This work constructs exact quiescent solitons of the resonant nonlinear Schr & ouml;dinger equation with nonlinear chromatic dispersion and nine distinct self-phase-modulation laws. Using an enhanced direct algebraic method, we derive bright, dark, singular, and straddled solitons and classify their existence domains via explicit parameter constraints. For the Kerr law, the model supports bright and singular solitons with amplitudes determined by dispersion parameters; for the power-law case, bright and singular families appear with characteristic hyperbolic profiles; and for elliptic-function constructions, Jacobian and Weierstrass forms reduce to solitons in the modulus-one limit. The analysis also yields the algebraic constraints required for physical realizability. Collectively, these results delineate when nonlinear chromatic dispersion, together with generalized self-phase modulation, produces stationary localized structures in quantum-optical and quantum-fluid settings.

This paper reports the observation of quiescent optical solitons in magneto-optic waveguides. The self-phase modulation structure is based on the one proposed by Kudryashov. Three algorithms have enabled this retrieval: the enhanced direct algebraic method, the extended auxiliary equation approach, and the new mapping scheme. Together, these methods have recovered a full spectrum of quiescent optical solitons. The parameter constraints for their existence are also included. A few numerical simulations demonstrate the analytical results.

This article delves into the dynamics of chirped solitons within the Triki–Biswas equation, particularly when coupled with multiplicative white noise. The study employs a novel mapping method as its primary analytical tool to investigate the behavior and properties of solitons under specified conditions. The research uncovers various new soliton solutions through meticulous analysis, including bright, singular, and straddled solitons. Importantly, each identified soliton solution distinctly corresponds to a specific chirp, revealing the intricate relationship between the soliton structure and its chirping phenomenon. A noteworthy aspect of this work is introducing and examining the governing model infused with this new structure, representing the first instance of such a study. This innovative approach enhances our comprehension of chirped solitons in the presence of noise. Further, it explores the dynamic behaviors of solitons influenced by external disturbances, offering valuable insights into nonlinear science and soliton theory.

This paper investigates the stochastic Fokas-Lenells Equation (FLE) with time-dependent coefficients and multiplicative white noise to model the propagation of chirped optical solitons in nonlinear dispersive media. The equation captures complex dynamics in realistic fiber systems, accounting for variable dispersion, Kerr nonlinearity, Raman effects, and stochastic perturbations arising from thermal and amplifier-induced fluctuations. Two analytical techniques, namely the extended simplest equation method and the enhanced direct algebraic method, are employed to derive exact soliton solutions. These methods offer a systematic and flexible analytic framework that efficiently handles stochastic and variable-coefficient systems, enabling closed-form solutions where conventional techniques often struggle. These include bright, dark, kink-shaped, singular, and elliptic solitons under specific parametric conditions. The richness of the obtained solution classes demonstrates the strength and versatility of the adopted approach in capturing diverse nonlinear wave profiles in noisy, inhomogeneous optical media. A key result is that the multiplicative noise affects only the soliton phase, preserving the amplitude and shape of the waveform. This demonstrates the inherent stability of soliton structures in noisy environments. The results highlight the analytical power of the proposed methods and provide deeper insights into robust pulse dynamics in realistic stochastic fiber systems, paving the way for future studies in dispersion-managed fibers, optical lattices, and structured photonic media.

The paper retrieves quiescent dispersive solitons in dispersion-flattened optical fibers having nonlinear chromatic dispersion and the Kerr law of self-phase modulation. The platform model is the Schr & ouml;dinger-Hirota equation. The enhanced direct algebraic method has made this retrieval possible. The intermediary functions are Jacobi's elliptic function and Weierstrass' elliptic function. The final results appear with parameter constraints for the existence of such solitons.

We study a stochastic generalized long–short wave resonance system with cubic–quintic nonlinearities, perturbation terms, and multiplicative white noise in the Itô sense. Using a traveling-wave reduction combined with the newly proposed sub-ODE method, we construct explicit families of solitary-wave and elliptic-function solutions, including elevation, depression, and singular branches. The analysis reveals how stochastic effects shrink the existence domains of these solutions, while a Hamiltonian linearization provides Vakhitov–Kolokolov- and Grillakis–Shatah–Strauss-type stability criteria. The results enrich nonlinear wave theory and offer insights relevant to solitary-wave applications in fiber optics and related technologies.

Quiescent optical solitons in a (3+1)-dimensional nonlinear Schrödinger equation governing magneto-optic waveguides are investigated in the presence of Kudryashovs nonlocal law nonlinearity, nonlinear chromatic dispersion, and generalized temporal evolution. By employing the Jacobi elliptic function expansion method and the extended auxiliary equation method, a wide class of exact analytical solutions is constructed. These solutions include bright, dark, singular, with solitary wave profiles recovered as limiting cases of the elliptic modulus. The influence of higher-order dispersion, nonlocal nonlinear response, and magneto-optic coupling on the amplitude, width, and stability of quiescent solitons is analyzed in detail. From the results we obtained, it was found that nonlinear chromatic dispersion and generalized time evolution have an effective role in controlling the localization and diffusion characteristics of solitons.

This study investigates the chiral nonlinear Schrödinger equation for current-dependent nonlinear wave propagation in the fractional quantum Hall setting when temporal evolution is represented by the conformable fractional derivative. This derivative is defined operationally as the ordinary first time derivative multiplied by a time-dependent power factor determined by the fractional order. The physical objective is to clarify how fractionally rescaled temporal evolution modifies coherent chiral excitations associated with quantum Hall edge-state dynamics. The novelty lies in a unified quartic auxiliary-equation classification based on the modified sub–ODE method rather than the derivation of isolated wave forms. A conformable traveling-wave reduction produces an amplitude problem whose admissible branches yield localized solitary waves, kink-type waves, singular structures, Jacobi elliptic waves, Weierstrass elliptic waves, and mixed solution families with explicit existence restrictions. The results are validated by direct substitution into the reduced and original models, together with first-integral and limiting-case checks. The analysis shows that, for fixed reduced coefficients, the fractional order changes the temporal scaling, phase evolution, and propagation trajectory while preserving the analytical amplitude family. This separation provides a useful mechanism for tuning the phase and propagation of chiral wave structures without altering their intrinsic analytical profile.

To represent Clifford-valued signals more efficiently in the time-frequency domain, we establish the notion of novel integral transform known as Clifford quadratic-phase wavelet transform (CQPWT) by invoking the convolution theory associated with the Clifford quadratic-phase Fourier transform (CQPFT). We begin our discussion by establishing the definition of CQPWT and some fundamental properties, few of them include linearity, translation, and parity. We then proceed to the derivation of some mathematical formulae including the orthogonality relation, inversion formula, and reproducing kernel by formulating the relationship between the CQPFT and Clifford Fourier transform (CFT) of an analyzing function. We then investigate the Heisenberg's and logarithmic uncertainty principles corresponding to the proposed transform. Finally, we conclude our discussion by displaying the validity of transform via illustrative examples.

The current paper retrieves quiescent optical solitons in magneto-optic waveguides with optical metamaterials with Kudryashov’s sextic power-law of self-phase modulation. The chromatic dispersion is rendered to be nonlinear, while the temporal evolution is taken to be in its generalized form. Both twin-core and multiple-core couplers are considered. The enhanced direct algebraic method is the integration algorithm for the recovery of such solitons.

This paper investigates the propagation of quiescent optical solitons in magneto-optic waveguides governed by Kudryashov's quintuple power-law of self-phase modulation, incorporating generalized temporal evolution and nonlinear chromatic dispersion. The governing model accounts for the interplay between higher-order nonlinear effects and dispersive properties, which are essential for understanding the complex dynamics of light in structured optical media. To obtain exact soliton solutions, we employ the direct algebraic method and an advanced version of the sub-ODE approach, providing a robust analytical framework for recovering a wide spectrum of soliton profiles. The derived solutions include bright, dark, and singular soliton structures. Additionally, numerical simulations are conducted to verify and illustrate the physical relevance of the analytical results, demonstrating the stability and dynamical behavior of the solitons under varying parametric conditions. The findings contribute to the theoretical foundation of magneto-optic waveguide systems and offer potential applications in advanced photonic technologies and optical communication systems.

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the θ-integral operator. Additionally, we investigate the compositional properties of θ-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics.

This paper presents a significant advancement in the study of stochastic partial differential equations within the context of nanobioscience, focusing on models that incorporate multiplicative noise. The main goal is to obtain precise solutions for these intricate equations, which is challenging due to the unpredictability brought about by stochastic elements. We utilize two advanced mathematical methods to achieve this: the refined Kudryashov’s technique and the exp (-Φ (ξ )) -expansion strategy. These methods allow us to derive various new exact solutions, such as kink-type wave solutions, straddled solitons, and singular solitons. A unique aspect of our research is the integration of a white noise term into our model, significantly enhancing the theoretical framework for stochastic partial differential equations in nanotechnology. This integration provides a closer match to actual phenomena and creates opportunities to investigate the behavior of nanoscale biological processes influenced by stochastic factors. The solutions we have identified highlight the value of our model in offering more profound insights into the complex behaviors observed in nanoscience systems, facilitating further theoretical and applied investigations in this cross-disciplinary area.

This study delves into the perturbed stochastic nonlinear Schr & ouml;dinger equation in an optical coupler for metamaterials, introducing a parabolic nonlocal law for refractive index nonlinearity, eighth-order dispersion, and multiplicative white noise. An enhanced version of Kudryashov's method, combined with the direct algebraic method, is utilized to examine stochastic effects in nonlinear optical systems. The findings reveal diverse solution types such as dark and singular solitons, Weierstrass and Jacobi doubly periodic solutions, and straddled solitons, showcasing the complex dynamics driven by nonlocal nonlinearity and higher-order dispersion under stochastic conditions. This pioneering work on the nonlinear Schr & ouml;dinger equation with a parabolic nonlocal nonlinearity and eighth-order dispersion investigates the impact of random fluctuations on nonlinear dynamics, advancing our understanding of light propagation at higher frequencies. The study aims to inspire further research into the behavior of light in unique materials and the development of more efficient light-based devices, leveraging the insights gained to innovate advanced communication tools and other optical devices, highlighting the response of these materials to random disturbances.

This article introduces a novel governing model characterized by stochastic long-short wave resonant equations with multiplicative white noise applicable in fields such as telecommunications and climate modeling. The study aims to explore chirped and chirp-free soliton solutions within this framework. Using Jacobi's elliptic function method as our primary methodology, we have successfully derived various soliton solutions, including solitary waves, singular solitons, and dark and bright soliton forms which can be relevant in optical communication and nonlinear optics. Significantly, our analysis has facilitated the extraction of both chirped and chirp-free solutions applicable to the model, marking a notable advancement in soliton research. Introducing this governing model is a pioneering endeavor in the field, distinguished by its ability to model the resonance interaction between long and short waves under the influence of multiplicative white noise. This aspect holds profound implications for the understanding and application of wave dynamics in stochastic environments such as in meteorology and fluid dynamics. To underscore our findings, the manuscript includes 3D and 2D graphical representations, effectively illustrating the impact of white noise on the wave profiles of the derived solitons. Our study broadens the theoretical landscape of soliton solutions and presents a significant step forward in practically examining wave resonance phenomena under stochastic conditions.

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.

Our study examines the impact of sixth-order dispersion and multiplicative white noise on the transmission of optical solitons within magneto-optic waveguides. Our findings offer valuable insights that are crucial for the advancement of optical communications. We demonstrate how these factors affect soliton behaviour by developing and analysing a stochastic version of the Sasa–Satsuma equation, which describes the dynamics of optical solitons in dispersive media. Our research introduces an enhanced analytical approach that addresses the complexities introduced by these perturbations, specifically by modifying the sub-ordinary differential equation (sub-ODE) method. The investigation reveals the substantial influence of sixth-order dispersion and multiplicative white noise on the quality and stability of soliton transmission in such waveguides. These findings emphasise the significant importance of considering these factors in designing and optimising optical communication systems, particularly those utilising the propagation of optical solitons. This research represents a substantial advancement in our understanding of and ability to mitigate the effects of higher-order dispersion and stochastic perturbations on soliton-based communication technology.

The article discusses how Kudryashov's proposed self-phase modulation scheme and nonlinear chromatic dispersion cause the evolution of quiescent optical solitons in magneto-optic waveguides. Provide a comprehensive understanding of the governing model; generalised temporal evolution is considered. The modified sub-ODE approach is employed to facilitate the recovery of such solitons. This leads to a complete range of optical solitons and the necessary conditions that must be met for these solitons to exist, which are also provided.

This paper recovers optical solitons solutions with polarization-mode dispersion in optical fibers. The governing model is the Kundu-Eckhaus equation. The adopted integration scheme is the generalized ϕ6-model expansion. The intermediary functions are the Jacobi’s elliptic functions and the Weierstrass’ elliptic functions. The parameter constraints for the existence of such solitons are also presented.

This paper investigates new optical soliton solutions to the complex Ginzburg-Landau equation in the presence of white noise, a fundamental model in nonlinear optics that describes soliton dynamics. The study focuses on nine distinct forms of self-phase modulation structures, each exhibiting unique nonlinear characteristics and dispersion properties. To derive the soliton solutions, the generalized G ' / G -expansion approach is employed, which is known for its effectiveness in handling nonlinear differential equations and extracting exact solutions systematically. Through this analytical framework, a variety of soliton profiles are retrieved, demonstrating the influence of nonlinear dispersion and gain-loss terms on soliton propagation. A key observation from the analysis is that the presence of white noise primarily affects the phase component of the solitons, while their amplitude remains intact. This result suggests that the robustness of the soliton amplitude against stochastic perturbations could have significant implications for practical optical communication systems and laser pulse propagation, where maintaining stable intensity profiles is crucial. The obtained results provide valuable insights into the interplay between noise and nonlinear wave dynamics, offering potential applications in fiber-optic communication, mode-locked lasers, and other areas of photonics where controlled soliton evolution is essential.•The paper investigates optical soliton solutions of the complex Ginzburg-Landau equation with white noise, focusing on nine forms of self-phase modulation with varied nonlinear and dispersive properties.•Using the generalized G ' / G -expansion method, the study derives exact soliton profiles, revealing how nonlinear dispersion and gain-loss mechanisms shape soliton dynamics.•A key finding shows that white noise affects soliton phase without disturbing amplitude, highlighting amplitude robustness with implications for optical communications and laser systems.