1. Introduction

jamp

Journal of Applied Mathematics and Physics

2327-4352 2327-4379

Scientific Research Publishing

10.4236/jamp.2025.1310194

jamp-146840

Articles

Physics Mathematics

Nonlinear Propagation of Coupled Pulses in Birefringent Optical Fibers

Mathurin

Fomekong

¹ Marcelle Nina Zambo

Abou’ou

¹ ² Oriel Loh

Ndichia

¹ Jean Roger

Bogning

² ³

aDepartment of Physics, Faculty of Science, University of Bamenda, Bamenda, Cameroon

aAfrican Optical Fiber Family, Camtel Kamkop, Bafoussam, Cameroon

aDepartment of Physics, Higher Teacher Training College, University of Bamenda, Bamenda, Cameroon

10 10 2025

13 10 3393 3430 19, August 2025 27, August 2025 27, October 2025

2014

This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

The purpose of this work is to study the nonlinear propagation of coupled pulses within birefringent optical fibers. To achieve this, an effective analytical technique using iB-function was applied. With this approach, we decoupled the nonlinear partial differential equations governing the wave propagation and solved them to obtain prototype solutions likely to propagate in the medium. A Mathlab program was then developed to validate the behavior of the propagated waves and analyze the resulting curves. The results revealed distinct complex phenomena such as stable propagation of coupled dark-bright vector soliton, energy exchange between the components during propagation to severe pulse broadening and conditions leading to group velocity walk-off. These findings provided a deeper insight into light propagation, improving its application for advanced optical communication systems and fiber-based devices.

Nonlinear Propagation Birefringent Optical Fiber Coupled Pulses iB-Function Coupled Nonlinear Partial Differential Equations

1. Introduction

Over time, optical fiber has progressively proven itself as the main and leading tool of modern communication. Universally, it has gained the attention of many researchers, as evidenced by the increase in various studies and works in this area [1] - [11] . It provides various possibilities in the field of information transport and signal transmission [12] - [16] . It is also being used more frequently in medicine, particularly in the field of Fibroscopy [17] - [20] . The advantages are multiple in the industry of new communication technologies. In this work, we align ourselves with the same dynamic to focus on the nonlinear propagation of coupled pulses within birefringent optical fiber. The general observation is that the wave propagation dynamics in optical fibers are often governed by highly complicated nonlinear partial differential equations, which are challenging to solve. While this is true for single-mode optical fiber, it is even more complex in birefringent optical fiber, which has the distinctive characteristic of propagating two signals simultaneously. As light pulses propagate through this fiber, nonlinear interactions can lead to complex phenomena, including pulse shaping, energy exchange, and phase modulation. Recent studies have examined several analytical and numerical approaches to model these relations, providing an understanding of the underlying mechanisms controlling pulse dynamics [21] - [32] . Consequently, the work done in this article consists of the nonlinear propagation of coupled pulses within birefringent optical fiber. All of this involves the outline of a scientific program aimed at facilitating this propagation. Nevertheless, before writing the program, it is essential to first obtain analytical sequences that are solutions to the equations influencing the wave propagation in the birefringent fiber (dispersive and strongly nonlinear coupled Schrödinger equations). Thus, the work initially consists of analytically determining prototype solutions of the coupled partial differential Schrödinger-type equations, using these solutions to initialize the carefully devised numerical program, and subsequently making observations and comments. It should be noted that solving coupled equations, which is crucial to this work, is not easy, especially when using direct integration methods, so the iB-functions were selected along with ansatz solutions to decouple these equations in order to obtain the solutions [32] - [35] . This choice is motivated by the fact that its analytical sequence embodies several types of waves, namely solitary waves and other types of waves and this fact imposes a strong mathematical content. The work here is organized as follows: In Section 2, a brief overview of the iB-functions is provided (methods). Section 3 focuses on seeking signal or prototype solutions. Section 4 contains detailed analysis and comments related to the curves obtained through Mathlab program and at the end of this article, there are reviews of the details of the work performed.

2. iB-Functions 2.1. Presentation of iB-Functions

iB-functions are versatile mathematical tools with diverse properties, widely applied to solve nonlinear problems in physics and mathematics. These functions exist in two distinct forms: a primary form and a secondary form, with the primary form being hyperbolic form and the secondary form being trigonometric form.

The primary form that will be widely used in this work is defined by $J_{n, m} (\sum_{i = 0}^{p} a_{i} x_{i}) = \frac{\sinh^{m} (\sum_{i = 0}^{p} a_{i} x_{i})}{\cosh^{n} (\sum_{i = 0}^{p} a_{i} x_{i})}$ , where $J_{n, m} (\sum_{i = 0}^{p} a_{i} x_{i})$ represent the implicit form of the function, $\sinh^{m} (\sum_{i = 0}^{p} a_{i} x_{i}) / \cosh^{n} (\sum_{i = 0}^{p} a_{i} x_{i})$ the explicit form of the function, $a_{i} (i = 0, 1, 2, \dots, p)$ represent the parameters associated to the independent variable $x_{i} (i = 0, 1, 2, \dots, p)$ , the couple $(n, m) \in R^{2}$ indicated the power of the function. In a more precise way, $n$ is the power of $\cosh (\sum_{i = 0}^{p} a_{i} x_{i})$ and $m$ is the power of $\sinh (\sum_{i = 0}^{p} a_{i} x_{i})$ .

In practice, the iB-function with one variable is mostly used to establish its properties. It is defined by

$J_{n, m} (α x) = \frac{\sinh^{m} (α x)}{\cosh^{n} (α x)} .$ (1)

Equation (1) obeys the transformation

$J_{n, m} (i α x) = i^{m} T_{n, m} (α x),$ (2)

where $T_{n, m} (α x) = \frac{\sin^{m} (α x)}{\cos^{n} (α x)}$ , with $i^{2} = - 1$ and $T_{n, m} (α x)$ , the secondary form of the iB-function, so called trigonometric form.

<xref ref-type="bibr" rid="scirp.146840-"></xref>2.2. Derivatives Properties of the Function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi> J </mi> </mstyle> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi> n </mi> </mstyle> <mo> , </mo> <mstyle mathvariant="bold" mathsize="normal"> <mi> m </mi> </mstyle> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mstyle mathvariant="bold" mathsize="normal"> <mi> x </mi> </mstyle> </mrow> <mo> ) </mo> </mrow> </mrow> </math>

Common properties of the derivatives include

$\frac{d J_{n, m}}{d x} = m α J_{n - 1, m - 1} - n α J_{n + 1, m + 1},$ (3)

$\begin{matrix} \frac{d^{2} J_{n, m}}{d x^{2}} = a m [(m - 1) J_{n - 2, m - 2} - (n - 1) J_{n, m}] \\ - a n [(m + 1) J_{n, m} - (n + 1) J_{n + 2, m + 2}], \end{matrix}$ (4)

If the function $J_{n, m} (α x)$ is a function of single variable, the derivative at other $k$ of $J_{n, m} (α x)$ with the respect to the variable is given by

$\frac{d^{k} J_{n, m}}{d x^{k}} = m α \frac{d^{k - 1} J_{n - 1, m - 1}}{d x^{k - 1}} - n α \frac{d^{k - 1} J_{n + 1, m + 1}}{d x^{k - 1}} .$ (5)

For the function of parameter $J_{n, m} (α x)$ and independent variable $x$ for the given reals $n, m$ and $p$ , we have the following relations:

${J^{'}}_{n + p, m + p} = {J^{'}}_{2 p, 2 p} J_{n - p, m - p} + J_{2 p, 2 p} {J^{'}}_{n - p, m - p},$ (6)

${J^{'}}_{n + p, m + p} J_{n - p, m - p} - {J^{'}}_{n - p, m - p} J_{n + p, m + p} = J_{2 p, 2 p} {J^{'}}_{n - p, m - p}^{2},$ (7)

and

$\frac{{J^{'}}_{2 p, 2 p}}{J_{2 p, 2 p}} = \frac{{J^{'}}_{n - p, m - p}^{2}}{J_{n - p, m - p}^{2}} .$ (8)

<xref ref-type="bibr" rid="scirp.146840-"></xref>2.3. Some Properties Linking the Functions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi> J </mi> </mstyle> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi> n </mi> </mstyle> <mo> , </mo> <mstyle mathvariant="bold" mathsize="normal"> <mi> m </mi> </mstyle> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi> i </mi> <mi> x </mi> </mstyle> </mrow> <mo> ) </mo> </mrow> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi> T </mi> </mstyle> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi> n </mi> </mstyle> <mo> , </mo> <mstyle mathvariant="bold" mathsize="normal"> <mi> m </mi> </mstyle> </mrow> </msub> <mrow> <mo> ( </mo> <mstyle mathvariant="bold" mathsize="normal"> <mi> x </mi> </mstyle> <mo> ) </mo> </mrow> </mrow> </math>

$J_{2 n, 2 n} (i x) = {(- 1)}^{n} T_{2 n, 2 n} (x),$ (9)

$J_{n p, m p} (i x) = i^{m p} T_{n p, m p} (x), n, m, p \in R$ (10)

and

$| J_{n, m} (i x) | = | T_{n, m} (x) | .$ (11)

2.4. Some Usual Transformations

The simple implicit Bogning functions are functions of the form $J_{n, 0}$ and $J_{n, 1}$ , $n \in R$ . They are known as simple because, they do not admit any more fractional transformation. Some usual properties of the simple implicit Bogning functions are given as follows:

$J_{2 n + 2, 2} = J_{2 n, 0} - J_{2 n + 2, 0},$ (12)

$J_{n + 3, 3} = J_{n + 1, 1} - J_{n + 3, 1},$ (13)

$J_{2 n + 3, 3} = J_{2 n + 1, 1} - J_{2 n + 3, 1},$ (14)

$J_{2 n + 1, 2 n + 1} = J_{1, 1} {(1 - J_{2, 0})}^{n},$ (15)

and

$J_{n, m} + J_{n, m}^{2} + J_{n, m}^{3} + \dots + J_{n, m}^{p} = \frac{J_{n, m} - J_{n, m}^{p + 1}}{1 - J_{n, m}} = \frac{J_{n, m} - J_{n (p + 1), m (p + 1)}}{1 - J_{n, m}} .$ (16)

3. Forms of Solutions and Solving the Equation

The coupled nonlinear partial differential equations that govern the propagation of solitary waves in birefringent optical fiber is given by

$\frac{\partial U}{\partial z} + β_{1} \frac{\partial U}{\partial t} + i \frac{β_{2}}{2} \frac{\partial^{2} U}{\partial t^{2}} + \frac{α_{1}}{2} U - i γ ({| U |}^{2} + B {| V |}^{2}) U = 0,$ (17)

$\frac{\partial V}{\partial z} + β_{3} \frac{\partial V}{\partial t} + i \frac{β_{4}}{2} \frac{\partial^{2} V}{\partial t^{2}} + \frac{α_{2}}{2} V - i γ ({| V |}^{2} + B {| U |}^{2}) V = 0,$ (18)

where, $U (z, t)$ and $V (z, t)$ are the envelop of the wave, $β_{1}$ and $β_{3}$ are the coefficients of chromatic dispersion of order one, $β_{2}$ and $β_{4}$ are the coefficients of chromatic dispersion of order two, $α_{1}$ and $α_{2}$ are the coefficients of dissipation, $γ$ is the coefficient of nonlinearity, $z$ is the spatial variable and $t$ is the time variable.

By using a change of variable $ξ = α z - α_{o} t$ , (19)

$U (z, t)$ and $V (z, t)$ become respectively $U (ξ)$ and $V (ξ)$ .

Equations (17) and (18) become respectively,

$(α - β_{1} α_{o}) U_{ξ} + i \frac{β_{2} α_{0}^{2}}{2} U_{ξ ξ} + \frac{α_{1}}{2} U - i γ ({| U |}^{2} + B {| V |}^{2}) U = 0,$ (20)

and

$(α - β_{3} α_{o}) V_{ξ} + i \frac{β_{4} α_{0}^{2}}{2} V_{ξ ξ} + \frac{α_{2}}{2} V - i γ ({| V |}^{2} + B {| U |}^{2}) V = 0.$ (21)

So, solutions to the above equations will be searched in the form

$U (ξ) = a J_{n, m} (ξ),$ (22)

and

$V (ξ) = b J_{n, m} (ξ),$ (23)

where, $J_{n, m} (ξ) = \frac{\sinh^{m} (ξ)}{\cosh^{n} (ξ)}$ is the implicit Bogning function (iBf), $ξ$ represents an independent variable, the pair $(n, m) \in R^{2}$ indicates the power of the function, $a$ and $b$ are arbitrary non zero constants.

Now, considering Equations (20) and (21) above, we can evaluate the different terms:

$U_{ξ} = a {J^{'}}_{n, m} (ξ),$ (24)

$U_{ξ ξ} = a {J^{″}}_{n, m} (ξ),$ (25)

${| U |}^{2} = {| a |}^{2} J_{2 n, 2 m} (ξ)$ (26)

and

${| U |}^{2} U = {| a |}^{2} a J_{3 n, 3 m} (ξ) .$ (27)

Similarly, we can obtain expressions for $V_{ξ}, V_{ξ ξ}, {| V |}^{2}$ and ${| V |}^{2} V$ . (28)

<xref ref-type="bibr" rid="scirp.146840-"></xref>3.1. Main Equation of Range of Coefficients

Taking into account the ansatz (22) and (23) in Equations (20) and (21) imposes the calculation of its different terms. For this purpose, we obtain after successive derivatives calculations, Equations (29) and (30) below

$\begin{array}{l} (α - β_{1} α_{o}) [m J_{n - 1, m - 1} (ξ) - n J_{n + 1, m + 1} (ξ)] \\ + i \frac{β_{2} α_{0}^{2}}{2} [m (m - 1) J_{n - 2, m - 2} (ξ) + n (n + 1) J_{n + 2, m + 2} (ξ) \\ - [m (n - 1) + n (m + 1)] J_{(n, m)} (ξ)] + \frac{α_{1}}{2} J_{n, m} (ξ) \\ - i γ ({| a |}^{2} J_{2 n, 2 m} (ξ) + B {| b |}^{2} J_{2 n^{'}, 2 m^{'}} (ξ)) \cdot J_{n, m} (ξ) = 0, \end{array}$ (29)

and

$\begin{array}{l} (α - β_{3} α_{o}) [m^{'} J_{n^{'} - 1, m^{'} - 1} (ξ) - n^{'} J_{n^{'} + 1, m^{'} + 1} (ξ)] \\ + i \frac{β_{4} α_{0}^{2}}{2} [m^{'} (m^{'} - 1) J_{n^{'} - 2, m^{'} - 2} (ξ) + n^{'} (n^{'} + 1) J_{n^{'} + 2, m^{'} + 2} (ξ) \\ - [m^{'} (n^{'} - 1) + n^{'} (m^{'} + 1)] J_{(n^{'}, m^{'})} (ξ)] + \frac{α_{2}}{2} J_{n^{'}, m^{'}} (ξ) \\ - i γ ({| b |}^{2} J_{2 n^{'}, 2 m^{'}} (ξ) + B {| a |}^{2} J_{2 n, 2 m} (ξ)) J_{n^{'}, m^{'}} (ξ) = 0. \end{array}$ (30)

For $a \neq 0, b \neq 0$ , setting $λ_{1} = α - β_{1} α_{o}, λ_{2} = \frac{β_{2} α_{0}^{2}}{2}, λ_{3} = α - β_{3} α_{o}$ and $λ_{4} = \frac{β_{4} α_{0}^{2}}{2}$ ,

Equations (29) and (30) are written

$\begin{array}{l} λ_{1} [m J_{n - 1, m - 1} (ξ) - n J_{n + 1, m + 1} (ξ)] \\ + i λ_{2} [m (m - 1) J_{n - 2, m - 2} (ξ) + n (n + 1) J_{n + 2, m + 2} (ξ) \\ - [m (n - 1) + n (m + 1)] J_{(n, m)} (ξ)] + \frac{α_{1}}{2} J_{n, m} (ξ) \\ - i γ ({| a |}^{2} J_{3 n, 3 m} (ξ) + B {| b |}^{2} J_{2 n^{'} + n, 2 m^{'} + m} (ξ)) = 0, \end{array}$ (31)

and

$\begin{array}{l} λ_{3} [m^{'} J_{n^{'} - 1, m^{'} - 1} (ξ) - n^{'} J_{n^{'} + 1, m^{'} + 1} (ξ)] + \frac{α_{2}}{2} J_{n^{'}, m^{'}} (ξ) \\ + i λ_{4} [m^{'} (m^{'} - 1) J_{n^{'} - 2, m^{'} - 2} (ξ) + n^{'} (n^{'} + 1) J_{n^{'} + 2, m^{'} + 2} (ξ) \\ - [m^{'} (n^{'} - 1) + n^{'} (m^{'} + 1)] J_{(n^{'}, m^{'})} (ξ)] \\ - i γ {| b |}^{2} J_{3 n^{'}, 3 m^{'}} (ξ) - i γ B {| a |}^{2} J_{2 n + n^{'}, 2 m + m^{'}} (ξ) = 0. \end{array}$ (32)

Equations (31) and (32) are the main coefficient range of equations that will be used for the search of all the solutions.

<xref ref-type="bibr" rid="scirp.146840-"></xref>3.2. Field of Possible Solutions: Pairs <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi> n </mi> </mstyle> <mo> , </mo> <mstyle mathvariant="bold" mathsize="normal"> <mi> m </mi> </mstyle> </mrow> <mo> ) </mo> </mrow> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <msup> <mi> n </mi> <mo> ′ </mo> </msup> </mstyle> <mo> , </mo> <mstyle mathvariant="bold" mathsize="normal"> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mstyle> </mrow> <mo> ) </mo> </mrow> </mrow> </math> for Which Certain Terms of Equations (31) and (32) Can Be Grouped Together

The field of possible solutions of Equations (31) and (32) are given by the values of $n; m; n^{'}$ and $m^{'}$ for which some of their terms are grouped together.

So, we look for the pairs $(n, m)$ and $(n^{'}, m^{'})$ such that if $U_{i} J_{n, m}$ and $U_{j} J_{n^{'}, m^{'}}$ are two terms of Equations (31) and (32), we have simultaneously $n = n^{'}$ and $m = m^{'}$ , so that

${\begin{cases} n - 1 = 3 n \\ m - 1 = 3 m \end{cases} \Rightarrow {\begin{cases} n = - \frac{1}{2} \\ m = - \frac{1}{2} \end{cases} \Rightarrow n = m = - \frac{1}{2},$

${\begin{cases} n + 1 = 3 n \\ m + 1 = 3 m \end{cases} \Rightarrow {\begin{cases} n = \frac{1}{2} \\ m = \frac{1}{2} \end{cases} \Rightarrow n = m = \frac{1}{2},$

${\begin{cases} n - 2 = 3 n \\ m - 2 = 3 m \end{cases} \Rightarrow {\begin{cases} n = - 1 \\ m = - 1 \end{cases} \Rightarrow n = m = - 1,$

${\begin{cases} n + 2 = 3 n \\ m + 2 = 3 m \end{cases} \Rightarrow {\begin{cases} n = 1 \\ m = 1 \end{cases} \Rightarrow n = m = 1$

and

${\begin{cases} n - 1 = 3 n \\ m - 1 = 3 m \end{cases} \Rightarrow {\begin{cases} n = 0 \\ m = 0 \end{cases} \Rightarrow n = m = 0.$

So, the pairs for which certain terms of Equations (31) and (32) are grouped together gives fixed couples $(n, m) = (- 1, - 1), (0, 0), (1, 1), (- \frac{1}{2}, - \frac{1}{2})$ and $(\frac{1}{2}, \frac{1}{2})$ . (33)

Similarly, we obtain the fixed couples $(n^{'}, m^{'}) = (- 1, - 1), (0, 0), (1, 1), (- \frac{1}{2}, - \frac{1}{2})$ and $(\frac{1}{2}, \frac{1}{2})$ . (34)

Apart from the fixed pairs, we obtain constraint relations which link the pairs

$\begin{array}{l} {\begin{cases} n^{'} = n - 2 \\ m^{'} = m - 2 \end{cases}; {\begin{cases} n^{'} = n + 1 \\ m^{'} = m + 1 \end{cases}; {\begin{cases} n^{'} = n - 1 \\ m^{'} = m - 1 \end{cases}; {\begin{cases} n^{'} = n + 2 \\ m^{'} = m + 2 \end{cases}; \\ {\begin{cases} n^{'} = 2 n^{'} + n \\ m^{'} = 2 m^{'} + m \end{cases}; {\begin{cases} n^{'} = n \\ m^{'} = m \end{cases} and {\begin{cases} n^{'} = 3 n \\ m^{'} = 3 m \end{cases} \end{array}$ (35)

These relationships above offer good alternatives regarding the resolution of those equations because they establish the links directly between the couples $(n, m)$ and $(n^{'}, m^{'})$ .

<xref ref-type="bibr" rid="scirp.146840-"></xref>3.3. Decoupling Equations (31) and (32) and Resolution

1) Taking: ${\begin{cases} n = n^{'} \\ m = m \end{cases}$ ,

then Equations (31) and (32) become respectively

and

$\begin{array}{l} λ_{3} [m^{'} J_{n^{'} - 1, m^{'} - 1} (ξ) - n^{'} J_{n^{'} + 1, m^{'} + 1} (ξ)] \\ + i λ_{4} [m^{'} (m^{'} - 1) J_{n^{'} - 2, m^{'} - 2} (ξ) + n^{'} (n^{'} + 1) J_{n^{'} + 2, m^{'} + 2} (ξ) \\ - [m^{'} (n^{'} - 1) + n^{'} (m^{'} + 1)] J_{(n^{'}, m^{'})} (ξ)] + \frac{α_{2}}{2} J_{n^{'}, m^{'}} (ξ) \\ - i γ ({| b |}^{2} J_{3 n^{'}, 3 m^{'}} (ξ) + B {| a |}^{2} J_{3 n^{'}, 3 m^{'}} (ξ)) = 0. \end{array}$ (37)

The values of the pairs for which certain terms of Equations (36) and (37) group together without modifying the structure of Equations (31) and (32) are $(n, m) = (n^{'}, m^{'}) = (1, 1)$ and $(n, m) = (n^{'}, m^{'}) = (- 1, - 1)$ .

From Equations (36) and (37), we obtain respectively

$\begin{array}{l} λ_{1} J_{o, o} (ξ) + (\frac{α_{1}}{2} - 2 i λ_{2}) J_{1, 1} (ξ) + (2 i λ_{2} - i γ {| a |}^{2} - i γ B {| b |}^{2}) J_{3, 3} (ξ) \\ - λ_{1} J_{2, 2} (ξ) = 0, \end{array}$ (38)

and

$\begin{array}{l} λ_{3} (J_{o, o} (ξ) - J_{2, 2} (ξ)) + (\frac{α_{2}}{2} - 2 i λ_{4}) J_{1, 1} (ξ) \\ + (2 i λ_{4} - i γ {| b |}^{2} - i γ B {| a |}^{2}) J_{3, 3} (ξ) = 0. \end{array}$ (39)

Using the following relation,

$J_{3, 3} = J_{1, 1} \cdot J_{2, 2} = J_{1, 1} (1 - J_{2, 0}) = J_{1, 1} - J_{3, 1},$ (40)

Equations (38) and (39) become respectively

$\begin{array}{l} λ_{1} (J_{o, o} (ξ) - J_{2, 2} (ξ)) + (\frac{α_{1}}{2} - i γ {| a |}^{2} - i γ B {| b |}^{2}) J_{1, 1} (ξ) \\ + (2 i λ_{2} - i γ {| a |}^{2} - i γ B {| b |}^{2}) J_{3, 1} (ξ) = 0, \end{array}$ (41)

and

$\begin{array}{l} λ_{3} (J_{o, o} (ξ) - J_{2, 2} (ξ)) + (\frac{α_{2}}{2} - 2 i λ_{4}) J_{1, 1} (ξ) \\ + (2 i λ_{4} - i γ {| b |}^{2} - i γ B {| a |}^{2}) J_{3, 3} (ξ) = 0. \end{array}$ (42)

Equations (41) and (42) are verified if and only if

$λ_{1} = λ_{3} = 0 \Rightarrow β_{1} = β_{3} = \frac{α}{α_{0}},$ (43)

$2 i λ_{2} - i γ {| a |}^{2} - i γ B {| b |}^{2} = 0 \Rightarrow {| a |}^{2} + B {| b |}^{2} = \frac{2 λ_{2}}{γ}$ (44)

and

$2 i λ_{4} - i γ {| b |}^{2} - i γ B {| a |}^{2} = 0 \Rightarrow {| b |}^{2} + B {| a |}^{2} = \frac{2 λ_{4}}{γ} .$ (45)

Solving Equations (44) and (45), we obtain the following

$| a | = α_{o} \sqrt{\frac{β_{2} - B β_{4}}{γ (1 - B^{2})}}, γ \neq 0, B \neq 1$ (46)

and

$| b | = α_{o} \sqrt{\frac{β_{4}}{γ} - \frac{B (β_{2} - B β_{4})}{γ (1 - B^{2})}}, γ \neq 0, B \neq 1.$ (47)

Equally,

$\frac{α_{1}}{2} - i γ {| a |}^{2} - i γ B {| b |}^{2} = 0 \Rightarrow \frac{α_{1}}{2} = i γ ({| a |}^{2} + B {| b |}^{2}),$ (48)

and

$\frac{α_{2}}{2} - 2 i λ_{4} = 0 \Rightarrow \frac{α_{2}}{2} = i γ ({| b |}^{2} + B {| a |}^{2}) .$ (49)

Solving Equations (48) and (49), we obtain the following

$\frac{α_{1}}{2} = i α_{o}^{2} (β_{2}),$ (50)

and

$\frac{α_{2}}{2} = i α_{o}^{2} (β_{4}) .$ (51)

From which, we obtain the following constraints $α_{o} = \sqrt{\frac{- i α_{1}}{2 β_{2}}}$ and $α = α_{o} β_{1}$ . (52)

Thus, the solution obtained is of the form

$U (z, t) = α_{o} \sqrt{\frac{β_{2} - B β_{4}}{γ (1 - B^{2})}} \exp (i θ) J_{1, 1} (α z - α_{o} t),$ (53)

and

$\begin{array}{l} V (z, t) = α_{o} \sqrt{\frac{β_{4}}{γ} - \frac{B (β_{2} - B β_{4})}{γ (1 - B^{2})}} \exp (i θ) J_{1, 1} (α z - α_{o} t), \\ with θ \in ℝ, β_{2} \cdot γ ≻ 0, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0, α_{o} = \sqrt{\frac{- i α_{1}}{2 β_{2}}} and α = α_{o} β_{1} . \end{array}$ (54)

These solutions above are indeed the exact solutions of Equations (31) and (32) when the constraints joining the coefficients of the terms are taken into account such that Equations (17) and (18) become

$\frac{\partial U}{\partial z} + \frac{α}{α_{o}} \frac{\partial U}{\partial t} + i \frac{β_{2}}{2} \frac{\partial^{2} U}{\partial t^{2}} + i α_{o}^{2} β_{2} U - i γ ({| U |}^{2} + B {| V |}^{2}) U = 0,$ (55)

and

$\frac{\partial V}{\partial z} + \frac{α}{α_{o}} \frac{\partial V}{\partial t} + i \frac{β_{4}}{2} \frac{\partial^{2} V}{\partial t^{2}} + i α_{o}^{2} β_{4} V - i γ ({| V |}^{2} + B {| U |}^{2}) V = 0.$ (56)

An approach similar to the previous case gives for $(n, m) = (n^{'}, m^{'}) = (- 1, - 1)$ , the following solutions

$U (z, t) = α_{o} \sqrt{\frac{β_{2} - B β_{4}}{γ (1 - B^{2})}} \exp (i θ) J_{- 1, - 1} (α z - α_{o} t),$ (57)

and

$\begin{array}{l} V (z, t) = α_{o} \sqrt{\frac{β_{4}}{γ} - \frac{B (β_{2} - B β_{4})}{γ (1 - B^{2})}} \exp (i θ) J_{- 1, - 1} (α z - α_{o} t), \\ with θ \in ℝ, β_{2} \cdot γ ≻ 0, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0, α_{o} = \sqrt{\frac{- i α_{1}}{2 β_{2}}} and α = α_{o} β_{1} . \end{array}$ (58)

2) Taking: ${\begin{cases} n^{'} = n - 1 \\ m^{'} = m - 1 \end{cases}$ or ${\begin{cases} n = n^{'} + 1 \\ m = m^{'} + 1 \end{cases}$ ,

then Equations (31) and (32) become respectively

and

Equations (59) and (60) are range equations decoupled by the constraint relations $n^{'} = n - 1; m^{'} = m - 1$ . Equation (59) depends exclusively on pairs $(n, m)$ and Equation (60) exclusively on pairs $(n^{'}, m^{'})$ . Thus, the preponderant pairs $(n, m)$ for which certain terms of Equation (59) are grouped together are $(1, 1); (0, 0)$ and $(\frac{1}{2}, \frac{1}{2})$ .

Similarly, the leading pairs $(n^{'}, m^{'})$ for which certain terms of Equation (60) group together are $(- 1, - 1); (0, 0)$ and $(- \frac{1}{2}, - \frac{1}{2})$ . Our concern is to have two non-trivial solutions and we subsequently search for the solutions for the case $(n, m) = (1, 1) \Rightarrow (n^{'}, m^{'}) = (- 1, - 1)$ .

Equations (59) and (60) give respectively

$\begin{array}{l} λ_{1} (J_{o, o} (ξ) - J_{2, 2} (ξ)) + (\frac{α_{1}}{2} - 2 i λ_{2} - i γ B {| b |}^{2}) J_{1, 1} (ξ) \\ + (2 i λ_{2} - i γ {| a |}^{2}) J_{3, 3} (ξ) = 0, \end{array}$ (61)

and

$\begin{array}{l} λ_{3} J_{o, o} (ξ) - λ_{3} J_{- 2, - 2} (ξ) + (\frac{α_{2}}{2} - 2 i λ_{4} - i γ B {| a |}^{2}) J_{- 1, - 1} (ξ) \\ + (2 i λ_{4} - i γ {| b |}^{2}) J_{- 3, - 3} (ξ) = 0. \end{array}$ (62)

Using the following relation,

$J_{3, 3} = J_{1, 1} \cdot J_{2, 2} = J_{1, 1} (1 - J_{2, 0}) = J_{1, 1} - J_{3, 1},$ (63)

Equations (61) and (62) become respectively

$\begin{array}{l} λ_{1} (J_{o, o} (ξ) - J_{2, 2} (ξ)) + (2 i λ_{2} - i γ {| a |}^{2}) J_{3, 1} (ξ) \\ + (\frac{α_{1}}{2} - i γ {| a |}^{2} - i γ B {| b |}^{2}) J_{1, 1} (ξ) = 0, \end{array}$ (64)

and

Equations (64) and (65) are verified if and only if $a \neq 0, b \neq 0$ , we have the following relations

$λ_{1} = λ_{3} = 0 \Rightarrow β_{1} = β_{3} = \frac{α}{α_{0}},$ (66)

$2 i λ_{2} - i γ {| a |}^{2} = 0 \Rightarrow | a | = α_{0} \sqrt{\frac{β_{2}}{γ}}$ , that’s $a = α_{o} \sqrt{\frac{β_{2}}{γ}} \exp (i θ), β_{2} \cdot γ ≻ 0, α_{o} ≻ 0$ (67)

and

$2 i λ_{4} - i γ {| b |}^{2} = 0 \Rightarrow | b | = α_{0} \sqrt{\frac{β_{4}}{γ}}$ , that is $b = α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ), β_{4} \cdot γ ≻ 0, α_{o} ≻ 0$ . (68)

Equally,

$\frac{α_{1}}{2} - i γ {| a |}^{2} - i γ B {| b |}^{2} = 0 \Rightarrow \frac{α_{1}}{2} = i γ ({| a |}^{2} + B {| b |}^{2}),$ (69)

and

$\frac{α_{2}}{2} - i γ {| b |}^{2} - i γ B {| a |}^{2} = 0 \Rightarrow \frac{α_{2}}{2} = i γ ({| b |}^{2} + B {| a |}^{2}),$ (70)

Solving Equations (69) and (70), we obtain the following

$\frac{α_{1}}{2} = i α_{o}^{2} (β_{2} + B β_{4}),$ (71)

and

$\frac{α_{2}}{2} = i α_{o}^{2} (β_{4} + B β_{2}) .$ (72)

From which, we obtain the following constraints $α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}}$ and $α = α_{o} β_{1}$ . (73)

Thus, the solutions obtained are of the form

$U (z, t) = α_{o} \sqrt{\frac{β_{2}}{γ}} \exp (i θ) J_{1, 1} (α z - α_{o} t),$ (74)

and

$\begin{array}{l} V (z, t) = α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) J_{- 1, - 1} (α z - α_{o} t), β_{2} \cdot γ ≻ 0, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0 \\ with α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}} and α = α_{o} β_{1} . \end{array}$ (75)

These solutions above are indeed the exact solutions of Equations (31) and (32) when the constraints joining the coefficients of the terms are taken into account such that, Equations (17) and (18) becomes

$\frac{\partial U}{\partial z} + \frac{α}{α_{o}} \frac{\partial U}{\partial t} + i \frac{β_{2}}{2} \frac{\partial^{2} U}{\partial t^{2}} + i α_{o}^{2} (β_{2} + B β_{4}) U - i γ ({| U |}^{2} + B {| V |}^{2}) U = 0,$ (76)

and

$\frac{\partial V}{\partial z} + \frac{α}{α_{o}} \frac{\partial V}{\partial t} + i \frac{β_{4}}{2} \frac{\partial^{2} V}{\partial t^{2}} + i α_{o}^{2} (β_{4} + B β_{2}) V - i γ ({| V |}^{2} + B {| U |}^{2}) V = 0.$ (77)

Using $(n, m) = (1, 0) \Rightarrow (n^{'}, m^{'}) = (- 1, 0)$ , Equations (59) and (60) give respectively:

$\begin{array}{l} - λ_{1} J_{2, 1} (ξ) + i λ_{2} (2 J_{3, 2} (ξ) - J_{1, 0} (ξ)) + \frac{α_{1}}{2} J_{1, 0} (ξ) - i γ {| a |}^{2} J_{3, 0} (ξ) \\ - i γ B {| b |}^{2} J_{1, - 2} (ξ) = 0, \end{array}$ (78)

and

$λ_{3} J_{0, 1} (ξ) + (\frac{α_{2}}{2} + i λ_{4}) J_{- 1, 0} (ξ) - i γ {| b |}^{2} J_{- 3, 0} (ξ) - i γ B {| a |}^{2} J_{- 1, 2} (ξ) = 0.$ (79)

Considering the transformation $J_{3, 2} = J_{1, 0} - J_{3, 0}$ in Equation (78) above, we obtain:

$- λ_{1} J_{2, 1} (ξ) - (2 i λ_{2} + i γ {| a |}^{2}) J_{3, 0} (ξ) + (\frac{α_{1}}{2} + i λ_{2}) J_{1, 0} (ξ) - i γ B {| b |}^{2} J_{1, - 2} (ξ) = 0,$ (80)

and

$λ_{3} J_{0, 1} (ξ) + (\frac{α_{2}}{2} + i λ_{4}) J_{- 1, 0} (ξ) - i γ {| b |}^{2} J_{- 3, 0} (ξ) - i γ B {| a |}^{2} J_{- 1, 2} (ξ) = 0.$ (81)

Equations (80) and (81) are checked if we have the following relationship

$λ_{1} = λ_{3} = 0 \Rightarrow β_{1} = β_{3} = \frac{α}{α_{0}},$ (82)

$\frac{α_{1}}{2} + i λ_{2} = 0 \Rightarrow α_{1} = - i α_{o}^{2} (β_{2}),$ (83)

$\frac{α_{2}}{2} + i λ_{4} = 0 \Rightarrow α_{2} = - i α_{o}^{2} (β_{4}) .$ (84)

From which, we obtain the following constraints $α_{o} = \sqrt{\frac{i α_{1}}{β_{2}}}$ and $α = α_{o} β_{1}$ . (85)

$i γ B {| b |}^{2} = 0 \Rightarrow γ \to 0$ since $b \neq 0$ , (86)

$i γ B {| a |}^{2} = 0 \Rightarrow γ \to 0$ since $a \neq 0$ , (87)

and

$2 λ_{2} + γ {| a |}^{2} = α_{o}^{2} β_{2} + γ {| a |}^{2} \Rightarrow {| a |}^{2} = - \frac{α_{o}^{2} β_{2}}{γ}$ $.$ (88)

Finally, we obtain from Equation (88)

$| a | = α_{0} \sqrt{\frac{- β_{2}}{γ}}, a = α_{o} \sqrt{\frac{- β_{2}}{γ}} \exp (i θ), β_{2} \cdot γ ≺ 0, α_{o} ≻ 0, γ \neq 0,$ (89)

Thus, the solution obtained is of the form

$\begin{array}{l} U (z, t) = α_{o} \sqrt{\frac{- β_{2}}{γ}} \exp (i θ) J_{1, 0} (α z - α_{o} t), \\ with θ \in R, β_{2} \cdot γ ≺ 0, α_{o} ≻ 0, α_{o} = \sqrt{\frac{i α_{1}}{β_{2}}} and α = α_{o} β_{1} . \end{array}$ (90)

and

$V (z, t) = b \exp (i θ) J_{- 1, 0} (α z - α_{o} t), with b \neq 0,$ (91)

This solution is indeed the exact solution of Equations (31) and (32), when the constraints joining the coefficients of the terms are taken into account such that (17) and (18) are modified as follows:

$\frac{\partial U}{\partial z} + \frac{α}{α_{o}} \frac{\partial U}{\partial t} + i \frac{β_{2}}{2} \frac{\partial^{2} U}{\partial t^{2}} - i α_{o}^{2} β_{2} U = 0,$ (92)

and

$\frac{\partial V}{\partial z} + \frac{α}{α_{o}} \frac{\partial V}{\partial t} + i \frac{β_{4}}{2} \frac{\partial^{2} V}{\partial t^{2}} - i α_{o}^{2} β_{4} V = 0.$ (93)

The above partial differential equations characterize an optical fiber with weak nonlinearity.

Equations (59) and (60) give respectively

$λ_{1} J_{- 1, 0} (ξ) + i λ_{2} J_{0, 1} (ξ) + \frac{α_{1}}{2} J_{0, 1} (ξ) - i γ {| a |}^{2} J_{0, 3} (ξ) - i γ B {| b |}^{2} J_{- 2, 1} (ξ) = 0,$ (94)

and

$\begin{array}{l} - λ_{3} J_{- 1, - 2} (ξ) + 2 i λ_{4} J_{- 2, - 3} (ξ) + \frac{α_{2}}{2} J_{0, - 1} (ξ) - i γ {| b |}^{2} J_{0, - 3} (ξ) \\ - i λ_{4} J_{0, - 1} (ξ) - i γ B {| a |}^{2} J_{2, - 1} (ξ) = 0, \end{array}$ (95)

Considering the transformation $J_{- 2, - 3} = J_{0, - 3} - J_{0, - 1}$ in Equation (95) above, we obtain

$λ_{1} J_{- 1, 0} (ξ) + i λ_{2} J_{0, 1} (ξ) + \frac{α_{1}}{2} J_{0, 1} (ξ) - i γ {| a |}^{2} J_{0, 3} (ξ) - i γ B {| b |}^{2} J_{- 2, 1} (ξ) = 0,$ (96)

and

$\begin{array}{l} - λ_{3} J_{- 1, - 2} (ξ) + (\frac{α_{2}}{2} - 3 i λ_{4}) J_{0, - 1} (ξ) + (2 i λ_{4} - i γ {| b |}^{2}) J_{0, - 3} (ξ) \\ - i γ B {| a |}^{2} J_{2, - 1} (ξ) = 0. \end{array}$ (97)

Equations (96) and (97) are checked if we have the following relationship

$λ_{1} = λ_{3} = 0 \Rightarrow β_{1} = β_{3} = \frac{α}{α_{0}},$ (98)

$\frac{α_{1}}{2} - i λ_{2} = 0 \Rightarrow α_{1} = i α_{o}^{2} (β_{2}),$ (99)

$\frac{α_{2}}{2} - 3 i λ_{4} = 0 \Rightarrow α_{2} = 3 i α_{o}^{2} (β_{4}) .$ (100)

From which, we obtain the following constraints $α_{o} = \sqrt{\frac{- i α_{2}}{3 β_{4}}}$ and $α = α_{o} β_{1}$ . (101)

$i γ B {| b |}^{2} = 0 \Rightarrow γ \to 0$ since $b \neq 0$ , (102)

$i γ B {| a |}^{2} = 0 \Rightarrow γ \to 0$ since $a \neq 0$ , (103)

and

$2 λ_{4} - γ {| b |}^{2} = α_{o}^{2} β_{4} - γ {| b |}^{2} \Rightarrow {| b |}^{2} = \frac{α_{o}^{2} β_{4}}{γ} .$ (104)

Finally, we obtain from Equation (104)

$| b | = α_{0} \sqrt{\frac{β_{4}}{γ}}, b = α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ), β_{4} \cdot γ ≻ 0, α_{o} ≻ 0, γ \neq 0,$ (105)

Thus, the solution obtained is of the form

$U (z, t) = a \exp (i θ) J_{0, 1} (α z - α_{o} t), with a \neq 0,$ (106)

and

$\begin{array}{l} V (z, t) = α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) J_{0, - 1} (α z - α_{o} t), \\ with θ \in R, β_{4} \cdot γ ≻ 0, α_{o} ≻ 0, α_{o} = \sqrt{\frac{- i α_{2}}{3 β_{4}}} and α = α_{o} β_{1} . \end{array}$ (107)

$\frac{\partial U}{\partial z} + \frac{α}{α_{o}} \frac{\partial U}{\partial t} + i \frac{β_{2}}{2} \frac{\partial^{2} U}{\partial t^{2}} + i α_{o}^{2} β_{2} U = 0,$ (108)

and

$\frac{\partial V}{\partial z} + \frac{α}{α_{o}} \frac{\partial V}{\partial t} + i \frac{β_{4}}{2} \frac{\partial^{2} V}{\partial t^{2}} + 3 i α_{o}^{2} β_{4} V = 0.$ (109)

The above partial differential equations characterize an optical fiber with weak nonlinearity.

3) Taking: ${\begin{cases} n^{'} = n + 1 \\ m^{'} = m + 1 \end{cases}$ or ${\begin{cases} n = n^{'} - 1 \\ m = m^{'} - 1 \end{cases}$ ,

then Equations (31) and (32) becomes respectively

and

$\begin{array}{l} λ_{3} [m^{'} J_{n^{'} - 1, m^{'} - 1} (ξ) - n^{'} J_{n^{'} + 1, m^{'} + 1} (ξ)] \\ - i γ ({| b |}^{2} J_{3 n^{'}, 3 m^{'}} (ξ) + B {| a |}^{2} J_{3 n^{'} - 2, 3 m^{'} - 2} (ξ)) + \frac{α_{2}}{2} J_{n^{'}, m^{'}} (ξ) \\ + i λ_{4} [m^{'} (m^{'} - 1) J_{n^{'} - 2, m^{'} - 2} (ξ) + n^{'} (n^{'} + 1) J_{n^{'} + 2, m^{'} + 2} (ξ) \\ - [m^{'} (n^{'} - 1) + n^{'} (m^{'} + 1)] J_{(n^{'}, m^{'})} (ξ)] = 0. \end{array}$ (111)

Equations (110) and (111) are range equations decoupled by the constraint relations $n^{'} = n + 1; m^{'} = m + 1$ . Equation (110) depends exclusively on pairs $(n, m)$ and Equation (111) exclusively on pairs $(n^{'}, m^{'})$ . Thus, the preponderant pairs $(n, m)$ for which certain terms of Equation (110) are grouped together are $(- 1, - 1); (0, 0)$ and $(- \frac{1}{2}, - \frac{1}{2})$ .

Similarly, the leading pairs $(n^{'}, m^{'})$ for which certain terms of Equation (111) group together are $(1, 1); (0, 0)$ and $(\frac{1}{2}, \frac{1}{2})$ . Our concern is to have two non-trivial solutions, and we subsequently search for the solutions for the case $(n, m) = (- 1, - 1) \Rightarrow (n^{'}, m^{'}) = (1, 1)$ .

Equations (110) and (111) give respectively

$\begin{array}{l} λ_{1} (J_{o, o} (ξ) - J_{- 2, - 2} (ξ)) + (\frac{α_{1}}{2} - 2 i λ_{2} - i γ B {| b |}^{2}) J_{- 1, - 1} (ξ) \\ + (2 i λ_{2} - i γ {| a |}^{2}) J_{- 3, - 3} (ξ) = 0, \end{array}$ (112)

and

$\begin{array}{l} λ_{3} J_{o, o} (ξ) - λ_{3} J_{2, 2} (ξ) + (\frac{α_{2}}{2} - 2 i λ_{4} - i γ B {| a |}^{2}) J_{1, 1} (ξ) \\ + (2 i λ_{4} - i γ {| b |}^{2}) J_{3, 3} (ξ) = 0. \end{array}$ (113)

Using the following relation,

$J_{3, 3} = J_{1, 1} \cdot J_{2, 2} = J_{1, 1} (1 - J_{2, 0}) = J_{1, 1} - J_{3, 1},$ (114)

Equations (112) and (113) become respectively

and

$\begin{array}{l} λ_{3} J_{o, o} (ξ) - λ_{3} J_{2, 2} (ξ) + (\frac{α_{2}}{2} - i γ {| b |}^{2} - i γ B {| a |}^{2}) J_{1, 1} (ξ) \\ - (2 i λ_{4} - i γ {| b |}^{2}) J_{3, 1} (ξ) = 0. \end{array}$ (116)

Equations (115) and (116) are verified if and only if $a \neq 0, b \neq 0$ , we have the following relations

$λ_{1} = λ_{3} = 0 \Rightarrow β_{1} = β_{3} = \frac{α}{α_{0}},$ (117)

$2 i λ_{2} - i γ {| a |}^{2} = 0 \Rightarrow | a | = α_{0} \sqrt{\frac{β_{2}}{γ}}$ , that’s $a = α_{o} \sqrt{\frac{β_{2}}{γ}} e^{i θ}, β_{2} \cdot γ ≻ 0, α_{o} ≻ 0$ (118)

and

$2 i λ_{4} - i γ {| b |}^{2} = 0 \Rightarrow | b | = α_{0} \sqrt{\frac{β_{4}}{γ}}$ , that’s $b = α_{o} \sqrt{\frac{β_{4}}{γ}} e^{i θ}, β_{4} \cdot γ ≻ 0, α_{o} ≻ 0.$ (119)

Equally,

$\frac{α_{1}}{2} - i γ {| a |}^{2} - i γ B {| b |}^{2} = 0 \Rightarrow \frac{α_{1}}{2} = i γ ({| a |}^{2} + B {| b |}^{2}),$ (120)

and

$\frac{α_{2}}{2} - i γ {| b |}^{2} - i γ B {| a |}^{2} = 0 \Rightarrow \frac{α_{2}}{2} = i γ ({| b |}^{2} + B {| a |}^{2}),$ (121)

Solving Equations (120) and (121), we obtain the following

$\frac{α_{1}}{2} = i α_{o}^{2} (β_{2} + B β_{4}),$ (122)

and

$\frac{α_{2}}{2} = i α_{o}^{2} (β_{4} + B β_{2}) .$ (123)

From which, we obtain the following constraints $α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}}$ and $α = α_{o} β_{1}$ . (124)

Thus, the solutions obtained are of the form

$U (z, t) = α_{o} \sqrt{\frac{β_{2}}{γ}} \exp (i θ) J_{- 1, - 1} (α z - α_{o} t),$ (125)

and

$\begin{array}{l} V (z, t) = α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) J_{1, 1} (α z - α_{o} t), β_{2} \cdot γ ≻ 0, γ \neq 0, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0 \\ with α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}} and α = α_{o} β_{1} . \end{array}$ (126)

and

One benefit of using iB-functions here is their ability to smoothly convert hyperbolic forms to trigonometric forms. This means we can find corresponding trigonometric solutions by simply mapping $α \leftarrow i α$ and $α_{0} \leftarrow i α_{0}$ , such that $i^{2} = - 1$ in our existing solutions.

Since we have many solutions, we’ll just show a few examples by using the transformation

$J_{n, m} (i x) = i^{m} T_{n, m} (x)$ with $T_{n, m} (x) = \frac{\sin^{m} (x)}{\cos^{n} (x)}$ . (129)

1) Taking: ${\begin{cases} n = n^{'} \\ m = m \end{cases}$ ,

The solution obtained is of the form

$U (z, t) = i α_{o} \sqrt{\frac{β_{2} - B β_{4}}{γ (1 - B^{2})}} \exp (i θ) \tan (α z - α_{o} t), B \neq 1, γ \neq 0,$ (130)

and

$\begin{array}{l} V (z, t) = i α_{o} \sqrt{\frac{β_{4}}{γ} - \frac{B (β_{2} - B β_{4})}{γ (1 - B^{2})}} \exp (i θ) \tan (α z - α_{o} t), B \neq 1, γ \neq 0, \\ with θ \in ℝ, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0, α_{o} = \sqrt{\frac{- i α_{1}}{2 β_{2}}} and α = α_{o} β_{1} . \end{array}$ (131)

The solution obtained is of the form

$U (z, t) = - i α_{o} \sqrt{\frac{β_{2} - B β_{4}}{γ (1 - B^{2})}} \exp (i θ) cotan (α z - α_{o} t), B \neq 1, γ \neq 0,$ (132)

and

$\begin{array}{l} V (z, t) = - i α_{o} \sqrt{\frac{β_{4}}{γ} - \frac{B (β_{2} - B β_{4})}{γ (1 - B^{2})}} \exp (i θ) cotan (α z - α_{o} t), \\ with θ \in ℝ, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0, α_{o} = \sqrt{\frac{- i α_{1}}{2 β_{2}}} and α = α_{o} β_{1} . \end{array}$ (133)

2) Taking: ${\begin{cases} n^{'} = n - 1 \\ m^{'} = m - 1 \end{cases}$ or ${\begin{cases} n = n^{'} + 1 \\ m = m^{'} + 1 \end{cases}$ ,

The solutions obtained are of the form

$U (z, t) = i α_{o} \sqrt{\frac{β_{2}}{γ}} \exp (i θ) \tan (α z - α_{o} t), β_{2} \cdot γ ≻ 0,$ (134)

and

$\begin{array}{l} V (z, t) = - i α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) cotan (α z - α_{o} t), α_{o} ≻ 0, β_{4} \cdot γ ≻ 0 \\ with α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}} and α = α_{o} β_{1} . \end{array}$ (135)

The solutions obtained are of the form

$\begin{array}{l} U (z, t) = α_{o} \sqrt{\frac{- β_{2}}{γ}} \exp (i θ) \sec (α z - α_{o} t), \\ with θ \in R, β_{2} \cdot γ ≺ 0, α_{o} ≻ 0, α_{o} = \sqrt{\frac{i α_{1}}{β_{2}}} and α = α_{o} β_{1} . \end{array}$ (136)

and

$V (z, t) = b \exp (i θ) \cos (α z - α_{o} t), with b \neq 0,$ (137)

The solutions obtained are of the form

$U (z, t) = i a \exp (i θ) \sin (α z - α_{o} t), a \neq 0, with a \neq 0,$ (138)

and

$\begin{array}{l} V (z, t) = - i α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) cosec (α z - α_{o} t), \\ with θ \in R, β_{4} \cdot γ ≻ 0, α_{o} ≻ 0, α_{o} = \sqrt{\frac{- i α_{2}}{3 β_{4}}} and α = α_{o} β_{1} . \end{array}$ (139)

3) Taking: ${\begin{cases} n^{'} = n + 1 \\ m^{'} = m + 1 \end{cases}$ or ${\begin{cases} n = n^{'} - 1 \\ m = m^{'} - 1 \end{cases}$ ,

The solutions obtained are of the form

$U (z, t) = - i α_{o} \sqrt{\frac{β_{2}}{γ}} \exp (i θ) cotan (α z - α_{o} t), β_{2} \cdot γ ≻ 0,$ (140)

and

$\begin{array}{l} V (z, t) = i α_{o} \sqrt{\frac{β_{4}}{γ}} \exp (i θ) \tan (α z - α_{o} t), γ \neq 0, α_{o} ≻ 0, β_{4} \cdot γ ≻ 0 \\ with α_{o} = \sqrt{\frac{- i α_{1}}{2 (β_{2} + B β_{4})}} and α = α_{o} β_{1}, θ \in R . \end{array}$ (141)

<xref ref-type="bibr" rid="scirp.146840-"></xref>4. Numerical Results and Discussion

For each modification made in the initial birefringent optical fiber, a propagation study to assess the practical feasibility of the solutions obtained is done. But because of the large volume of this work, a few solutions have been chosen to focus on:

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in Equations (53) and (54) can be illustrated, as shown in Figure 1 and Figure 2 .

Figure 1 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 1. 3D-intensity profiles showing the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

General Observations and Interpretations (Both Profiles):

In summary, the plots demonstrate the successful formation and stable propagation of coupled dark solitons in a birefringent optical fiber. The pulses nonlinear interaction forces them to propagate together as a single unit, despite the fiber’s implicit birefringence, focusing on the strong role of nonlinearity in maintaining the pulse structure and order.

Figure 2 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 2. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

Observations and Interpretations:

The first two graphs (horizontally) show the real and imaginary sections of the pulse amplitudes, U(ξ) and V(ξ), while the last two graphs (horizontally) show the residuals of the corresponding equations.

The Topmost Graphs (horizontally) U(ξ) and V(ξ): Real and Imaginary Sections ( Figure 2 )

The Lower Graphs (Horizontally): Residual of U and V Equation

Interpretation: As a result, these plots strongly suggest the numerical investigation of coupled dark solitons propagating in a birefringent optical fiber.

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in Equations (57) and (58) can be illustrated, as shown in Figure 3 and Figure 4 .

Figure 3 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 3. 3D-intensity profiles displaying the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

General Observations and Interpretations (Both Profiles):

The plots display the intensity profiles of two optical components against propagation distance (z) and time (t). The x and y-axes are similar to the previous case, while the z-axis ranging from 0 to 100 (likely arbitrary units, e.g. Watts per metre square or Watts per kilometre square), with the color bars showing the intensity values. Both profiles are deep blue, signifying very low or zero intensity everywhere except where the pulses are present.

Figure 4 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 4. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

As a summary, the above profiles indicate the propagation of two distinct, localized pulses through a birefringent optical fiber with no strong coupling due to their relative temporal position changing. The most prominent feature here is the group velocity walk-off between the two coupled components, where they gradually separate in time as they propagate along the fiber. Walk-off can be a significant issue in optical communication systems, leading to pulse broadening, if not properly managed.

Observations and Interpretations ( Figure 4 ):

The top two graphs (first row) show the real and imaginary sections of the pulse amplitudes, U(ξ) and V(ξ), while the bottom two graphs (second row) show the residuals of the corresponding equations.

The Graphs on the First Row U(ξ) and V(ξ): Real and Imaginary Sections

In summary, the fact that both U and V components display very similar bright soliton profiles and are perfectly centered at ξ = 0 indicates that these are coupled bright solitons that are propagating together without significant walk-off.

The Graphs on the Second Row: Residual of U and V Equation

In brief, the exceptionally low residual away from the peak indicates that the numerical solution is highly accurate in the “tails” of the pulse. The significant spike at ξ = 0 implies that the numerical method faces challenges in accurately capturing the rapid changes in amplitude.

Overall, these graphs indicate firmly the numerical investigation of coupled bright solitons propagating in a birefringent optical fiber.

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in Equations (74) and (75) can be drawn, as shown in Figure 5 and Figure 6 .

Figure 5 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 5. Intensity profiles displaying the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

General Observations and Interpretations (Both Profiles):

In summary, this set of charts illustrates a compelling case of a highly asymmetric and distinct reaction between the two components, propagating in birefringent optical fiber. A very high-intensity, localized second component appears to propagate stably, while a much weaker first component displays a strong periodic temporal transformation, likely induced by nonlinear interactions with the dominant first component. There is an absence of a significant temporal walk-off between the two components, despite the birefringence.

Figure 6 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 6. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

Observations and Interpretations:

The Graphs (Top Row) U(ξ) and V(ξ): Real and Imaginary Sections

This case is particularly intriguing as it combines features of both dark and bright solitons, showing a mixed-type coupled solution.

The Graphs (Bottom Row): Residual of U and V Equation

Interpretation: The very low residual away from the “dip” indicates that the numerical solution is highly accurate in those regions. The observations and interpretations for the V equation’s residual are similar to those for bright solitons in previous cases. The overall extremely low residuals away from the peak suggest a good quality numerical solution for this component as well.

As a summary, these graphs strongly demonstrate the numerical investigation of mixed-type coupled solitons (specifically, a coupled dark-bright soliton pair) propagating in birefringent optical fiber.

Figure 7 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 7. Intensity profiles displaying the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

This case is a typical example of how different types of solitons can be coupled and propagate together in birefringent media, showcasing the infinite variety of nonlinear phenomena in optical fibers.

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in system Equations (90) and (91) can be drawn, as shown in Figure 7 and Figure 8 .

General Observations and Interpretation (Both Profiles):

Figure 8 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 8. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

As a summary, this case demonstrates a highly uneven coupled pulse propagation where one component ( ${| V (z, t) |}^{2}$ ) is dominant in intensity and propagates as a remarkably stable, localized pulse, probably a fundamental soliton. The other component ( ${| U (z, t) |}^{2}$ ) exists as a very weak, diffused, and likely dispersive background, with minimal visible interaction or influence from the dominant pulse. This setup effectively reduces the coupled system to the propagation of a single, powerful pulse in a birefringent fiber, with negligible involvement from the orthogonal polarization.

Observations and Interpretations:

The Graphs (Top Row) U(ξ) and V(ξ): Real and Imaginary Sections

The Graph (Bottom Row): Residual of U and V Equation

Figure 9 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 9. Intensity profiles displaying the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

In summary, these plots reveal a complex framework of mixed-type coupled solitons, specifically a higher-order dark soliton coupled with a bright soliton. The U component forms an extended dark-type structure, while the V component is a localized bright pulse. Their perfect arrangement at ξ = 0 is a distinctive feature of dark-bright vector solitons or more complex coupled structures. The presence of these higher residuals suggests that while a solution has been found, its numerical accuracy might be limited in certain areas, possibly due to the intricate nature of the solution itself.

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in Equations (106) and (107) can be illustrated, as shown in Figure 9 and Figure 10 .

General Observations and Interpretation (Both Profiles):

Figure 10 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 10. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

As a summary, these plots illustrate a structure where one of the coupled pulse components ( ${| V (z, t) |}^{2}$ ) is significantly dominant in intensity, propagating as a wide and stable pulse. The first component ( ${| U (z, t) |}^{2}$ ) is extremely weak and appears as a negligible background, possibly declining.

Observations and Interpretations:

The Top Graph (First Row) U(ξ) and V(ξ): Real and Imaginary Sections

The Bottom Graph (Second Row): Residual of U and V Equation

Figure 11 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 11. Intensity profiles displaying the propagation of coupled pulses <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mrow> <mo> | </mo> <mrow> <mi> V </mi> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> | </mo> </mrow> </mrow> <mn> 2 </mn> </msup> </mrow> </math>, in birefringent optical fibers for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

As a result, this case demonstrates a mixed-type coupled soliton solution, where the U component is a bright soliton and the V component is a higher-order dark soliton (or dark-dark bound state). Their coordination at ξ = 0 is crucial, indicating a stable bound state, a form of dark-bright vector soliton. The very high residuals, particularly for the V component, emphasize the significant numerical challenges in accurately representing this complex coupled system, especially for the extended dark structure.

The nonlinear coupled partial differential Equations (17) and (18) are decoupled and solved such that the solutions obtained in Equations (125) and (126) can be plotted, as shown in Figure 11 and Figure 12 .

General Observations and Interpretation (Both Profiles):

This case shows a riveting inversion of the asymmetric behaviour seen in Case 3.

Figure 12 <xref ref-type="bibr" rid="scirp.146840-"></xref>Figure 12. Graphs representing the numerical solution of coupled pulses U(ξ) and V(ξ), propagating in birefringent optical fibers, <math xmlns="http://www.w3.org/1998/Math/MathML"><?xml version="1.0" encoding="UTF-16"?><mml:math xmlns:m="http://schemas.openxmlformats.org/officeDocument/2006/math" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal"> </mml:mi> </mml:math> </math>for the case <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mo> − </mo> <mn> 1 </mn> <mo> , </mo> <mo> − </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⇒ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> n </mi> <mo> ′ </mo> </msup> <mo> , </mo> <msup> <mi> m </mi> <mo> ′ </mo> </msup> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </math>.

In summary, these plots illustrate a compelling case of highly asymmetric coupled pulse propagation in a birefringent fiber, where a very high-intensity, localized first component propagates stably, while a much weaker second component displays strong periodic temporal modulation.

Observations and Interpretations:

The Top Graphs (First Row) U(ξ) and V(ξ): Real and Imaginary Sections

The Bottom Graphs (Second Row): Residual of U and V Equation

In summary, these diagrams describe a mixed-type coupled soliton solution, specifically a bright-dark vector soliton. The U component is a bright soliton, and the V component is a dark soliton. Their precise layout and concordance at ξ = 0 confirm that they form a stable bound state. The residual plots highlight the numerical challenges in accurately resolving the sharp features of both bright and dark solitons.

Across the six cases studied in the course of this work, we collectively obtained twelve figures, where Figure 1 , Figure 3 , Figure 5 , Figure 7 , Figure 9 and Figure 1 1 represent the intensities of the envelopes of the coupled signals resulting from the solutions obtained and Figure 2 , Figure 4 , Figure 6 , Figure 8 , Figure 1 0 and Figure 1 2 representing the graphs of the corresponding residuals equations. Under Case 1, we have Figure 1 , which shows the stable co-propagation of two coupled dark solitons in a birefringent fiber and Figure 2 , confirming the existence of two coupled dark solitons by showing their real parts as dips at the same point, with negligible imaginary parts and low residuals. As for Case 2, Figure 3 displays two distinct pulses propagating diagonally with a visible temporal separation (walk-off), indicating weak or no strong coupling and Figure 4 , showing the numerical solution of two coupled bright solitons centered at the same point, with negligible imaginary parts and low residuals everywhere except at the central peak. In Case 3, Figure 5 illustrates a highly asymmetric propagation, where a dominant, stable, high-intensity pulse coexists with a much weaker, periodically modulated wave without walk-off and Figure 6 , presenting a mixed-type coupled solution, showing one component as a bright soliton and the other as a dark soliton propagating together. As for Case 4, Figure 7 demonstrates an extreme power asymmetry where one component is a very stable, dominant pulse, while the other is a weak, negligible background and Figure 8 reveals a complex mixed-type solution where a higher-order dark soliton is coupled with a bright soliton, forming a stable bound state. For Case 5, Figure 9 illustrates a highly asymmetric case where one component is a stable, dominant pulse, and the other is an extremely weak, negligible background and Figure 1 0 confirms a mixed-type coupled soliton, with a bright soliton and a higher-order dark soliton forming a stable bound state. And therefore, Case 6 with Figure 1 1 shows a high-intensity, stable pulse coupled with a much weaker, periodically modulated component, similar to Case 3 but with the roles of the two components inverted and Figure 1 2 depicts a bright-dark vector soliton solution where one component is a bright soliton and the other is a dark soliton, forming a stable bound state.

Parametric Studies: Effect of Birefringence Parameters on Soliton Separation

A parametric study shows the effect of birefringence parameters on soliton behaviour ( Table 1 ).

Table 1 <xref ref-type="bibr" rid="scirp.146840-"></xref>Table 1. Effect of birefringence parameters on soliton separation.

Beta1-parameter	B-parameter	Max U-V Separation
0.08	0.6	0.750
0.08	0.8	0.667
0.08	1.0	0.600
0.10	0.6	0.625
0.10	0.8	0.556
0.10	1.0	0.500
0.12	0.6	0.500
0.12	0.8	0.444
0.12	1.0	0.400

This parameter represents the linear birefringence of the optical fiber. It quantifies the difference in group velocities between the two polarization components (U and V). The values for beta1 in the table are 0.08, 0.10, and 0.12. As beta1 increases (e.g., from 0.08 to 0.12) while B is kept constant, the Max U-V Separation decreases. For instance, when B is 0.6, increasing beta1 from 0.08 to 0.12 causes the separation to drop from 0.750 to 0.500. This suggests that a stronger linear birefringence might lead to a smaller maximum separation of the solitons.

This parameter represents the strength of cross-phase modulation (XPM) between the U and V polarization components. XPM describes how the phase of one component is affected by the intensity of the other. The values for B in the table are 0.6, 0.8, and 1.0. As B increases (e.g., from 0.6 to 1.0) while beta1 is kept constant, the Max U-V Separation decreases. For example, when beta1 is 0.08, increasing B from 0.6 to 1.0 reduces the separation from 0.750 to 0.600. This indicates that stronger cross-phase modulation leads to less separation between the U and V solitons.

This column quantifies the maximum temporal separation observed between the U and V solitons during their propagation. The values range from 0.400 to 0.750. The general trend observed from the table is that increasing either beta1 or B (or both) leads to a decrease in the maximum U-V separation. This suggests that both higher linear birefringence and stronger cross-phase modulation tend to reduce the maximum separation between the two polarization components。

Key Findings:

5. Conclusion

This work aimed to study the nonlinear propagation of coupled pulses in birefringent optical fibers. But facing the difficulty of going through a direct integration method to obtain the solutions of the coupled nonlinear partial differential equation that governs the dynamics of waves in these waveguides, the iB-function was chosen to build the solution. Once this choice was made, the form of the solution to be constructed was specified. At this level, some calculations were done, which allowed the identification of the range of possible solutions, essentially outlining a domain where solutions could be found. While not all pairs within this range resulted in non-trivial solutions, we believe that most existing solutions can be located there. The challenge lies in the patience required to verify the existence of valid solutions. Analytical sequences were obtained that are solutions to the equations governing wave propagation in the birefringent fiber (dispersive and strongly nonlinear coupled Schrödinger equations). Finally, two families of solutions were found, which are the solitary waves and the trigonometric solutions for certain pairs, though not for all, as some pairs produced trivial solutions or even led to impossibilities. The principle used in this work was to adjust the properties of the fiber or waveguide to obtain coupled signals likely to propagate therein. Corresponding solutions were constructed numerically. The numerical solutions presented across the six cases obtained collectively illustrate the rich and diverse dynamics of coupled pulse propagation in birefringent optical fibers. They accurately capture complex phenomena such as stable propagation of a coupled dark soliton with minimal walk-off, indicating strong interaction between the components(Case 1), energy exchange between the components during propagation which reveals instability in maintaining their coupled state (Cases 3, 5 and 6), to severe pulse broadening and breakdown at high intensities (Cases 2, 3 and 4), and modulational instability, providing invaluable insights into the behavior of light in such nonlinear waveguides under various conditions, all influenced by the fiber’s birefringence and the initial pulse characteristics. This analysis could still consider higher-order dispersive effects or signal noise, which would provide additional information about the coupled pulse propagation. The consistency of these numerical observations with established theoretical solutions for these nonlinear optical effects enabled the validation of the accuracy of the numerical models employed. Moreover, a parametric study of the effect of birefringence parameters on soliton behaviour suggests that both higher linear birefringence parameter (beta 1) and stronger cross-phase modulation parameter (B) tend to reduce the maximum separation between the two polarization components, which is a strong indicator of group velocities walk-off. Walk-off can be a significant issue in optical communication systems, leading to pulse broadening if not properly managed. Regarding these results, we can confirm that our goal has been very well achieved because now it is easy for us to know exactly what types of signals we can use as propagating signal in birefringent optical fiber. A key recommendation from this work is to focus specifically on the manufacturing of optical fibers while considering the types of signals that can be best accommodated.

Funding

This research paper does not receive funding from any organization.

Acknowledgements

All authors approved the version of the manuscript to be published.

Data Availability Statement

All data that support the findings of this study are included within the article.

Authors’ Contributions

J. R. Bogning: Conception of the project, writing, printing, investigation, computation and numerical study, interpretation of results. M. Fomekong: Execution of the project, writing, printing, investigation, computation and numerical study, interpretation of results. M. N. Zambo Abou’o: Investigation, writing, printing, computation, verification of results. O. L. Ndichia: Writing, printing, investigation, computation, verification of results.

References 1

Paliwal, D.R. (2025) Optical Fiber Communication. International Journal of Scientific Research in Engineering and Management, 9, 1-9. >https://doi.org/10.55041/ijsrem47650

Mandeep, S., Debanuj, C., Suchita, Y., Sugeet, S., Karamdeep, S., Mrudula, K., Deepa, M.S.A., et al. (2024) Advancements in Optical Communication Research: A Review of India’s Progress. IEEE Photonics Journal, 99, 1-9.

Salama, R. and Al-Turjman, F. (2024) An Overview of Long-Distance Optical Fiber Communication. Journal of Biomedical Research&Environmental Sciences, 5, 1306-1312. >https://doi.org/10.37871/jbres2018

Agrawal, G.P. (2016) Optical Communication: Its History and Recent Progress. In: Al-Amri, M., El-Gomati, M. and Zubairy, M. Eds., Optics in Our Time, Springer International Publishing, 177-199. >https://doi.org/10.1007/978-3-319-31903-2_8

Khan, M.H.R. and Yesmin, L. (2001) Optical Fiber-Past, Present and Future: A Review. Khulna University Studies, 3, 1-10.

Zhao, Z. (2011) Past, Present and Future of Optical Fiber Communications. Acta Optica Sinica, 31, Article ID: 0900109. >https://doi.org/10.3788/aos201131.0900109

Alsharari, M., Aliqab, K., Ali, F. and Armghan, A. (2023) Integrated Free-Space Optics and Fiber Optic Network Performance Enhancement for Sustaining 5G High Capacity Communications. International Journal of Optics, 2023, Article ID: 8685686. >https://doi.org/10.1155/2023/8685686

Du, Y., Hu, G. and Bai, S. (2014) High Power All-Optical Gain-Clamped and Gain-Flattened Fiber Amplifier. Chinese Journal of Lasers, 41, Article ID: 1205003. >https://doi.org/10.3788/cjl201441.1205003

Ming-Jun, L. (2020) Optical Fiber Evolution Over the Past 5 Decades. Frontiers in Optics/Laser Science, OSA Technical Digest (Optica Publishing Group).

Motoharu, M. (2024) High-Power Optical Fiber Transmission Technologies for Radio-over-Fiber Networks. IEICE Transactions on Communications, 99, 1-11.

Zhong, Z.J., Liu, G.D., Yu, Y. and Chen, C. (2013) The High-Speed Video Signals Fiber Transmission Design Using of Optical Fiber in Photoelectric Tracking Equipment. Advanced Materials Research, 760, 298-301. >https://doi.org/10.4028/www.scientific.net/amr.760-762.298

Pant, D., Malik, D. and Dudeja, D. (2014) Recent Progress in Fiber-Optic Communication. Journal of Advance Research in Electrical&Electronics Engineering, 1, 6-9. >https://doi.org/10.53555/nneee.v1i3.250

Rottwitt, K. and Stentz, A.J. (2002) Raman Amplification in Lightwave Communication Systems. In: Kaminow, I.P. and Li, T., Eds., Optical Fiber Telecommunications IV-A, Elsevier, 213-257. >https://doi.org/10.1016/b978-012395172-4/50005-x

Liu, Y., Qi, Y., Cai, Y., Bao, X. and Gao, S. (2025) Recent Advances in Optical Fiber-Based Gas Sensors Utilizing Light-Induced Acoustic/Elastic Techniques. Photoacoutics, 43, Article ID: 100715. >https://doi.org/10.1016/j.pacs.2025.100715

Winzer, P.G. (2012) High-Capacity Optical Transmission: Moving towards Shannon’s Limit. IEEE Photonics Journal, 4, 157-167.

Taylor, J.R. (1996) Optical Solitons Theory and Experiment. Optical Engineering, 35, 2437-2439. >https://doi.org/10.1117/1.600816

Wurster, L.M. and Ginner, L. (2018) Endoscopic Optical Coherence Tomography with a Flexible Fiber Bundle. Journal of Biomedical Optics, 23, Article ID: 066001. >https://doi.org/10.1117/1.jbo.23.6.066001

Ochoa, M., Algorri, J.F., Roldán-Varona, P., Rodríguez-Cobo, L. and López-Higuera, J.M. (2021) Recent Advances in Biomedical Photonic Sensors: A Focus on Optical-Fibre-Based Sensing. Sensors, 21, Article 6469. >https://doi.org/10.3390/s21196469

Gu, R.Y., Mahalati, R.N. and Kahn, J.M. (2015) Design of Flexible Multi-Mode Fiber Endoscope. Optics Express, 23, 26905-26918. >https://doi.org/10.1364/oe.23.026905

Wu, G., Zhu, R., Lu, Y., Hong, M. and Xu, F. (2024) Optical Scanning Endoscope via a Single Multimode Optical Fiber. Opto-Electronic Science, 3, Article ID: 230041. >https://doi.org/10.29026/oes.2024.230041

Choksi, N. and Qian, L. (2024) Vector Theory of Optical Nonlinearities in Birefringent Fibers. Journal of Lightwave Technology, 42, 1-12.

Kumar, D.R. and Rao, B.P. (2012) Soliton Interaction in Birefringent Optical Fibers: Effects of Nonlinear Gain Devices. Optik, 123, 117-124. >https://doi.org/10.1016/j.ijleo.2011.03.009

Ahmed, K.K., Alsahafi, N.A., Ahmed, H.M., Boulaaras, S. and Osman, M.S. (2025) Diverse Solitons Wave Structures for Coupled NLSEs in Birefringent Fibers with Higher Nonlinearities Using the Modified Extended Mapping Algorithm. Scientific Reports, 15, Article No. 17047. >https://doi.org/10.1038/s41598-025-00668-1

Lakshmanan, M., Kanna, T. and Radhakrishnan, R. (2000) Shape-Changing Collisions of Coupled Bright Solitons in Birefringent Optical Fibers. Reports on Mathematical Physics, 46, 143-156. >https://doi.org/10.1016/s0034-4877(01)80018-5

Lozano-Crisostomo, N., Garcia-Melgarejo, J.C., Gomez-Correa, J.E., Gonzalez-Galicia, M.A. and Villegas-Martinez, B.M. (2023) Polarization Coherence Theorem in Birefringent Nonlinear Single-Mode Optical Fibers. Journal of Lightwave Technology, 41, 320-326. >https://doi.org/10.1109/jlt.2022.3213486

Tan, Z., Chen, X., Liu, N., Xu, J., Zhang, J., Zhu, Z., et al. (2024) Spatio‐Temporal Nonlinear Theory in Birefringent Microrings. Laser&Photonics Reviews, 18, Article ID: 2301281. >https://doi.org/10.1002/lpor.202301281

Yomba, E. and Ramchandra Nair, P. (2024) New Coupled Optical Solitons to Birefringent Fibers for Complex Ginzburg-Landau Equations with Hamiltonian Perturbations and Kerr Law Nonlinearity. Mathematics, 12, Article 3073. >https://doi.org/10.3390/math12193073

Balla, P. and Agrawal, G.P. (2018) Nonlinear Interaction of Vector Solitons inside Birefringent Optical Fibers. Physical Review A, 98, Article ID: 023822. >https://doi.org/10.1103/physreva.98.023822

Liu, J., Yi, Y. and Xu, T. (2025) Coupled-Mode Theory for Cross-Polarization Coupling in Optical Fibers. Journal of Lightwave Technology, 43, 1-10.

Jiang, Y., Tian, B., Liu, W., Sun, K. and Wang, P. (2013) Mixed-Type Solitons for the Coupled Higher-Order Nonlinear Schrödinger Equations in Multi-Mode and Birefringent Fibers. Journal of Modern Optics, 60, 629-636. >https://doi.org/10.1080/09500340.2013.798432

Bogning, J.R., Abou’ou, M.N.Z., Tchinda, C.R.N., Fomekong, M., Ndichia, O.L., Songna, S.M., et al. (2024) Influence of Waveguide Properties on Wave Prototypes Likely to Accompany the Dynamics of Four-Wave Mixing in Optical Fibers. Journal of Applied Mathematics and Physics, 12, 2601-2633. >https://doi.org/10.4236/jamp.2024.127155

Bogning, J.R. and Tchinda, C.N. (2021) Impact of the Properties of the Elliptical Birefringent Optical Fiber on the Nature and Propagation of Wave and Solitary Wave Solutions. International Journal of Scientific Engineering and Science, 5, 27-43.

Bogning, J.R. (2019) Mathematics for Nonlinear Physics: Solitary Wave in the Center of the Resolution of Dispersive Nonlinear Partial Differential Equations. USA Dorrance Publishing Co.

Bogning, J.R., Djeumen Tchaho, C. and Kofané, T. (2012) Generalization of the Bogning-Djeumen Tchaho-Kofané Method for the Construction of the Solitary Waves and the Survey of the Instabilities. Far East Journal of Dynamical Systems, 20, 101-119.

Ngouo Tchinda, C. and Roger Bogning, J. (2020) Solitary Waves and Property Management of Nonlinear Dispersive and Flattened Optical Fiber. American Journal of Optics and Photonics, 8, 27-43. >https://doi.org/10.11648/j.ajop.20200801.13