδ < 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \pm \frac{\sqrt{2} \sqrt{- α} ϵ}{\sqrt{μ_{1} + γ_{1} + τ_{1}}}) \\ (\times {(- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))}^{- 1}], \end{align}

(123)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \pm \frac{\sqrt{2} \sqrt{- α} ϵ}{\sqrt{μ_{2} + γ_{2} + τ_{2}}}) \\ (\times {(- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))}^{- 1}] . \end{align}

(124)

Upon choosing P = M, the dark soliton solution is obtained as

\begin{align} r (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α} \tanh [\sqrt{| δ ϵ |} ξ]}{\sqrt{μ_{1} + γ_{1} + τ_{1}}}], \end{align}

(125)

\begin{align} s (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α} \tanh [\sqrt{| δ ϵ |} ξ]}{\sqrt{μ_{2} + γ_{2} + τ_{2}}}] . \end{align}

(126)

If ϵ = 0, δ ≠ 0, the plane wave solution is obtained as

r (x, t) = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}],

(127)

s (x, t) = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}] .

(128)

According to set 2, the following solutions are retrieved.

If ϵδ > 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{1} + γ_{1} + τ_{1}} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}], \end{align}

(129)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{2} + γ_{2} + τ_{2}} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}] . \end{align}

(130)

If ϵδ < 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{1} + γ_{1} + τ_{1}}}) \\ (\times (- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))], \end{align}

(131)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{2} + γ_{2} + τ_{2}}}) \\ (\times (- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))] . \end{align}

(132)

To obtain the soliton solution, P = M is chosen; we get the singular soliton as

\begin{align} r (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α} \coth [\sqrt{| δ ϵ |} ξ]}{\sqrt{μ_{1} + γ_{1} + τ_{1}}}], \end{align}

(133)

\begin{align} s (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α} \coth [\sqrt{| δ ϵ |} ξ]}{\sqrt{μ_{2} + γ_{2} + τ_{2}}}] . \end{align}

(134)

If ϵ = 0, δ ≠ 0, the plane wave solution is obtained as

r (x, t) = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}],

(135)

s (x, t) = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}] .

(136)

According to set 3, the subsequent solutions are retrieved.

If ϵδ > 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})} \pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{1} + γ_{1} + τ_{1}} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}) \\ (\pm \frac{β_{1}^{2}}{9 \sqrt{2 δ ϵ} \sqrt{- α} \sqrt{{(μ_{1} + γ_{1} + τ_{1})}^{3}}} \frac{(M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}{(P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}], \end{align}

(137)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})} \pm \frac{\sqrt{2 δ ϵ} \sqrt{- α} (P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}{\sqrt{μ_{2} + γ_{2} + τ_{2}} (M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}) \\ (\pm \frac{β_{2}^{2}}{9 \sqrt{2 δ ϵ} \sqrt{- α} \sqrt{{(μ_{2} + γ_{2} + τ_{2})}^{3}}} \frac{(M \cos [\sqrt{δ ϵ} ξ] - P \sin [\sqrt{δ ϵ} ξ])}{(P \cos [\sqrt{δ ϵ} ξ] + M \sin [\sqrt{δ ϵ} ξ])}] . \end{align}

(138)

If ϵδ < 0,

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{1} + γ_{1} + τ_{1}}} (- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))) \\ (\pm \frac{β_{1}^{2}}{9 \sqrt{2} \sqrt{- α} \sqrt{{(μ_{1} + γ_{1} + τ_{1})}^{3}} δ} {(- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))}^{- 1}], \end{align}

(139)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\pm \frac{\sqrt{2} \sqrt{- α} δ}{\sqrt{μ_{2} + γ_{2} + τ_{2}}} (- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))) \\ (\pm \frac{β_{l}^{2}}{9 \sqrt{2} \sqrt{- α} \sqrt{{(μ_{2} + γ_{2} + τ_{2})}^{3}} δ} {(- \frac{\sqrt{| ϵ δ |}}{δ} (\frac{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) + M}{P \sinh (2 \sqrt{| ϵ δ |} ξ) + P \cosh (2 \sqrt{| ϵ δ |} ξ) - M}))}^{- 1}] . \end{align}

(140)

Upon choosing P = M, the dark singular combo soliton solution is obtained as

\begin{align} r (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α}}{\sqrt{μ_{1} + γ_{1} + τ_{1}}} \coth [\sqrt{| δ ϵ |} ξ]) \\ (\mp \frac{β_{1}^{2}}{9 \sqrt{2 | δ ϵ |} \sqrt{- α} \sqrt{{(μ_{1} + γ_{1} + τ_{1})}^{3}}} \tanh [\sqrt{| δ ϵ |} ξ]], \end{align}

(141)

\begin{align} s (x, t) & = e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\mp \frac{\sqrt{2 | δ ϵ |} \sqrt{- α}}{\sqrt{μ_{l} + γ_{l} + τ_{l}}} \coth [\sqrt{| δ ϵ |} ξ]) \\ (\mp \frac{β_{2}^{2}}{9 \sqrt{2 | δ ϵ |} \sqrt{- α} \sqrt{{(μ_{2} + γ_{2} + τ_{2})}^{3}}} \tanh [\sqrt{| δ ϵ |} ξ]] . \end{align}

(142)

If ϵ = 0, δ ≠ 0, the plane wave solution is obtained as

\begin{align} r (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{1}}{3 (μ_{1} + γ_{1} + τ_{1})}) \\ (\mp \frac{\sqrt{2} \sqrt{- α} P}{\sqrt{μ_{1} + γ_{1} + τ_{1}} (M + P ξ)}) \\ (\mp \frac{β_{1}^{2} (M + P ξ)}{9 \sqrt{2} \sqrt{- α} P {\sqrt{(μ_{1} + γ_{1} + τ_{1})}}^{3}}], \end{align}

(143)

\begin{align} s (x, t) & = - e^{i (- κ x + ω t + θ)} [- \frac{2 β_{2}}{3 (μ_{2} + γ_{2} + τ_{2})}) \\ (\mp \frac{\sqrt{2} \sqrt{- α} P}{\sqrt{μ_{2} + γ_{2} + τ_{2}} (M + P ξ)}) \\ (\mp \frac{β_{2}^{2} (M + P ξ)}{9 \sqrt{2} \sqrt{- α} P {\sqrt{(μ_{2} + γ_{2} + τ_{2})}}^{3}}] . \end{align}

(144)

The constraint conditions for the existence of soliton solutions given above using the $\frac{G^{'}}{G^{2}}$ –expansion method are α < 0, μ₁ + γ₁ + τ₁ > 0, and μ₂ + γ₂ + τ₂ > 0.

IV. COMPARISON

This section deals with the comparison of obtained results with some previously known ones. A detailed analysis is made on the basis of different values of parameters E, F, and G, which is represented through equation numbers of the present article and already exists in the literature, as shown in Table I.

TABLE I.

Comparison of current paper solutions with already existing solutions.

		Solutions by the	Solutions by the	Solutions by the
Parametric values	Solutions of current paper	JEF-expansion method⁶⁰	$(\frac{G^{'}}{G})$ method⁶¹	F-expansion method⁶²
F = 2, E, G, C = 0	(76), (77), (78), (79), (82), (83)	(29), (30)	(48), (49)	(84), (85)
F = 2, E, G, C = 0	(55), (56), (57), (58), (61), (62)	(31), (32)	(52), (53)	(88), (89)
F = 4, E, G, C = 0	(55), (56), (57), (58), (61), (62)	(33), (34)	…	…
G = 2, E, F, C = 0	(53), (54), (59), (60)	…	(62), (63)	(120), (121)
G = 2, E, F, C = 0	(74), (75), (80), (81)	…	(58), (59)	(118), (119)
F = 1, E, G, C = 0	(55), (56), (57), (58), (61), (62)	…	…	(108), (109)
F = 1, E, G, C = 0	(76), (77), (78), (79), (82), (83)	…	…	(110), (111)

		Solutions by the	Solutions by the	Solutions by the
Parametric values	Solutions of current paper	JEF-expansion method⁶⁰	$(\frac{G^{'}}{G})$ method⁶¹	F-expansion method⁶²
F = 2, E, G, C = 0	(76), (77), (78), (79), (82), (83)	(29), (30)	(48), (49)	(84), (85)
F = 2, E, G, C = 0	(55), (56), (57), (58), (61), (62)	(31), (32)	(52), (53)	(88), (89)
F = 4, E, G, C = 0	(55), (56), (57), (58), (61), (62)	(33), (34)	…	…
G = 2, E, F, C = 0	(53), (54), (59), (60)	…	(62), (63)	(120), (121)
G = 2, E, F, C = 0	(74), (75), (80), (81)	…	(58), (59)	(118), (119)
F = 1, E, G, C = 0	(55), (56), (57), (58), (61), (62)	…	…	(108), (109)
F = 1, E, G, C = 0	(76), (77), (78), (79), (82), (83)	…	…	(110), (111)

In Table I, the first column shows different values of E, F, and G, and the second column represents the equation numbers of the concerned article; third, fourth, and fifth columns denote the equation numbers of the articles.^60–62 It is observed that for different values of E, F, and G, most of our results match exactly the solutions^60–62 mentioned in Table I, which shows the effectiveness of our results as a generalized case of the mentioned published results. The values of E, F, and G, which are taken from Table I, satisfy their corresponding constraints.

From Table I, we find that the techniques implemented in this study, over all the other methods, provide further new computational solutions, including additional free parameters. Most of the obtained solutions in the literature are taken via these applied approaches as a particular case, and we receive some new solutions as well.

Remarks:

To the best of our knowledge, the obtained results in this article are new, except those mentioned in Table I.
The CPU time of the computation is ∼0.2189 s.

V. CONCLUSION

This paper reveals a plethora of solutions to the coupled NLSE in birefringent fibers with 4WM incorporating QC nonlinearity. The improved $\tan (\frac{ϕ (ξ)}{2})$ -expansion method, first integral method, and $\frac{G^{'}}{G^{2}}$ -expansion method are used to procure these soliton solutions. Dark, periodic, singular, dark-singular, and a plenty of other soliton solutions have been successfully yielded through this process. A detailed comparison of solutions with the already published results reveals that our techniques are not only reliable but also fruitful as they provide us with a bunch of new solutions. The results being reported in this work are an excellent addition to the existing literature. In the near future, we will modify the algorithms presented here to deal with different NLEEs when their coefficients are variables, for exhaling nonautonomous wave solutions.

AUTHORS’ CONTRIBUTIONS

All authors contributed equally to this work. All authors read and approved the final manuscript.

All persons who meet authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, design, analysis, writing, or revision of the manuscript. Furthermore, each author certifies that this material or similar material has not been and will not be submitted to or published in any other publication before.

ACKNOWLEDGMENTS

Emad E. Mahmoud acknowledges Taif University Researchers Supporting Project No. (TURSP-2020/20), Taif University, Taif, Saudi Arabia. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.

The authors declare that they have no competing interests.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Author(s)

Optical solitons in birefringent fibers with quadratic-cubic nonlinearity using three integration architectures

I. INTRODUCTION

A. Governing model

II. MATHEMATICAL PRELIMINARIES

III. EXTRACTION OF SOLITONS BY THE IMPROVED $t a n (\frac{ϕ (ξ)}{2})$ -EXPANSION METHOD

A. The first integral method

B. $\frac{G^{'}}{G^{2}}$ -Expansion method

IV. COMPARISON

V. CONCLUSION

AUTHORS’ CONTRIBUTIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

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Optical solitons in birefringent fibers with quadratic-cubic nonlinearity using three integration architectures

I. INTRODUCTION

A. Governing model

II. MATHEMATICAL PRELIMINARIES

III. EXTRACTION OF SOLITONS BY THE IMPROVED tanϕ(ξ)2-EXPANSION METHOD

A. The first integral method

B. G′G2-Expansion method

IV. COMPARISON

V. CONCLUSION

AUTHORS’ CONTRIBUTIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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III. EXTRACTION OF SOLITONS BY THE IMPROVED $t a n (\frac{ϕ (ξ)}{2})$ -EXPANSION METHOD

B. $\frac{G^{'}}{G^{2}}$ -Expansion method