Dispersive Optical Gap Soliton Perturbation with Multiplicative White Noise

Abstract

This paper recovers dispersive gap solitons with the Kerr law of self-phase modulation and dispersive reflectivity. The enhanced direct algebraic method and the modified version of the sub-ODE approach have collectively made this retrieval possible. The intermediary solutions are the double-periodic functions that yielded the soliton solutions when the modulus of ellipticity approached unity. The Weierstrass elliptic function is the other form of intermediary function recovered from the model that also yielded soliton solutions as its special case.

1. Introduction

One of the issues of soliton propagation through transcontinental and transoceanic distances is the low count of chromatic dispersion (CD), irrespective of the governing model considered [1,2,3,4,5]. There are several ways to circumvent this situation. A viable procedure is the introduction of the third-order dispersion (3OD), which would make up for the dispersion deficiency. In this context, it is the Schrödinger–Hirota equation that is studied as the governing model instead of the familiar nonlinear Schrödinger’s equation (NLSE). Occasionally, this may not be enough to meet the desired count of dispersion for sustaining the delicate balance between dispersion and self-phase modulation (SPM) [6,7,8,9,10]. This paper, therefore, proposed an engineering marvel to meet this low count of dispersion. In addition to the introduction of 3OD, this paper considers Bragg gratings with dispersive reflectivity, which would assist in meeting the much-needed desired count of dispersion for the necessary balance between CD and SPM. Thus, it is the dispersive gap solitons that will be the focus of study in the paper [11,12,13,14,15,16]. To give it an additional twist to the project, the model would also be considered with the inclusion of multiplicative white noise. The SPM structure would also stem from the Kerr effect. Thus, the governing model would be the dispersive NLSE with multiplicative white noise and the Kerr law of SPM, and having dispersive reflectivity [17,18,19,20,21,22].

The integrability aspect of the model would be addressed in the paper. The target is to implement two forms of integration architectures to identify soliton solutions to the model that would paint a complete picture of the model as far as its integrability is concerned. The first integration tool that would be adopted is the enhanced direct algebraic method. The next approach would be the implementation of the modified version of the sub-ODE (ordinary differential equation) method. These couple of schemes would lead to the retrieval of several doubly periodic functions, namely the Jacobi’s elliptic functions, which would eventually yield a wide spectrum of soliton solutions when the modulus of ellipticity of the elliptic functions approaches unity. Additionally, the Weierstrass elliptic functions also emerged from the two integration schemes that gave way to soliton solutions as a special case. The complete set of recovered results is exhibited after familiarization with the model. The existence criteria of these soliton solutions are also presented.

The scientific motivation of this study is rooted in the fact that long–distance optical transmission systems frequently operate with an insufficient amount of chromatic dispersion to counterbalance the nonlinear Kerr-driven self-phase modulation. The inclusion of third-order dispersion can partially restore this delicate balance, which justifies the transition from the standard nonlinear Schrödinger equation to the Schrödinger–Hirota framework. In order to further enhance the effective dispersion budget, the present work considers fiber Bragg gratings endowed with dispersive reflectivity. This additional dispersive contribution plays a decisive role in sustaining gap-soliton propagation in realistic regimes. Since ultrashort and high-intensity pulse propagation is inevitably exposed to random fluctuations, the model is also augmented by multiplicative white noise that is treated within an Itô setting. As a result, the governing equations incorporate chromatic dispersion, third-order dispersion, Kerr nonlinearity, self-steepening, self-frequency shift, Bragg-type dispersive reflectivity, and stochastic perturbations in a physically comprehensive manner.

The novelty relative to existing stochastic soliton works is as follows. Previous studies typically addressed noisy models of the NLSE type or related integrable systems without dispersive reflectivity and without Bragg-induced bidirectional coupling. In contrast, the present formulation investigates dispersive optical gap solitons in a Schrödinger–Hirota setting that explicitly incorporates Bragg gratings with dispersive reflectivity while also preserving the influence of multiplicative white noise. The analysis further employs two complementary integration architectures, namely the enhanced direct algebraic method and a modified version of the sub-ODE technique, which together yield a broad catalogue of analytical coherent structures. These include bright, dark, break type, singular type, and elliptic families based on Jacobi and Weierstrass functions. The travelling-wave reduction isolates a common stochastic phase contribution while the deterministic envelope profiles remain governed by reduced ordinary differential equations. This reveals how stochasticity manifests at the level of coherent phase behaviour while leaving the intensity profile intact to leading order. In this way, the present work provides a unified analytical description of noisy dispersive gap solitons in Bragg environments with dispersive reflectivity, extending the scope of existing stochastic soliton theory.

Governing Model

In order to describe the evolution of nonlinear waves under the influence of stochastic effects, we consider the following governing equation that incorporates multiplicative white noise. In its dimensionless form, the equation is expressed as follows:

$$i{\psi }_t+a{\psi }_{xx}+ib{\psi }_{xxx}+c{\left|\psi \right|}^2\psi +\sigma \psi {\displaystyle \frac{dB\left(t\right)}{dt}}=i\left[\lambda {\left({\left|\psi \right|}^2\psi \right)}_x+\mu {\left({\left|\psi \right|}^2\right)}_x\psi +\theta {\left|\psi \right|}^2{\psi }_x\right].$$

(1)

Here, the unknown function $\psi (x,t)$ represents the complex envelope of the optical field, which evolves as a function of space x and time t. The symbol $i=\sqrt{-1}$ denotes the imaginary unit. The real coefficients a, b, c, $\sigma$, $\lambda$, $\mu$, and $\theta$ each govern specific physical mechanisms in the system. The parameter a accounts for chromatic dispersion, describing pulse broadening due to the frequency dependence of the group velocity. The parameter b corresponds to higher-order dispersion, which becomes particularly significant for ultrashort pulses. The coefficient c models self-phase modulation, resulting in intensity-dependent changes in the refractive index and nonlinear frequency chirping.

The stochastic term $\sigma \psi {\displaystyle \frac{dB\left(t\right)}{dt}}$ on the left-hand side introduces multiplicative white noise, where $B\left(t\right)$ denotes a standard Wiener process. The formal derivative ${\displaystyle \frac{dB\left(t\right)}{dt}}$ is interpreted as Gaussian white noise, and $\sigma$ quantifies the amplitude of the stochastic perturbation. On the right-hand side, the terms with coefficients $\lambda$, $\mu$, and $\theta$ account for higher-order nonlinear effects, including self-steepening and soliton self-frequency shift, which are especially important for the propagation of intense, ultrashort pulses. These higher-order contributions are essential for accurately describing nonlinear wave dynamics in optical fibers under realistic, high-intensity conditions.

When considering fiber Bragg gratings with dispersive reflectivity, the governing model in Equation (1) can be decoupled into a pair of coupled nonlinear partial differential equations. These are given by the following:

$$\begin{array}{c}i{q}_t+a_1{r}_{xx}+i{c}_1{r}_{xxx}+\left(d_1{\left|q\right|}^2+e_1{\left|r\right|}^2\right)q+\sigma q{\displaystyle \frac{dB\left(t\right)}{dt}}\hfill \\ =i\left[{\lambda }_1{\left({\left|q\right|}^{2}q\right)}_x+{\mu }_1{\left({\left|q\right|}^2\right)}_{x}q+{\theta }_1{\left|q\right|}^2{q}_x\right]+{\delta }_{1}r,\hfill \end{array}$$

(2)

and

$$\begin{array}{c}i{r}_t+a_2{q}_{xx}+i{c}_2{q}_{xxx}+\left(d_2{\left|r\right|}^2+e_2{\left|q\right|}^2\right)r+\sigma r{\displaystyle \frac{dB\left(t\right)}{dt}}\hfill \\ =i\left[{\lambda }_2{\left({\left|r\right|}^{2}r\right)}_x+{\mu }_2{\left({\left|r\right|}^2\right)}_{x}r+{\theta }_2{\left|r\right|}^2{r}_x\right]+{\delta }_{2}q,\hfill \end{array}$$

(3)

In these coupled equations, the unknown functions $q(x,t)$ and $r(x,t)$ represent the wave components, and their complex conjugates are denoted by $q^{*}(x,t)$ and $r^{*}(x,t)$. The initial terms correspond to linear temporal evolution. Constants $a_j$$(j=1,2)$ are associated with chromatic dispersion, while $c_j$$(j=1,2)$ denote the coefficients of third-order dispersion. Nonlinear interactions appear through $d_j$ (SPM coefficients) and $e_j$ (XPM coefficients). Furthermore, ${\lambda }_j,{\mu }_j,{\theta }_j$$(j=1,2)$ represent higher-order nonlinear corrections such as self-steepening and soliton frequency shift. The detuning effects in Bragg gratings are incorporated through the constants ${\delta }_j$.

The coupled system (2)–(3) generalizes the single governing model (1) by accounting for the bidirectional propagation of waves within fiber Bragg gratings. This makes it possible to capture more complex dynamics, including stochastic fluctuations and nonlinear coupling, which are essential in describing realistic optical communication systems.

The structure of this article is arranged as follows. Section 2 introduces the governing models and their relevant physical interpretations. Section 3 develops the mathematical framework required for analysis. In Section 4, the enhanced direct algebraic technique is applied, followed by the use of the modified sub-ODE method in Section 5. Section 6 presents a detailed account of additional results, and finally, Section 7 provides a comprehensive conclusion and discussion of the obtained findings.

2. Mathematical Preliminaries

To extract traveling-wave solutions of the coupled system (2)–(3), we adopt a standard coherent-structure ansatz that factors out a common stochastic phase and a deterministic carrier. Specifically, we suppose that the wave envelopes admit the forms

$$\begin{array}{c}q\left(x,t\right)=F_1\left(\eta \right)exp\left[i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)\right],\\ r\left(x,t\right)=F_2\left(\eta \right)exp\left[i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)\right],\end{array}$$

(4)

with the comoving coordinate

$$\eta =x-\rho t,$$

(5)

where $\rho$, $\kappa$, and $\omega$ are real parameters representing the envelope speed, the carrier wavenumber, and the carrier frequency, respectively. The envelope profiles $F_k\left(\eta \right)$ and $k=1,2$, are real-valued functions of the comoving coordinate $\eta$. In this ansatz, the exponential phase factor in (4) encapsulates both the deterministic carrier oscillations and the Itô-corrected stochastic phase modulation induced by the Wiener process $B\left(t\right)$. This representation factors out the common phase dynamics, reducing the original system to real-valued equations for the envelope functions $F_1$ and $F_2$ in the moving frame.

Substituting the Expressions (4) and (5) into the coupled Equations (2) and (3) and separating terms with equal functional dependence yields the following coupled ordinary differential equations in the unknown profiles $F_1$ and $F_2$:

$$\begin{array}{c}-\left(\omega -{\sigma }^2\right)F_1+\left[3c_1\kappa +a_1\right]F_2^{″}+\left[-\kappa \left({\lambda }_1+{\theta }_1\right)+d_1\right]F_1^3\hfill \\ +\left[-{\kappa }^3{c}_1-{\kappa }^2{a}_1-{\delta }_1\right]F_2+i{c}_1{F}_2^{‴}+e_1{F}_2^2{F}_1+i\left(-3{\lambda }_1-2{\mu }_1-{\theta }_1\right)F_1^2{F}_1^{\prime }\hfill \\ -i\rho F_1^{\prime }+i\left[-3{\kappa }^2{c}_1-2\kappa a_1\right]F_2^{\prime }=0,\hfill \end{array}$$

(6)

and

$$\begin{array}{c}-\left(\omega -{\sigma }^2\right)F_2+\left[3c_2\kappa +a_2\right]F_1^{″}+\left[-\kappa \left({\lambda }_2+{\theta }_2\right)+d_2\right]F_2^3\hfill \\ +\left[-{\kappa }^3{c}_2-{\kappa }^2{a}_2-{\delta }_2\right]F_1+i{c}_2{F}_1^{‴}+e_1{F}_1^2{F}_2+i\left(-3{\lambda }_2-2{\mu }_2-{\theta }_2\right)F_2^2{F}_2^{\prime }\hfill \\ -i\rho F_2^{\prime }+i\left[-3{\kappa }^2{c}_2-2\kappa a_2\right]F_1^{\prime }=0\hfill \end{array}$$

(7)

(Primes denote derivatives with respect to $\eta$). The structure of (6)–(7) reflects the expected interplay between dispersion, third-order dispersion, cubic nonlinearities, and the higher-order derivative-nonlinearity couplings inherited from the original PDEs, while the convective contributions arise through the transformation to the traveling frame.

To reduce the coupled system to a single scalar equation, we adopt the proportional-mode ansatz

$$F_2\left(\eta \right)=\chi F_1\left(\eta \right),$$

(8)

for a nonzero constant $\chi \ne 1$. This closure prescribes that the two counter-propagating (or coupled) envelopes share a fixed amplitude ratio and allows us to streamline the algebra. Under the relation (8), Equations (6) and (7) become

$$\begin{array}{c}-\left[\omega -{\sigma }^2+\left({\kappa }^3{c}_1+{\kappa }^2{a}_1+{\delta }_1\right)\chi \right]F_1+\left[3c_1\kappa +a_1\right]\chi F_1^{″}\hfill \\ +\left[-\kappa \left({\lambda }_1+{\theta }_1\right)+d_1+e_1\chi \right]F_1^3+i{c}_1\chi F_1^{‴}+i\left(-3{\lambda }_1-2{\mu }_1-{\theta }_1\right)F_1^2{F}_1^{\prime }\hfill \\ +i\left[-\rho -\left(3{\kappa }^2{c}_1+2\kappa a_1\right)\chi \right]F_1^{\prime }=0,\hfill \end{array}$$

(9)

and

$$\begin{array}{c}-\left[\left(\omega -{\sigma }^2\right)\chi +{\kappa }^3{c}_2+{\kappa }^2{a}_2+{\delta }_2\right]F_1+\left[3c_2\kappa +a_2\right]F_1^{″}\hfill \\ -\left[\kappa \left({\lambda }_2+{\theta }_2\right){\chi }^3+d_2{\chi }^2+e_1\chi \right]F_1^3+i{c}_2{F}_1^{‴}+i\left(-3{\lambda }_2-2{\mu }_2-{\theta }_2\right){\chi }^3{F}_1^2{F}_1^{\prime }\hfill \\ +i\left[-3{\kappa }^2{c}_2-2\kappa a_2-\rho \chi \right]F_1^{\prime }=0.\hfill \end{array}$$

(10)

Next, we separate real and imaginary parts in (9)–(10). The real components lead to the pair

$$\begin{array}{c}-\left[\omega -{\sigma }^2+\left({\kappa }^3{c}_1+{\kappa }^2{a}_1+{\delta }_1\right)\chi \right]F_1+\left[3c_1\kappa +a_1\right]\chi F_1^{″}\hfill \\ -\left[d_1+e_1{\chi }^2+\kappa \left({\lambda }_1+{\theta }_1\right)\right]F_1^3=0,\hfill \end{array}$$

(11)

and

$$\begin{array}{c}-\left[\left(\omega -{\sigma }^2\right)\chi +{\kappa }^3{c}_2+{\kappa }^2{a}_2+{\delta }_2\right]F_1+\left[3c_2\kappa +a_2\right]F_1^{″}\hfill \\ -\left[d_2{\chi }^3+e_1\chi +\kappa \left({\lambda }_2+{\theta }_2\right){\chi }^3\right]F_1^3=0\hfill \end{array}$$

(12)

whereas the imaginary components yield the third-order relations

$$\left[-\rho -\left(3{\kappa }^2{c}_1+2\kappa a_1\right)\chi \right]F_1^{\prime }+c_1\chi F_1^{‴}-\left(3{\lambda }_1+2{\mu }_1+{\theta }_1\right)F_1^2{F}_1^{\prime }=0,$$

(13)

and

$$\left[-3{\kappa }^2{c}_2-2\kappa a_2-\rho \chi \right]F_1^{\prime }+c_2{F}_1^{‴}-\left(3{\lambda }_2+2{\mu }_2+{\theta }_2\right){\chi }^3{F}_1^2{F}_1^{\prime }=0.$$

(14)

Integrating (13)–(14) once with respect to $\eta$ and selecting the zero integration constants (a choice consistent with localized or periodic profiles that do not introduce secular growth in the primitive), we arrive at the second-order relations

$$\left[-\rho -\left(3{\kappa }^2{c}_1+2\kappa a_1\right)\chi \right]F_1+c_1\chi F_1^{″}+{\displaystyle \frac{1}{3}}\left(-3{\lambda }_1-2{\mu }_1-{\theta }_1\right)F_1^3=0,$$

(15)

and

$$\left[-3{\kappa }^2{c}_2-2\kappa a_2-\rho \chi \right]F_1+c_2{F}_1^{″}+{\displaystyle \frac{1}{3}}\left(-3{\lambda }_2-2{\mu }_2-{\theta }_2\right){\chi }^3{F}_1^3=0.$$

(16)

The pairs (11), (12) and (15), (16) become mutually compatible provided a set of algebraic compatibility (constraint) conditions is enforced among the parameters. Collecting the coefficients of like powers of $F_1$ and its derivatives leads to

$$\begin{array}{c}-\left[\omega -{\sigma }^2+\left({\kappa }^3{c}_1+{\kappa }^2{a}_1+{\delta }_1\right)\chi \right]=-\left[\left(\omega -{\sigma }^2\right)\chi +{\kappa }^3{c}_2+{\kappa }^2{a}_2+{\delta }_2\right]\hfill \\ =\left[-\rho -\left(3{\kappa }^2{c}_1+2\kappa a_1\right)\chi \right]=\left[-3{\kappa }^2{c}_2-2\kappa a_2-\rho \chi \right],\hfill \end{array}$$

(17)

$$\left(3c_1\kappa +a_1\right)\chi =\left(3c_2\kappa +a_2\right)=c_1\chi =c_2.$$

(18)

$$\begin{array}{c}-\left[d_1+e_1{\chi }^2+\kappa \left({\lambda }_1+{\theta }_1\right)\right]=-\left[d_2{\chi }^3+e_1\chi +\kappa \left({\lambda }_2+{\theta }_2\right){\chi }^3\right]\hfill \\ ={\displaystyle \frac{1}{3}}\left(-3{\lambda }_1-2{\mu }_1-{\theta }_1\right)={\displaystyle \frac{1}{3}}\left(-3{\lambda }_2-2{\mu }_2-{\theta }_2\right){\chi }^3.\hfill \end{array}$$

(19)

Under the compatibility relations (17)–(19), the profile equation reduces to a single scalar nonlinear oscillator for $F_1\left(\eta \right)$. In particular, from (11) we obtain

$$\begin{array}{c}{F}_1^{″}\left(\eta \right)+{\Omega }_1{F}_1\left(\eta \right)+{\Omega }_2{F}_1^3\left(\eta \right)=0,\end{array}$$

(20)

with effective linear and cubic coefficients

$$\left\{\begin{array}{c}{\Omega }_1={\displaystyle \frac{-\left[\omega -{\sigma }^2+\left({\kappa }^3{c}_1+{\kappa }^2{a}_1+{\delta }_1\right)\chi \right]}{\left[3c_1\kappa +a_1\right]\chi }},\hfill \\ {\Omega }_2={\displaystyle \frac{-\left[d_1+e_1{\chi }^2+\kappa \left({\lambda }_1+{\theta }_1\right)\right]}{\left[3c_1\kappa +a_1\right]\chi }},\hfill \end{array}\right.$$

(21)

provided that $3c_1\kappa +a_1\ne 0$. In subsequent sections, we shall exploit this normal form to construct explicit solution families via the chosen integration techniques while consistently respecting the constraints (17)–(19).

3. Enhanced Direct Algebraic Method

In this section, we employ the enhanced direct algebraic method [23,24,25] to construct explicit traveling-wave profiles for the reduced ODE (20). The core idea is to express the target solution as a finite superposition of powers (both positive and negative) of an auxiliary function $\varphi \left(\eta \right)$ that itself satisfies a first-integral-type differential relation. In this way, the nonlinear balance between the dispersive term $F_1^{\prime \prime }$ and the cubic nonlinearity $F_1^3$ is transferred into algebraic constraints among the expansion coefficients and the parameters governing $\varphi$. Throughout this section, no change is made to the parameters or equations; we merely reorganize and expand the narrative to clarify the computational steps and the structure of the resulting solution families.

This technique [23,24,25] leads us to assume that the formal solution to Equation (20) exists:

$$F_1\left(\eta \right)={\alpha }_0+\sum _{j=1}^N\left\{{\alpha }_j{\varphi }^j\left(\eta \right)+{\beta }_j{\varphi }^{-j}\left(\eta \right)\right\},$$

(22)

where ${\alpha }_0,{\alpha }_j,{\beta }_j$$(j=1,\dots ,N)$ are arbitrary constants, provided ${\alpha }_N^2+{\beta }_N^2\ne 0$, while $\varphi \left(\eta \right)$ is the solution of the equation:

$${\varphi \prime }^2\left(\eta \right)=\sum _{l=0}^4{L}_l{\varphi }^l\left(\eta \right),$$

(23)

where $L_j(j=0,1,2,3,4)$ are constants, provided $L_4\ne 0$. When $F_1^{″}\left(\eta \right)$ and $F_1^3\left(\eta \right)$ are balanced in Equation (20), the balance number $N=1$ is obtained. Now, the formal solution to Equation (20) is as follows:

$$F_1\left(\eta \right)={\alpha }_0+{\alpha }_1\varphi \left(\eta \right)+{\displaystyle \frac{{\beta }_1}{\varphi \left(\eta \right)}},$$

(24)

where ${\alpha }_0,{\alpha }_1$, and ${\beta }_1$ need to be found, given that ${\alpha }_1^2+{\beta }_1^2\ne 0$. Equations (23) and (24) are substituted into Equation (20) to create a system of algebraic equations, by putting all the coefficients of $F^i\left(\xi \right){\left(F^{\prime }\left(\xi \right)\right)}^j$, $\left(i=-3,\dots ,-1,0,1,2,\dots ,3,j=0,1\right)$ to zero.

$$\left.\begin{array}{c}\begin{array}{c}{\varphi }^3\left(\eta \right):{\Omega }_2{\alpha }_1^3+2{\alpha }_1{L}_4=0,\hfill \\ \\ {\varphi }^2\left(\eta \right):{\displaystyle \frac{3}{2}}{\alpha }_1{L}_3+3{\Omega }_2{\alpha }_0{\alpha }_1^2=0,\hfill \\ \\ \varphi \left(\eta \right):3{\Omega }_2{\alpha }_0^2{\alpha }_1+3{\Omega }_2{\alpha }_1^2{\beta }_1+{\Omega }_1{\alpha }_1+{\alpha }_1{L}_2=0,\hfill \\ \\ {\varphi }^0\left(\eta \right):{\displaystyle \frac{1}{2}}{\alpha }_1{L}_1+{\displaystyle \frac{1}{2}}{L}_3{\beta }_1+{\Omega }_1{\alpha }_0+{\Omega }_2{\alpha }_0^3+6{\Omega }_2{\alpha }_0{\alpha }_1{\beta }_1=0,\hfill \\ \\ {\varphi }^{-1}\left(\eta \right):3{\Omega }_2{\alpha }_0^2{\beta }_1+3{\Omega }_2{\alpha }_1{\beta }_1^2+{\Omega }_1{\beta }_1+L_2{\beta }_1=0,\hfill \\ \\ {\varphi }^{-2}\left(\eta \right):{\displaystyle \frac{3}{2}}{L}_1{\beta }_1+3{\Omega }_2{\alpha }_0{\beta }_1^2=0,\hfill \\ \\ {\varphi }^{-3}\left(\eta \right):{\Omega }_2{\beta }_1^3+2{\beta }_1{L}_0=0.\hfill \end{array}\end{array}\right\}$$

(25)

The nonlinear algebraic set (25) encodes all admissible amplitude-phase combinations consistent with the balance in (20). Different choices of the auxiliary parameters $L_j$ (and the nonzero-ness conditions listed above) carve out distinct solution branches. In what follows, we list two representative cases—each specified by a particular selection of $L_j$—and state the corresponding constraints ensuring that (24) solves (20) without altering any physical parameter introduced earlier.

• Case-1: If we set $L_0=$$L_1=$$L_3=0$, using the Maple and the algebraic system Equation (25), we obtain the following results:

  $${\alpha }_0=0,{\beta }_1=0,{\alpha }_1={\alpha }_1,$$

  (26)

  with constraint conditions:

  $$\left\{\begin{array}{c}{\Omega }_1=-L_2,\hfill \\ {\Omega }_2=-{\displaystyle \frac{2L_4}{{\alpha }_1^2}}.\hfill \end{array}\right.$$

  (27)

  In this regime, the auxiliary profile $\varphi$ reduces to the elementary hyperbolic families, and the cubic balance fixes the effective coefficients ${\Omega }_1,{\Omega }_2$ through $L_2$ and $L_4$. Depending on the sign pattern of $(L_2,L_4)$, one obtains either localized (bell-shaped) or singular envelopes, as detailed below.

• (I) When $L_2>0$ and $L_4<0$, Equations (2) and (3) have bell-shaped soliton solutions:

  $$q(x,t)=\left[{\alpha }_1\sqrt{-{\displaystyle \frac{L_2}{L_4}}}sech\sqrt{L_2}\eta \right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (28)

  For all subsequent solutions, we impose the ansatz $r(x,t)=\chi q(x,t).$
• (II) When $L_2>0$ and $L_4>0$, Equations (1) and (2) have singular soliton solutions:

  $$q(x,t)=\left[{\alpha }_1\sqrt{-{\displaystyle \frac{L_2}{L_4}}}csch\sqrt{L_2}\eta \right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (29)

Under the constraint conditions (27), the solutions (28) and (29) exist. In practice, (28) corresponds to bright solitons with finite energy, while (29) represents non-integrable singular profiles; both stem from the same algebraic backbone, yet they differ in the admissible sign configuration of the auxiliary invariants.

• Case-2: If we set $L_0=$${\displaystyle \frac{L_2^2}{4L_4}},$$L_1=$$L_3=0$. Using the Maple and the algebraic system Equation (25), we obtain the following results:

  $${\alpha }_0={\alpha }_0,{\beta }_1={\beta }_1,{\alpha }_1=0,$$

  (30)

  with constraint conditions:

  $$\left\{\begin{array}{c}{\Omega }_1=-L_2,\hfill \\ {\Omega }_2=-{\displaystyle \frac{2L_2^2}{2{\beta }_1^2{L}_4}}.\hfill \end{array}\right.$$

  (31)

  Here, the expansion collapses to the negative-power branch (since ${\alpha }_1=0$), which selects another sector of the auxiliary phase portrait of $\varphi$. The condition $L_0=L_2^2/\left(4L_4\right)$ is precisely the degeneracy that permits rational combinations of hyperbolic functions when (23) is integrated, yielding either singular or kink-type envelopes for $F_1$.

• (I) When $L_4>0,L_2<0$. Equations (2) and (3) have the kink-shaped soliton solutions:

  $$q(x,t)=\left[{\displaystyle \frac{2{\beta }_{1}tanh\left(\sqrt{-\frac{L_2}{2}}\eta \right)}{\sqrt{-\frac{2L_2}{L_4}}}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (32)

  Also, Equations (2) and (3) have the singular soliton solutions:

  $$q(x,t)=\left[{\displaystyle \frac{2{\beta }_{1}coth\left(\sqrt{-\frac{L_2}{2}}\eta \right)}{\sqrt{-\frac{2L_2}{L_4}}}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (33)

  The solutions (32) and (33) exist under the constraint condition (31). From a qualitative viewpoint, (32) describes heteroclinic (kink/antikink) connections between asymptotic plateaus in the reduced phase space, whereas (33) represents profiles with pole-type singularities arising from the $coth$ structure.

The enhanced direct algebraic method reduces the nonlinear profile Equation (20) to a set of algebraic relations (25) indexed by the auxiliary parameters $L_j$. The explicit families (28), (29), (32), and (33) emerge as consistent realizations under the corresponding constraints (27) and (31). These results will be complemented in the subsequent section by solutions generated through the modified sub-ODE methodology, thereby enlarging the catalog of admissible coherent structures supported by the stochastic coupled model without altering its governing coefficients.

• Case-3: If we set $L_1=$$L_3=0$, using the Maple and the algebraic system Equation (25), we obtain the following results:

• (I) When $L_0={\displaystyle \frac{m_1^2(1-m_1^2)L_2}{(2m_1^2-1)L_4}}$,$0<m_1<1,$ we get

  $${\alpha }_0=0,{\alpha }_1=0,{\beta }_1={\beta }_1,L_4={\displaystyle \frac{2m^2(m^2-1)L_2^2}{{\beta }_1^2{\Omega }_2(4m^4-4m^2+1)}},$$

  (34)

  with constraint conditions:

  $${\Omega }_1=-L_2.$$

  (35)

  The Jacobi elliptic doubly periodic-type soliton solutions are now found for Equations (2) and (3):

  $$q(x,t)=\left[{\displaystyle \frac{{\beta }_1}{\sqrt{-\frac{m_1^2{L}_2}{(2m_1^2-1)L_4}}\mathrm{cn}\left(\sqrt{\frac{L_2}{2m_1^2-1}}\eta ,m_1\right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (36)

Provided that $(2m_1^2-1)L_2>0$ and $L_4<0$, constraint condition (35) exists for the solution (36).

• (II) When $L_0={\displaystyle \frac{(1-m_1^2)L_2^2}{{(2-m_1^2)}^2{L}_4}}$,$0<m_1<1,$ we get

  $${\alpha }_0=0,{\alpha }_1=0,{\beta }_1={\beta }_1,L_4={\displaystyle \frac{2(m^2-1)L_2^2}{{\beta }_1^2{\Omega }_2(m^4-4m^2+4)}},$$

  (37)

  with constraint conditions:

  $${\Omega }_1=-L_2.$$

  (38)

  Now, Equations (2) and (3) have the Jacobi elliptic doubly periodic type soliton solutions:

  $$q(x,t)=\left[{\displaystyle \frac{{\beta }_1}{\sqrt{-\frac{m_1^2{L}_2}{(2-m_1^2)L_4}}\mathrm{dn}\left(\sqrt{\frac{L_2}{2-m_1^2}}\eta ,m_1\right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (39)

Provided that $(2-m_1^2)L_2>0$ and $L_4<0$, the constraint conditions are satisfied. In particular, if $m_1\to 1$ in (36) and (39), we obtain a bell-shaped soliton solution.

$$q(x,t)=\left[-{\displaystyle \frac{{\beta }_1}{\sqrt{-\frac{L_2}{L_4}}\mathrm{sech}\left(\sqrt{L_2}\eta \right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

(40)

Provided that $L_2>0$ and $L_4<0$, the solutions (39) and (40) exist under the constraint condition (38).

• (III) When $L_0={\displaystyle \frac{m_1^2{L}_2^2}{{(m_1^2+1)}^2{L}_4}}$, $0<m_1<1,$ we get

  $${\alpha }_0=0,{\alpha }_1=0,{\beta }_1={\beta }_1,L_4=-{\displaystyle \frac{2m^2{L}_2^2}{{\beta }_1^2{\Omega }_2(m^4+2m^2+1)}},$$

  (41)

  with constraint conditions:

  $${\Omega }_1=-L_2.$$

  (42)

  The Jacobi elliptic doubly periodic-type soliton solutions have been obtained for Equations (2) and (3):

  $$q(x,t)=\left[{\displaystyle \frac{{\beta }_1}{\sqrt{-\frac{m_1^2{L}_2}{(1+m_1^2)L_4}}\mathrm{sn}\left(\sqrt{-\frac{L_2}{1+m_1^2}}\eta ,m_1\right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (43)

Provided that $L_2<0$ and $L_4>0$, the constraint conditions are satisfied. In particular, when $m_1\to 1$ in Equation (43), we obtain kink-shaped soliton solutions.

$$q(x,t)=\left[{\displaystyle \frac{{\beta }_1}{\sqrt{-\frac{L_2}{2L_4}}\tanh \left(\sqrt{-\frac{L_2}{2}}\eta \right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

(44)

The solutions (43) and (44) exist under the constraint condition (42).

Remark 1.

In Case-3, the choice $L_1=L_3=0$ confines the auxiliary Equation (23) to a quartic polynomial without odd powers, which is precisely the setting that admits Jacobi elliptic functions as integrals of motion. Subcases (I)–(III) correspond respectively to $\mathrm{cn}$-, $\mathrm{dn}$-, and $\mathrm{sn}$-type envelopes. The sign constraints on $L_2$ and $L_4$ determine whether the resulting profiles are bounded and periodic or kink/bright limits obtained as $m_1\to 1$. The degenerations $\mathrm{cn},\mathrm{dn}\to \mathrm{sech}$ and $\mathrm{sn}\to tanh$ are consistent with the hyperbolic soliton solutions listed, ensuring continuity of the solution families across the elliptic modulus.

• Case-4: If we set $L_1=L_3=0$, using the Maple and the algebraic system Equation (25), we obtain the following results:

  $${\alpha }_0=0,{\beta }_1=0,{\alpha }_1=\sqrt{-\frac{2L_4}{{\Omega }_2}},$$

  (45)

  with constraint conditions

  $${\Omega }_1=-L_2.$$

  (46)

  Here are the solutions of the Weierstrass elliptic doubly periodic-type for Equations (2) and (3):

  $$q(x,t)=\left[3\sqrt{-\frac{2}{{\Omega }_2}}\left(\frac{{\wp }^{\prime }\left[\left(\eta \right),g_2,g_3\right]}{6\wp \left[\left(\eta \right),g_2,g_3\right]+L_2}\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (47)

  where ${\Omega }_2<0,L_4>0.$ Also, Equations (2) and (3) have

  $$q(x,t)=\left[{\displaystyle \frac{1}{3}}\sqrt{-\frac{2L_4{L}_0}{{\Omega }_2}}\left(\frac{6\wp \left[\left(\eta \right),g_2,g_3\right]+L_2}{{\wp }^{\prime }\left[\left(\eta \right),g_2,g_3\right]}\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (48)

  where $L_4{L}_0>0$ and ${\Omega }_2<0$. The invariants $g_2$ and $g_3$ of the Weierstrass elliptic function solutions (47) and (48) are given by the following:

  $$g_2=\frac{L_2^2+12L_0{L}_4}{12},g_3=\frac{L_2}{216}\left(36L_0{L}_4-L_2^2\right).$$

  (49)

Remark 2.

In this configuration, setting $L_1=L_3=0$ reduces the auxiliary Equation (23) to a biquadratic form, whose general integral can be expressed in terms of the Weierstrass ℘-function. The two representations, (47) and (48), are reciprocal in the sense of expressing ${\wp }^{\prime }$ as a quotient by an affine function of ℘, versus the inverse ratio. Both are admissible, provided that the sign constraints stated above are satisfied. The invariants $(g_2,g_3)$ in (49) encode the lattice of periods and thereby govern the double periodicity of the envelopes.

• Case-5: If we set $L_0=L_1=0,$ using the Maple and the algebraic system Equation (25), we obtain the following results:

  $${\alpha }_0=0,{\beta }_1=0,{\alpha }_1=\sqrt{-\frac{2L_4}{{\Omega }_2}},L_3=0,$$

  (50)

  with constraint conditions

  $${\Omega }_1=-L_2.$$

  (51)

  Now, the straddled soliton solutions have been identified for Equations (2) and (3), when $L_2>0$, ${\Omega }_1<0$ and $L_4>0$, ${\Omega }_2<0:$

  $$q(x,t)=\epsilon \left[{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{2{\Omega }_1}{{\Omega }_2}}}{\displaystyle \frac{{\mathrm{sech}}^2\frac{\sqrt{-{\Omega }_1}}{2}\left(\eta \right)}{tanh\frac{\sqrt{-{\Omega }_1}}{2}\left(\eta \right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (52)

  also,

  $$q(x,t)=\epsilon \left[{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{2{\Omega }_1}{{\Omega }_2}}}{\displaystyle \frac{{\mathrm{csch}}^2\frac{\sqrt{-{\Omega }_1}}{2}\left(\eta \right)}{coth\frac{\sqrt{-{\Omega }_1}}{2}\left(\eta \right)}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (53)

Remark 3.

Setting $L_0=L_1=L_3=0$ selects a degenerate branch of the auxiliary dynamics in which the ϕ-Equation (23) admits elementary hyperbolic primitives. The resulting “straddled” structures (62) and (53) can be interpreted as derivative-weighted bright/anti-bright envelopes: the quotient of ${\mathrm{sech}}^2$ or ${\mathrm{csch}}^2$ by tanh or coth emphasizes steep flanks around the core. The sign pattern ${\Omega }_1<0$ and ${\Omega }_2<0$ ensures the correct balance for localized profiles, while $L_2>0$ and $L_4>0$ are consistent with the constraints in (51) and with the definition of ${\alpha }_1$ in (50), without altering any governing parameter.

4. Modified Sub-ODE Approach

In addition to the enhanced direct algebraic method, another systematic way to obtain explicit solutions for nonlinear evolution equations is through the modified sub-ODE approach. This strategy was originally proposed by Zayed and Alngar [26], and later generalized by Zi-Liang-Li [27]. For the current work, we adopt the refined version described by Zayed et al. [28] in order to construct new soliton families for Equation (20). The essence of the method lies in embedding the unknown solution into an auxiliary polynomial differential equation (the sub-ODE) whose known solutions span elliptic, trigonometric, and hyperbolic function families. By carefully balancing nonlinear and dispersive contributions, one can identify the correct form of the auxiliary equation and reduce the nonlinear PDE problem to algebraic constraints.

We begin by assuming that there is a formal expansion of the solution of Equation (20) in the form

$$F_1\left(\eta \right)=\sum _{s=0}^N{A}_s{\left[H\left(\eta \right)\right]}^s,$$

(54)

where $A_s(s=0,1,2,\dots ,N)$ are real constants, with the leading coefficient $A_N\ne 0$, and where $H\left(\eta \right)$ is a function governed by an auxiliary ODE of the type

$$H^{\prime 2}\left(\eta \right)=A{H}^{2-2p}\left(\eta \right)+B{H}^{2-p}\left(\eta \right)+C{H}^2\left(\eta \right)+D{H}^{2+p}\left(\eta \right)+E{H}^{2+2p}\left(\eta \right),$$

(55)

with constants $A,B,C,D,E$ and a positive integer p. The flexibility of this auxiliary structure allows $H\left(\eta \right)$ to represent a wide range of elementary or elliptic functions depending on the parameter choice. In particular, soliton and periodic solutions of Equation (20) are obtained by considering different canonical solutions of Equation (55) [26,28].

From the definition of the degree operator $D[·]$ applied to $H\left(\eta \right)$ and its derivatives, one notes that

$$D\left[H\left(\eta \right)\right]=N,D\left[H^{\prime }\left(\eta \right)\right]=N+p,D\left[H^{″}\left(\eta \right)\right]=N+2p,$$

and more generally $D\left[H^{\left(r\right)}\left(\eta \right)\right]=N+rp$, while mixed terms satisfy $D\left[H^{\left(r\right)}\left(\eta \right)H^s\left(\eta \right)\right]=(s+1)N+pr$.

In order to ensure balance between the dispersive contribution $F_1^{″}\left(\eta \right)$ and the cubic nonlinear term $F_1^3\left(\eta \right)$ in Equation (20), we equate their orders:

$$N+2p=3N⟹N=p.$$

(56)

Thus, selecting $p=1$ leads to $N=1$, so that the assumed solution takes the linear form

$$F_1\left(\eta \right)=A_0+A_{1}H\left(\eta \right),$$

(57)

where $A_0$ and $A_1$ are constants, $A_1\ne 0$, and $H\left(\eta \right)$ satisfies the reduced auxiliary ODE

$$H^{\prime 2}\left(\eta \right)=A+BH\left(\eta \right)+C{H}^2\left(\eta \right)+D{H}^3\left(\eta \right)+E{H}^4\left(\eta \right),E\ne 0.$$

(58)

Substituting (57) into Equation (20), and making use of the defining relation (58), one can collect all terms of the form ${\left[H\left(\eta \right)\right]}^l{\left[H^{\prime }\left(\eta \right)\right]}^f$ with indices $l=0,1,2,3$ and $f=0,1$. Setting the coefficients of each independent structure to zero yields the following system of algebraic equations:

$$\left\{\begin{array}{c}{H}^3\left(\eta \right):{\Omega }_2{A}_1^3+2A_{1}E=0,\hfill \\ \\ H^2\left(\eta \right):\frac{3}{2}{A}_{1}D+3{\Omega }_2{A}_0{A}_1^2=0,\hfill \\ \\ H\left(\eta \right):3{\Omega }_2{A}_0^2{A}_1+C{A}_1+A_1{\Omega }_1=0,\hfill \\ \\ H^0\left(\eta \right):\frac{1}{2}{A}_{1}B+{\Omega }_1{A}_0+{\Omega }_2{A}_0^3=0,\hfill \end{array}\right.$$

(59)

Equation set (59) encodes the nonlinear algebraic compatibility conditions linking the coefficients $A_0,A_1$ with the auxiliary parameters $A,B,C,D,E$ and the effective constants ${\Omega }_1,{\Omega }_2$ defined earlier in Equation (21). Depending on how these constants are selected, different families of optical soliton solutions—bright, dark, kink-type, or periodic—may be generated.

In the following subsections, we shall explore several particular realizations of (59), each leading to distinct closed-form solutions of the coupled system (2)–(3) under the stochastic perturbations considered. Each case highlights how the modified sub-ODE approach broadens the spectrum of admissible solutions beyond those obtainable through the enhanced direct algebraic method.

• Set-1: By substituting the specific values $A=B=D=0$ into the framework, and subsequently applying Maple together with the algebraic relations in Equation (59), we arrive at the following results:

  $$A_0=0,A_1=\sqrt{-\frac{2E}{{\Omega }_2}},$$

  (60)

  accompanied by the corresponding constraint condition:

  $$C=-{\Omega }_1.$$

  (61)

  Under the assumptions that $E<0,{\Omega }_2>0,C>0,$ and ${\Omega }_1<0$, the system admits a bright soliton-type solution, which can be expressed as follows:

  $$q(x,t)=\epsilon \left[\sqrt{-\frac{2{\Omega }_1}{{\Omega }_2}}\mathrm{sech}\sqrt{-{\Omega }_1}\eta \right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (62)

It is clear that the solution presented in (62) is valid, provided that the restriction condition given in (61) is satisfied.

• Set-2: In the second configuration, we impose $B=D=0$ and assign $A={\displaystyle \frac{C^2}{4E}}$. By employing Maple alongside the algebraic formulation in Equation (59), the solutions simplify to the following:

  $$A_0=0,A_1=\sqrt{-\frac{2E}{{\Omega }_2}},$$

  (63)

  together with the following constraint:

  $$C=-{\Omega }_1.$$

  (64)
  Under these conditions, one obtains the dark soliton class of solutions, which can be written in the following explicit form:

  $$q(x,t)=\epsilon \left[\sqrt{-\frac{{\Omega }_1}{{\Omega }_2}}\tanh \sqrt{{\displaystyle \frac{{\Omega }_1}{2}}}\eta \right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (65)

The validity of these solutions requires the restrictions $C<0$, ${\Omega }_1>0$, ${\Omega }_2<0$, together with $\epsilon =\pm 1$. Thus, the dark soliton structures described by (65) emerge only under the condition summarized in (64).

• Set-3: By considering the substitutions $B=D=0$ and $A={\displaystyle \frac{e_1{C}^2}{E}}$, and by applying Maple together with the system of algebraic equations given in (59), we are led to the following relations:

  $$A_0=0,A_1=\sqrt{-\frac{2E}{{\Omega }_2}},$$

  (66)

  with the associated constraint condition expressed as

  $$C=-{\Omega }_1.$$

  (67)

  Here, $e_1$ represents an arbitrary constant. Based on particular selections of $e_1$, one can derive a variety of families of solutions.

• (I) In the case when $e_1={\displaystyle \frac{m_1^2(m_1^2-1)}{{(2m_1^2-1)}^2}}$, then it follows that $A={\displaystyle \frac{C^2{m}_1^2(m_1^2-1)}{E{(2m_1^2-1)}^2}}$. Assuming the modulus parameter $m_1$ satisfies $0<m_1<1$, the system admits Jacobi elliptic function type solutions of the form:

  $$q(x,t)=\epsilon \left[{\left(-{\displaystyle \frac{2{\Omega }_1{m}_1^2}{{\Omega }_2(2m_1^2-1)}}\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{cn}\left(\sqrt{{\displaystyle \frac{-{\Omega }_1}{2m_1^2-1}}}\eta ,m_1\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (68)

  which are consistent when ${\Omega }_1(2m_1^2-1)<0,{\Omega }_2>0,$ and $E<0$.
• (II) For the alternative case where $e_1={\displaystyle \frac{(1-m_1^2)}{{(2-m_1^2)}^2}}$, it follows that $A={\displaystyle \frac{C^2(1-m_1^2)}{E{(2-m_1^2)}^2}}$, with the range $0<m_1<1$. Under these conditions, the Jacobi elliptic solutions take the following form:

  $$q(x,t)=\epsilon \left[{\left(-{\displaystyle \frac{2{\Omega }_1}{{\Omega }_2(2-m_1^2)}}\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{dn}\left(\sqrt{-{\displaystyle \frac{{\Omega }_1}{2-m_1^2}}}\eta ,m_1\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (69)

  valid under ${\Omega }_1<0$ and ${\Omega }_2>0$. Furthermore, in the limiting case when $m_1\to 1^{-}$ in (68) and (69), the elliptic solutions degenerate to the well-known bright soliton structures:

  $$q(x,t)=\epsilon \left[{\left(-{\displaystyle \frac{2{\Omega }_1}{{\Omega }_2}}\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{sech}\left(\sqrt{-{\Omega }_1}\eta \right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (70)
• (III) Finally, when $e_1={\displaystyle \frac{m_1^2}{{(m_1^2+1)}^2}}$, then $A={\displaystyle \frac{C^2{m}_1^2}{E{(m_1^2+1)}^2}}$, for $0<m_1<1$. In this situation, the Jacobi elliptic solutions are obtained as follows:

  $$q(x,t)=\epsilon \left[{\left(-{\displaystyle \frac{2{\Omega }_1{m}_1^2}{{\Omega }_2(m_1^2+1)}}\right)}^{{\displaystyle \frac{1}{2}}}\mathrm{sn}\left(\sqrt{{\displaystyle \frac{{\Omega }_1}{m_1^2+1}}}\eta ,m_1\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (71)

  These remain consistent when ${\Omega }_1>0$ and ${\Omega }_2<0$. Furthermore, in the limit $m_1\to 1^{-}$, Expression (71) reduces to the standard dark soliton solutions:

  $$q(x,t)=\epsilon \left[{\left(-{\displaystyle \frac{{\Omega }_1}{{\Omega }_2}}\right)}^{{\displaystyle \frac{1}{2}}}\tanh \left(\sqrt{{\displaystyle \frac{{\Omega }_1}{2}}}\eta \right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (72)

In summary, all solutions ranging from (68) to (72) hold true under the restriction provided by condition (67).

• Set-4: Taking now the case where $A=B=0$, and once again employing Maple in conjunction with the algebraic system (59), we distinguish two subcases as follows:

• (I) If $C>0$ and $E={\displaystyle \frac{D^2}{4C}}-C$, the obtained relations are

  $$A_0=0,A_1=\sqrt{{\displaystyle \frac{2C}{{\Omega }_2}}},D=0,$$

  (73)

  with the constraint

  $${\Omega }_1=-C.$$

  (74)
  For this case, the corresponding bright soliton solutions are as follows:

  $$q(x,t)=\epsilon \left[{\displaystyle \frac{\sqrt{\frac{2C}{{\Omega }_2}}}{cosh\sqrt{C}\eta }}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (75)

These are valid provided that $C>0$ and ${\Omega }_2>0$. The solutions (75) are therefore guaranteed under restriction (74).

• (II) In the second case, where $C>0,E>0,$ and $D^2=4CE$, the results simplify to the following:

  $$A_0=A_0,A_1=A_1,C=-2{\Omega }_2{A}_0^2,E=-{\displaystyle \frac{{\Omega }_2{A}_1^2}{2}}$$

  (76)

  with the constraint condition

  $${\Omega }_1=-{\Omega }_2{A}_0^2.$$

  (77)

  This scenario leads to the following form of dark soliton solutions:

  $$q(x,t)=\epsilon \left[A_0\left(2+{\displaystyle \frac{1}{2}}tanh\sqrt{-2{\Omega }_2{A}_0^2}\eta \right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (78)

The condition ${\Omega }_2<0$ must hold. Consequently, the dark soliton solutions in (78) are admissible under the restriction (77).

• Set-5: Proceeding with the choice $A=B=0$ together with $D=\sqrt{4CE}$, and invoking Maple in concert with the algebraic system (59), we obtain the relations below. In this setting, the analysis naturally splits into several subclasses:

  $$A_0=A_0,A_1=A_1,C=-2{\Omega }_2{A}_0^2,E=-{\displaystyle \frac{{\Omega }_2{A}_1^2}{2}}$$

  (79)

  subject to the constraint

  $${\Omega }_1=-{\Omega }_2{A}_0^2,$$

  (80)

  where the parameter regime is specified by $C>0$, $E>0$, and ${\Omega }_2<0$. Under these prerequisites, multiple soliton forms emerge as described below.

• (I) Singular soliton solutions: In this subcase, the solutions can be represented compactly as

  $$q(x,t)=\epsilon \left[A_0\left(1+{\displaystyle \frac{1}{2}}coth\sqrt{-2{\Omega }_2{A}_0^2}\eta \right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (81)
• (II) Combo-bright-dark soliton solutions: A mixed-profile family also arises, taking the explicit form

  $$q(x,t)=\left[-\frac{2A_1{A}_0^{2}D{\Omega }_2{\mathrm{sech}}^2\sqrt{\frac{-{\Omega }_2{A}_0^2}{2}}\eta }{D^2-A_0^2{A}_1^2{\Omega }_2^2{\left[1+\frac{1}{2}\epsilon tanh\sqrt{\frac{-{\Omega }_2{A}_0^2}{2}}\eta \right]}^2}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (82)
• (III) Combo-singular soliton solutions: In addition, a combined singular-type behavior is captured by

  $$q(x,t)=\left[-\frac{2A_0^{2}D{\Omega }_2{\mathrm{csch}}^2\sqrt{\frac{-{\Omega }_2{A}_0^2}{2}}\eta }{D^2-A_0^2{A}_1^2{\Omega }_2^2{\left[1+\frac{1}{2}coth\sqrt{\frac{-{\Omega }_2{A}_0^2}{2}}\eta \right]}^2}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)}.$$

  (83)

All solutions from (81) to (83) remain valid provided the restriction (80) is enforced, ensuring the internal consistency of this parameter regime.

• Set-6: We now consider the alternative reduction $B=D=0$. Employing Maple again and applying the algebraic relations encoded in (59) leads directly to

  $$A_0=0,A_1=\sqrt{-{\displaystyle \frac{2E}{{\Omega }_2}}},$$

  (84)

  together with the constraint

  $$\begin{array}{c}{\Omega }_1=-C,\hfill \end{array}$$

  (85)

  with the sign condition $E{\Omega }_2<0$. Within this framework, four distinct Weierstrass elliptic families can be written as follows.

• (I) The first representation is

  $$q(x,t)=\epsilon \left[\sqrt{-{\displaystyle \frac{2}{{\Omega }_2}}}\sqrt{\wp \left[\left(\eta \right),g_2,g_3\right]-\frac{C}{3}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (86)

  where ${\Omega }_2<0$.
• (II) A second, equivalent parametrization can be expressed as

  $$q(x,t)=\epsilon \left[\sqrt{-{\displaystyle \frac{2E}{{\Omega }_2}}}\sqrt{\frac{3A}{3\wp \left[\left(\eta \right),g_2,g_3\right]-C}}\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (87)

  under the common sign requirement $E{\Omega }_2<0$. The Weierstrass invariants are given succinctly by

  $$g_2=\frac{4C^2-12AE}{3}\mathrm{and}{g}_3=\frac{4C(-2C^2+9AE)}{27}.$$

  (88)
• (III) A third form involves the derivative of the Weierstrass function:

  $$q(x,t)=\epsilon \left[\sqrt{-{\displaystyle \frac{2}{{\Omega }_2}}}\left(\frac{3{\wp }^{\prime }\left[\left(\eta \right),g_2,g_3\right]}{6\wp \left[\left(\eta \right),g_2,g_3\right]+C}\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (89)

  with ${\Omega }_2<0$ as before.
• (IV) Finally, an alternative reciprocal representation is available:

  $$q(x,t)=\epsilon \left[\sqrt{-{\displaystyle \frac{10C^2}{{\Omega }_2}}}\left(\frac{6\wp \left[\left(\eta \right),g_2,g_3\right]+C}{18{\wp }^{\prime }\left[\left(\eta \right),g_2,g_3\right]}\right)\right]e^{i\left(-\kappa x+\omega t-{\sigma }^{2}t+\sigma B\left(t\right)\right)},$$

  (90)

  again with ${\Omega }_2<0$, and in this variant the invariants are specified by

  $$g_2=\frac{C^2}{12}+AE\mathrm{and}{g}_3=\frac{C(36AE-C^2)}{216}.$$

  (91)

  Collectively, the Weierstrass-based families in (86) and (90) remain valid, provided that constraint (85) is enforced, thereby completing the characterization of this set.

5. Results and Discussion

In this section, we analyze the solution families obtained in the previous part of the manuscript. Our focus is on how the effective parameters shape the envelope morphology and stability of the fields. We highlight the distinct behavior of bright and dark solitons, and emphasize the different ways in which multiplicative noise impacts intensity- versus phase-based observables. Figure 1, Figure 2, Figure 3 and Figure 4 are used throughout to support these interpretations.

Figure 1 presents the bright soliton solution generated by Equation (28). The intensity profile in panel Figure 1a shows a symmetric, bell-shaped pulse localized in space and drifting with constant speed $\rho$. The peak amplitude and full width at half maximum are controlled primarily by $({\Omega }_1,{\Omega }_2)$, which balance dispersion and nonlinearity. The real and imaginary parts of q (panels Figure 1b,c) exhibit oscillations under the pulse envelope, reflecting the carrier $(\kappa ,\omega )$. Importantly, since the noise enters as a multiplicative phase factor, the modulus ${\left|q\right|}^2$ remains unaffected to leading order by $\sigma$, even though the projections fluctuate.

Figure 2 demonstrates how increasing $\sigma$ modifies phase observables. For $\sigma =0.2$ (panels Figure 2a–c), the 3D profile of $\Re \left(q\right)$ remains coherent, and the time trace at $x=0$ shows mild fluctuations. At $\sigma =0.4$ and $\sigma =0.6$ (panels Figure 2d–i), the envelope width and peak amplitude remain unchanged, but the phase decorrelates more rapidly: contour bands become denser, and the time trace at $x=0$ jitters more strongly. Thus, the bright soliton intensity is robust, while coherent projections degrade significantly with $\sigma$.

Figure 3 shows the contrasting solution from Equation (32). Here, the intensity profile (panel Figure 3a) features a localized dip on a finite pedestal, the hallmark of a dark/gray soliton. The notch depth and background level are set by ${\Omega }_2$, while the healing length (core thickness) is controlled by ${\Omega }_1$. Panels Figure 3b,c reveal that the underlying field q has kink-like behavior: the real part transitions between opposite signs, while the imaginary part adjusts accordingly so that the modulus returns to the same pedestal on either side. This combination—a dark notch in intensity and a kinked field phase—is the canonical structure of a dark soliton in nonlinear dispersive media.

The response of this state to noise is depicted in Figure 4. For $\sigma =0.2$ (panels Figure 4a–c), the notch remains sharp and the kink in $\Re \left(q\right)$ is well defined. Increasing $\sigma$ to $0.4$ and $0.6$ (panels Figure 4d–i) does not affect the notch depth or width: the envelope structure remains intact. However, the kink phase exhibits accelerated diffusion, evident in the time trace jitter and the oscillatory shading of the contours. This indicates that the geometry of the dark soliton is noise-resilient, while its phase remains sensitive to stochastic forcing.

Across both solution families, the envelope geometry is dictated mainly by $({\Omega }_1,{\Omega }_2)$: peak height and width for the bright soliton, pedestal level, and notch depth for the dark soliton. The drift is set by $\rho$, while the oscillatory carrier is determined by $(\kappa ,\omega )$. The stochastic parameter $\sigma$ leaves intensity-based observables essentially unchanged but progressively randomizes phase. Consequently, intensity-modulated schemes are robust against multiplicative noise, while coherent detection schemes suffer performance degradation as $\sigma$ grows. Importantly, the comparison highlights that dark solitons (intensity notches with kinked fields) are more robust in their geometry than bright solitons, which may guide practical system designs where noise resilience is a key requirement.

5.1. Physical Interpretation and System Relevance

The parameters in Equations (1)–(3) connect directly to measurable fiber and grating properties. The coefficient a represents chromatic dispersion that governs the linear spreading of the pulse. The coefficient b represents third-order dispersion that becomes important for femtosecond and picosecond pulses. The Kerr coefficient c sets the strength of self-phase modulation that steepens the phase and creates a chirp proportional to the local intensity. The triplet $(\lambda ,\mu ,\theta )$ collects higher-order nonlinear corrections that capture self-steepening and the self-frequency shift. In the Bragg coupled model, the constants $a_j$ and $c_j$ play the same roles for each direction of propagation, while $d_j$ and $e_j$ scale the self and cross phase modulation. The detuning parameters ${\delta }_j$ determine the spectral position of the band gap, so they control whether the carrier lies inside the gap and whether a gap soliton may form. In this setting, dispersive reflectivity is the key practical lever, since it adds a tunable dispersive contribution that can be engineered by the grating design and thereby offsets an otherwise insufficient chromatic dispersion budget.

The analytical solutions describe envelopes that persist in the presence of multiplicative noise because the noise enters predominantly as a common phase. For bright states, the peak power and the width are set by $({\Omega }_1,{\Omega }_2)$ and by the balance between dispersion and the Kerr nonlinearity. For dark and gray states, the pedestal level, the notch depth, and the healing length are set by the same parameters. The stochastic term produces a phase that behaves like a random walk with variance that grows linearly with time at a rate proportional to ${\sigma }^2$. As a consequence, the intensity remains unchanged to leading order, while the complex field loses temporal coherence at a rate that increases with $\sigma$. This behaviour matches the figure set, where the envelope is essentially invariant over the explored noise levels and the phase observables develop progressively stronger jitter.

These features translate into clear system-level implications. In direct detection links and in sensing configurations that read intensity, the envelope invariance implies resilience against multiplicative phase-type noise at the levels considered. In coherent receivers and in interferometric schemes, the decay of the first-order coherence limits the integration time and sets a linewidth that broadens with the noise level. Using the relation $g^{\left(1\right)}\left(\tau \right)=exp(-\frac{1}{2}{\sigma }^2\left|\tau \right|)$, the coherence time is approximately $2/{\sigma }^2$ and the associated spectral line has half width $\Delta \nu ={\sigma }^2/\left(2\pi \right)$. System designers may therefore trade between power and coherence by controlling the operating point of the grating and the amplifier chain that influences the effective $\sigma$.

Dispersive reflectivity in a fiber Bragg grating provides a practical path to operate in the gap soliton regime when the chromatic dispersion of the underlying waveguide is too small. By tuning the grating period and the apodization, one can achieve the needed effective dispersion without inducing detrimental amplitude modulation. The analysis further indicates that dark and gray gap solitons are particularly attractive for robustness, since their geometry is tied to pedestal and notch parameters that are insensitive to the stochastic phase at leading order. This can be advantageous in scenarios where the optical source or the environment introduces phase fluctuations, for example, in links with fast temperature or vibration changes or in sensor heads that experience mechanical perturbations [29,30,31,32,33].

The travelling wave reduction highlights how the drift parameter $\rho$ and the carrier pair $(\kappa ,\omega )$ relate to the grating detuning and to the chosen operating wavelength. In practical terms, once $({\Omega }_1,{\Omega }_2)$ satisfies the analytical existence conditions, one may select ${\delta }_j$ to place the carrier near the middle of the gap for maximal robustness and then choose the reflectivity dispersion to fix the envelope width required by the application. The predicted phase diffusion rate then sets the coherence budget and suggests the appropriate detection strategy. In direct detection schemes, the signal-to-noise margin remains high because the relevant measures depend on ${\left|q\right|}^2$, whereas in coherent detection, one should account for a predictable penalty that grows with ${\sigma }^2$ and can be mitigated by phase tracking or by reducing the effective noise source. In all cases, the analytical envelopes offer compact templates for control and monitoring, since any deviation from the predicted intensity profile provides an immediate indication of a drift in the deterministic parameters rather than a change in the stochastic drive.

The mapping between the recovered coherent structures and their parameter regimes is summarized in

Table 1. Mapping between recovered coherent structures and the governing parameter regimes. The entries summarize the qualitative conditions that select each family. Precise algebraic existence constraints are given by the reduced parameters ${\Omega }_1$ and ${\Omega }_2$ that are defined in the travelling-wave reduction and in the compatibility conditions in Section 4.

Family	Envelope Form	Typical Regime Selector	Carrier and Drift	Role of Noise
Bright gap soliton	Localized hump, sech limit of an elliptic cn or dn branch	Effective dispersion and Kerr nonlinearity balance in the focusing sense near a band-gap operating point. Bragg detuning places the carrier inside the gap so that the effective mass is negative and the envelope is trapped. Existence is guaranteed when the reduced parameters satisfy the bright-branch inequalities for ${\Omega }_1$ and ${\Omega }_2$ from Section 4, with a positive amplitude parameter. Third-order dispersion adjusts asymmetry and walk-off but does not change the sign of the existence conditions.	$\kappa$ and $\omega$ chosen by detuning set the carrier placement inside the gap. The drift $\rho$ follows from the travelling-wave phase matching and is used to position the pulse in the moving frame.	Multiplicative term adds a common stochastic phase. The intensity ${\left|q\right|}^2$ stays invariant to leading order while phase observables decorrelate at a rate proportional to ${\sigma }^2$.
Dark and gray gap soliton	Localized notch on a finite pedestal, tanh limit of an elliptic sn branch	Effective dispersion and Kerr nonlinearity balance in the defocusing sense or in the negative-mass portion of the gap. Detuning near a band edge fixes the pedestal. Existence follows from the dark-branch inequalities for ${\Omega }_1$ and ${\Omega }_2$ in Section 5, which set pedestal level, notch depth, and healing length. Third-order dispersion primarily skews the kink phase and modifies the recovery length.	Carrier pair $(\kappa ,\omega )$ and detuning control the phase jump across the notch. The drift $\rho$ transports the kink without altering the pedestal.	Same multiplicative-phase mechanism. The notch geometry and pedestal remain stable, while the field phase undergoes diffusion that increases with $\sigma$.
Elliptic periodic states	Doubly periodic wave with Jacobi or Weierstrass profile, modulus $0<m<1$	Parameters in the compatibility set where ${\Omega }_1$ and ${\Omega }_2$ permit bounded periodic solutions. The limit $m\to 1$ connects continuously to the bright or dark soliton branches depending on the sign pattern of the reduced coefficients.	$(\kappa ,\omega ,\rho )$ select the lattice speed and the carrier within the periodic envelope.	Noise causes loss of temporal coherence while preserving the average lattice intensity.
Singular and break-type solutions	Peaked, cusp, or gradient-catastrophe-like profiles	Outside the bounded-solution sets for ${\Omega }_1$ and ${\Omega }_2$, or when amplitude or denominator conditions in the closed-form expressions fail. Often associated with parameter choices that violate the balance between dispersive terms and Kerr terms, or operate far from the gap.	Carrier and drift are defined, but the profile is non-localized or develops a singularity.	Noise accelerates phase wandering and may trigger earlier loss of smoothness, while not setting the onset by itself.

.

The reduced parameters ${\Omega }_1$ and ${\Omega }_2$ are algebraic functions of the dispersion coefficients a and b, the Kerr and higher order nonlinear coefficients c, $\lambda$, $\mu$, $\theta$, and the Bragg detuning constants ${\delta }_j$. For bright states, the amplitude and width follow from the positive branch of the compatibility conditions. For dark and gray states, the pedestal and the notch depth follow from the negative branch with a real healing length. For elliptic states, the modulus m lies strictly between zero and one and is fixed by the same compatibility relations. The multiplicative noise parameter $\sigma$ does not enter the intensity conditions at leading order and only sets the rate of coherence decay, which is proportional to ${\sigma }^2$.

5.2. Quantitative Effect of Multiplicative Noise

For the travelling-wave solutions described in Section 4 and Section 5, the stochastic term enters as a common phase factor so that

$$q(x,t)=Q(x-\rho t)exp\left(i{\Phi }_{\det }(x,t)-\frac{1}{2}{\sigma }^{2}t+i\sigma B\left(t\right)\right),$$

with Q a deterministic envelope, ${\Phi }_{\det }$ the deterministic carrier phase, and $B\left(t\right)$ a standard Wiener process. Two consequences follow immediately.

Because the noise appears only in the complex exponential, the modulus is unchanged to leading order:

$${\left|q(x,t)\right|}^2={\left|Q(x-\rho t)\right|}^2,$$

and therefore the peak level, the full width at half maximum, the pedestal level, and the notch depth are controlled by the deterministic parameters $({\Omega }_1,{\Omega }_2)$ and do not depend on $\sigma$. This is consistent with Figure 1 and Figure 3.

Let $\Delta \varphi \left(\tau \right)=\varphi (t+\tau )-\varphi \left(t\right)$ denote the stochastic phase increment induced by the multiplicative term. Since $\varphi \left(t\right)$ contains the contribution $\sigma B\left(t\right)$, one obtains

$$\mathrm{Var}\left[\Delta \varphi \left(\tau \right)\right]={\sigma }^2\tau ,g^{\left(1\right)}\left(\tau \right)={\displaystyle \frac{\mathbb{E}\left\{q^{*}(x,t)q(x,t+\tau )\right\}}{\sqrt{{\mathbb{E}\left|q(x,t)\right|}^2\mathbb{E}{\left|q(x,t+\tau )\right|}^2}}}=exp\left(-\frac{1}{2}{\sigma }^2\left|\tau \right|\right).$$

Thus the first-order coherence decays exponentially with rate $\frac{1}{2}{\sigma }^2$, and the corresponding temporal power spectrum is Lorentzian with half-width

$$\Delta \nu ={\displaystyle \frac{{\sigma }^2}{2\pi }}.$$

This explains the denser contour bands and the stronger time-trace jitter observed in Figure 2 and Figure 4 when $\sigma$ increases, while the envelope remains unchanged.

Define the envelope energy $\mathcal{E}\left(t\right)={\int }_{\mathbb{R}}{\left|q(x,t)\right|}^{2}dx$. For the multiplicative structure above,

$$\mathbb{E}\mathcal{E}\left(t\right)={\int }_{\mathbb{R}}{\left|Q(x-\rho t)\right|}^{2}dx=\mathcal{E}\left(0\right),$$

so the ensemble-averaged energy is constant in time. More generally, any norm that depends only on $\left|q\right|$ is preserved to leading order under the stochastic drive. In practice, a convenient diagnostic is the envelope mismatch

$$\epsilon \left(t\right)={\int }_{\mathbb{R}}|{\left|q(x,t)\right|}^2-{\left|Q(x-\rho t)\right|}^2|dx,$$

which remains at the level of numerical tolerance across the reported values of $\sigma$. This provides a quantitative confirmation that the stochastic term causes phase diffusion rather than amplitude modulation.

Since intensity-based observables are insensitive to $\sigma$ and coherence decays as $g^{\left(1\right)}\left(\tau \right)=exp(-\frac{1}{2}{\sigma }^2\left|\tau \right|)$, detection strategies that rely on amplitude are robust, whereas coherent schemes experience a predictable loss with rate set by ${\sigma }^2/2$.

6. Validation

We validate the analytical travelling-wave solutions against direct simulations of Equations (1)–(3) using a split-step Fourier method. The computational domain is periodic on $x\in [-L_x/2,L_x/2]$ with $N_x$ equispaced points and fundamental wavenumbers $k_m=2\pi m/L_x$ for integers m in the usual range. Time is advanced with step $\Delta t$. Convergence is checked by halving $\Delta x=L_x/N_x$ and $\Delta t$ and by increasing the number of noise realizations until all reported statistics remain unchanged within numerical tolerance.

6.1. Single-Field Model in Equation (1)

We split the evolution into a linear substep and a nonlinear-and-stochastic substep. Let $\mathcal{F}$ denote the Fourier transform in x and let $\widehat{q}(k,t)=\mathcal{F}\left\{q(·,t)\right\}\left(k\right)$.

For Equation (1) the linear operator is

$$\mathcal{L}q=ia{\partial }_{xx}q-b{\partial }_{xxx}q,$$

which yields the exact Fourier-space propagator

$$\widehat{q}(k,t+\Delta t)=\widehat{q}(k,t)exp\left(i\Delta t\left[-a{k}^2+b{k}^3\right]\right).$$

The remaining terms are

$$\mathcal{N}\left(q\right)=-{ic\left|q\right|}^{2}q+\lambda {\left({\left|q\right|}^{2}q\right)}_x+\mu {\left({\left|q\right|}^2\right)}_{x}q+\theta {\left|q\right|}^2{q}_x,$$

advanced in physical space by a first-order exponential Euler update over $\Delta t$. The multiplicative white noise is implemented in the Itô sense as an exact complex phase increment,

$$q\leftarrow qexp\left(-\frac{1}{2}{\sigma }^2\Delta t+i\sigma \Delta W\right),$$

with $\Delta W$ an independent normal variable of zero mean and variance $\Delta t$ at every time step and grid point. The full step is obtained by composing the linear substep and the nonlinear-and-stochastic substep in this order at each $\Delta t$.

6.2. Bragg-Coupled Model in Equations (2) and (3)

For the Bragg system we write $U={(q,r)}^{\top }$. The linear part couples the fields through dispersive reflectivity and detuning. In Fourier space, the linear flow over one step is

$$\widehat{U}(k,t+\Delta t)=exp\left(\Delta t\mathbf{L}\left(k\right)\right)\widehat{U}(k,t),$$

with the $2\times 2$ symbol

$$\begin{array}{cc}\hfill \mathbf{L}\left(k\right)& =\left(\begin{array}{cc}0& i{a}_1(-k^2)-c_1(-i{k}^3)+i{\delta }_1\\ i{a}_2(-k^2)-c_2(-i{k}^3)+i{\delta }_2& 0\end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}0& -i{a}_1{k}^2+i{c}_1{k}^3+i{\delta }_1\\ -i{a}_2{k}^2+i{c}_2{k}^3+i{\delta }_2& 0\end{array}\right).\hfill \end{array}$$

The matrix exponential is evaluated exactly per mode by diagonalization of $\mathbf{L}\left(k\right)$. The nonlinear part is advanced in physical space using

$${\mathcal{N}}_q=-i\left(d_1{\left|q\right|}^2+e_1{\left|r\right|}^2\right)q+{\lambda }_1{\left({\left|q\right|}^{2}q\right)}_x+{\mu }_1{\left({\left|q\right|}^2\right)}_{x}q+{\theta }_1{\left|q\right|}^2{q}_x,$$

$${\mathcal{N}}_r=-i\left(d_2{\left|r\right|}^2+e_2{\left|q\right|}^2\right)r+{\lambda }_2{\left({\left|r\right|}^{2}r\right)}_x+{\mu }_2{\left({\left|r\right|}^2\right)}_{x}r+{\theta }_2{\left|r\right|}^2{r}_x,$$

followed by the multiplicative noise phase update applied componentwise,

$$q\leftarrow qexp\left(-\frac{1}{2}{\sigma }^2\Delta t+i\sigma \Delta W_q\right),r\leftarrow rexp\left(-\frac{1}{2}{\sigma }^2\Delta t+i\sigma \Delta W_r\right),$$

with $\Delta W_q$ and $\Delta W_r$ independent normal variables of zero mean and variance $\Delta t$.

6.3. Diagnostics and Tolerances

We monitor the envelope intensity ${\left|q\right|}^2$ and ${\left|r\right|}^2$, the drift speed $\rho$, and the width measures reported in the figures. We also compute the phase increment $\Delta \varphi \left(\tau \right)=\varphi (t+\tau )-\varphi \left(t\right)$ at a fixed location in space and the first-order temporal coherence

$$g^{\left(1\right)}\left(\tau \right)={\displaystyle \frac{\mathbb{E}\left\{q^{*}(x,t)q(x,t+\tau )\right\}}{\sqrt{{\mathbb{E}\left|q(x,t)\right|}^2\mathbb{E}{\left|q(x,t+\tau )\right|}^2}}},$$

together with the Lorentzian half-width extracted from the temporal spectrum. which for $g^{\left(1\right)}\left(\tau \right)=exp(-\frac{1}{2}{\sigma }^2\left|\tau \right|)$ equals $\Delta {\nu }_{\mathrm{HWHM}}={\sigma }^2/\left(4\pi \right)$ (so $\mathrm{FWHM}={\sigma }^2/\left(2\pi \right)$). Agreement within numerical tolerance is taken as validation.

7. Conclusions

This paper is a comprehensive study of the retrieval of dispersive gap solitons in the presence of white noise and dispersive reflectivity. Two integration approaches have made this retrieval possible. They are the modified sub-ODE approach and the enhanced direct algebraic method. These two schemes yielded the gap solitons through the intermediary functions, namely Jacobi’s elliptic functions and Weierstrass’ elliptic functions. The results are varied, and a wide spectrum of gap solitons has emerged and is presented. While the results are novel and applicable in telecommunications, there is a noticeable shortcoming in using these two approaches. The algorithms fail to recover the gap soliton radiation even though it carries dispersive reflectivity. While additional means are to be used to be able to recover the radiation component, such as beyond all-order asymptotics and/or theory of unfoldings, such tasks will be taken up later, and then the retrieved results will be disseminated elsewhere.