In this paper, we study the nonlinear wave propagation of the fractional Joseph-Egri equation by means of the modified Sardar sub-equation method. The non-linear model under consideration includes Riemann-Liouville fractional derivatives, which are efficient in reflecting memory and heredity phenomena that are often present in complex physical media, such as polymers, gels, and biological tissues. With the aid of a proper traveling wave transformation, we reduce the non-linear partial differential equation to an ordinary differential equation. Finally, we use the modified Sardar sub-equation method to obtain various non-linear solutions, including bright soliton, dark soliton, dark singular soliton, periodic singular, and rational waves. By means of graphical analysis, we examine the effect of the fractional parameters alpha and beta on the non-linear solutions. In order to further validate the accuracy of the analytical results, numerical approximations of the solution have been carried out by employing the differential transform method, which has revealed an excellent agreement between the exact and numerical results with very small absolute errors. Moreover, an analysis of modulation instability is carried out in order to validate the stability of the background wave. The results have shown that the perturbation frequency is real for the considered parameter condition, thus validating that the system is modulationally stable. The results have revealed that fractional derivatives play an important role in governing wave attenuation, dispersion, and localization in viscoelastic media, thus providing more insight into nonlinear wave propagation in complex systems. The novelty of the study lies in the application of the modified Sardar sub-equation method, which provides a systematic and efficient framework for generating a wide variety of exact nonlinear wave solutions. Unlike many conventional analytical techniques that yield limited solution structures, the proposed approach enables the construction of bright, dark, singular, periodic, and rational wave solutions within a unified formulation.

Purpose This study develops a computationally oriented framework to obtain exact coherent structures for a stochastic coupled Klein–Gordon–Schrödinger (KGS) system driven by multiplicative amplitude noise in the Stratonovich sense. Design/methodology/approach A traveling-wave reduction, combined with a mean-field representation of the stochastic exponential modulation, converts the stochastic PDE model into a deterministic nonlinear ODE system. Two algorithmic solvers are then implemented: the Enhanced Direct Algebraic Method and the New Projective Riccati Equation Method, which systematically generate closed-form solutions together with their parameter admissibility constraints. Findings The procedure yields families of bright, dark, kink-type, singular, straddled, and periodic waves expressed via hyperbolic and elliptic functions. Explicit parameter regimes are derived to guarantee real-valuedness and boundedness, and to quantify how free parameters control amplitude, width, and propagation speed under noise modulation. Originality/value The paper contributes reproducible solution algorithms and a benchmark catalogue of exact waveforms for validating numerical solvers and computer-aided engineering workflows involving stochastic coupled-wave dynamics in nonlinear dispersive media.

The paraxial wave equation, a simplified version of the full electromagnetic wave formulation, is frequently employed to characterize optical signal transmission in these materials, facilitating an efficient and precise description of light propagation. This study examines the dynamical characteristics of the proposed model using both analytical and dynamical methods, specifically employing a bifurcation approach. Initially, wave transformation is applied to convert the proposed model into its conventional form. The system's dynamic properties are then examined using phase and Hamiltonian portraits, considering different parameter settings. We also find bright periodic, kink, bright-type, and dark-type bell wave, anti-kink, and solitary wave solutions by changing parameters, using their corresponding Hamiltonians, and integrating heteroclinic and homoclinic orbits. We also visually represent various characteristics by carefully choosing the right parameters. Moreover, the physical characteristics and computational results support the robustness and significance of the proposed theory.

This work examines the impact of initial stress on the spread of coupled thermo-mechanical waves in a semiconductor thermo-elastic medium using a β -order fractional derivative model. The study is undertaken utilizing three generalized thermo-elastic theories: Dual-Phase-Lag (DPL) model, Lord-Shulman (L-S) theory, and Refined Dual-Phase-Lag (RDPL) model. To accurately reflect the underlying physical processes, the governing equations for heat conduction, elastic deformation, and semiconductor carrier dynamics are fully coupled, taking into account both finite-speed thermal transport and carrier density effects. To simplify the mathematical treatment, a suitable nondimensionalization strategy is adopted, lowering the number of governing parameters and clarifying the structure of the problem. A transform technique is then utilized to transform the resulting system of coupled partial differential equations (PDEs) into an analogous system of ordinary differential equations (ODEs) that can be solved analytically. This approach produces solution for displacement, temperature, carrier density, and stress tensor components, allowing for a complete investigation of wave propagation properties inside the medium. Numerical simulations using graphical representations are used to investigate the physical significance of the theoretical concept. These computations provide a comparative examination of physical quantities with and without initial stress across the three thermo-elastic theories. The β -fractional derivative and initial stress significantly impact wave behavior, including propagation velocity, amplitude evolution, and attenuation characteristics. Furthermore, the analysis sheds further light on the interplay between thermal processes and carrier density effects in thermo-elastic materials, emphasizing the significance of these elements in precisely forecasting the dynamic response of modern semiconductor media.

In this work, we explore the evolution of non-static, radiating, shearing, and hyperbolically symmetric stellar configurations within the framework of general relativity by imposing the Euclidean condition. To model the interior spacetime, we consider a non-static hyperbolically symmetric fluid distribution undergoing gravitational collapse in the presence of anisotropic stresses and heat dissipation. We derive the corresponding gravitational field equations for the anisotropic matter content. We use the Euclidean condition, which simplifies the equations of motion by establishing a direct relationship between the metric functions and allows the construction of exact analytic stellar models. By smoothly matching the interior spacetime to the hyperbolic Vaidya geometry across the boundary hypersurface, the appropriate boundary conditions are obtained. Explicit expressions for the physical variables are derived, including the energy density, radial and tangential pressures, heat flux, anisotropic factor, and hyperbolic mass function. The effects of density inhomogeneity, dissipative heat flux, and pressure anisotropies on the dynamical evolution of hyperbolic stellar matter are analyzed in detail. The presented analytical solution may provide a useful application of the complexity factor in the dynamics of non-static hyperbolically symmetric matter distributions.

The current paper recovers a bright 1-soliton solution to the dispersive concatenation model that is considered with power-law of nonlinear self-phase modulation. The semi-inverse variational principle is applied to recover the soliton solution. The parameter constraints that naturally emerge from the analysis for the existence of bright solitons are also presented in the work. (c) 2026 L&H Scientific Publishing, LLC. All rights reserved.

The primary objective of this work is to investigate the complex dynamics of coupled dis-persionless equations formulated with the conformable derivative. Exact soliton solutions are first derived using the Jacobi elliptic function expansion method. These solutions are then used to train a data-driven model based on a Bayesian regularization-backpropagation neural network. The trained network is evaluated against the exact solutions through visual comparison using surface plots, contour plots, and corresponding error graphs. Furthermore, statistical metrics are computed for varying values of the conformable derivative parameter to assess prediction accuracy and robustness. The close agreement between predicted and exact solutions confirms the effectiveness of the proposed data-driven approach. This study highlights the potential of integrating analytical soliton solutions with neural network models for modeling nonlinear systems involving conformable derivatives.

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 investigates a stochastic Yajima-Oikawa long-wave-short-wave resonance system with multiplicative Stratonovich phase noise acting on the short-wave component. By embedding the Brownian motion into the carrier phase, we show that the stochastic perturbation is gauge-removable from the amplitude dynamics. Consequently, the traveling-wave reduction leads to a deterministic Duffing-type profile equation while the long-wave component is slaved to the short-wave intensity through an explicit algebraic relation. Using the associated Hamiltonian structure, we classify the admissible dynamical regimes and derive explicit solitary-wave solutions, kink-type amplitude waves with dark-notch intensity profiles on a nonzero background, and Jacobi elliptic periodic waves. The homoclinic, heteroclinic, and closed periodic trajectories of the reduced phase plane are directly connected with these wave families. We also perform a modulational-instability analysis of the continuous-wave background and prove that, for the present pure Stratonovich phase-noise model, the instability threshold and growth law coincide with those of the gauge-equivalent deterministic system. Finally, we derive phase-sensitive stochastic observables showing that Brownian phase noise produces exponential attenuation of the ensemble mean field and temporal correlation decay, whereas the short-wave intensity and the induced long-wave response remain deterministic. The results clarify how random carrier-phase fluctuations influence observable phase coherence without changing the leading-order coherent envelope mechanism.

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 study investigates the stochastic fractional new coupled Konno–Oono equation with external forced multiplicative noise, focusing on the chaotic nature, the influence of multiplicative noise intensity, and the fractionality parameter on exact soliton solutions. The proposed model is used to describe the complex phenomena in the magnetic field. To examine the exact solutions for the stochastic fractional new coupled Konno–Oono equation, the functional transformation method is applied. This method yields various forms of travelling wave solutions, including Jacobian elliptic, exponential, hyperbolic, and trigonometric functions, each dependent on different parameters. Additionally, we investigate the quasiperiodic and chaotic behavior of the system, including the formation of shock wave structures and sensitivity to initial conditions. The modulation instability is also investigated for the proposed model, which is important for quickly creating localized structures (such as solitons or rogue waves) by destabilizing unbroken waves when they are slightly disturbed, allowing for energy localization and spectrum widening. The findings in this study not only broaden but also enhance certain results from earlier research. Moreover, the influence of multiplicative noise on the analytical solutions of the stochastic system is illustrated through 3D diagrams and corresponding 2D path lines.

Abstract This article introduces new optical soliton solutions for a fractional version of the quadratic–cubic nonlinear Schrödinger equation that describes the transmission of optical pulses in fiber optic systems with superfast fibers. The solutions are obtained using the modified Sardar sub-equation method and the $$(\frac{1}{\varphi (\zeta )},\frac{\varphi ^{'}(\zeta )}{\varphi (\zeta )})$$ method. The model is transformed into a non-linear fractional partial differential equation with a non-integer order using the conformable derivative, which is an efficient fractional derivative. The proposed methods work by adding a new variable to the equation to convert its form into a non-linear equation with ordinary derivatives. A comparative analysis of the solutions is carried out, and the effect of varying the fractional parameter values on the behavior of the obtained solutions is investigated. The paper’s novelty lies in the fact that no previous articles have identified the new solutions obtained through the application of these two analytical methods. The acquired solutions include kink, bright, periodic, dark-bright, singular and dark-singular wave solutions, which are illustrated using several 3D and 2D graphs. Furthermore, to effectively validates the exactness of analytical solutions with an elevated precision, a numerical method known as differential transform method is carried out. The adopted approaches demonstrate notable performance and are suitable for solving other nonlinear partial differential equations that arise in the natural sciences.

This paper offers a thorough analytical and numerical analysis of nonlinear dynamics under the Ablowitz-Kaup-Newell-Segur (AKNS) water wave equation, particularly in its possible applicability to advanced fiber materials. Utilizing an appropriate wave transformation, the governing nonlinear partial differential equation is simplified into an ordinary differential equation, which is solved exactly using the generalized Riccati equation mapping method (GREMM). The closed-form solutions form the basis for a systematic study of the system's rich dynamical features such as equilibrium states, bifurcations, multi-stability, and chaos. Bifurcation analysis identifies the key transitions, demonstrating the routes from stable periodic oscillations to irregular and chaotic states. Numerical simulations also confirm these transitions, demonstrating the system's evolution between regular, quasi-periodic, and fully developed chaotic states. Illustrations like phase portraits, temporal evolution of series, Lyapunov exponents, Poincar & eacute; sections, and recurrence plots attest to the existence of chaos, fractal-like self-similarity, and huge sensitivity to initial conditions. Multi-stability, which is the coexistence of more than one attractor under the same parameter conditions, highlights the tenuous balance of nonlinear responses in complex systems. These results are most directly applicable to high-performance fiber materials, in which wave propagation, stress-strain instabilities, and nonlinear dynamic responses have a profound impact on performance and reliability. The revealed bifurcation and chaos mechanisms shed light on how fiber composites, smart fibers, and adaptive materials could switch between different mechanical states under different loads, impacts, or environmental inputs. Mapping the entire bifurcation topology of the system provides a predictive platform for designing fibers with improved vibration damping, impact resistance, and energy absorption properties. In general, this work demonstrates the strength of linking analytical exact solutions with numerical computations in revealing the subtle interaction among nonlinearity, parameter variation, and dynamic behavior, providing insightful viewpoints toward designing and controlling next-generation fiber materials and fiber-reinforced structures. The novelty of this work lies in the integration of exact analytical solutions with advanced chaos diagnostics, providing deeper insight into the interplay between nonlinearity and system dynamics, and extending beyond existing studies by offering both a broader solution spectrum and a comprehensive dynamical characterization.

In this paper, we explore novel nonlinear wave solutions of the combined Kairat-II-X equation using the artificial neural network approach. Analytical and approximate results are derived. For exact soliton solutions, the Hirota bilinear form of the governing model is constructed, and the Bilinear neural network method is used. Data-driven approximate results are obtained using the Levenberg–Marquardt artificial neural network approach. Multiple neural network frameworks, such as single- and double-hidden-layer networks, are used to derive analytical solutions with this method. We explore different types of solutions, such as opposite-wave interactions and breather solutions using single-hidden-layer networks, as well as rogue waves and bright–dark soliton solutions by utilizing double hidden layers. The wave characteristics are plotted graphically, and the intricate structure of these solutions is revealed. A data-based verification of the obtained soliton solutions is also performed using the Levenberg–Marquardt artificial neural network method. It has been shown that the computational scheme proposed in this paper is sufficiently effective and accurate through a fitness plot, regression analysis, performance curve, transition state analysis, as well as error histograms. This work also provides insight into the nature of complex wave phenomena in the Ocean wave environment. The feasibility and advantages of the proposed method are demonstrated by its ability to discover new-wave solutions with practical implications for nonlinear dynamics.

The presented paper discusses an extended (3+1)-dimensional generalized Kadomtsev-Petviashivli equation through the use of the Hirota bilinear method and the modified extended tanh method. Bilinear equations are derived using Hirota’s derivatives, which will result in the derivation of lump, breather, two-wave, and three-wave solutions. The obtained solutions have applications in fluid dynamics and are pertinent to the focus on wave and charged particle interactions in plasmas. Furthermore, the modified extended tanh method is applied to derive a solution, yielding hyperbolic, trigonometric, and rational solutions with adjustable parameters. The obtained results demonstrate the practicality, simplicity, and effectiveness of the proposed methods in wave characterization behaviors, and novel wave strategies are being introduced for a range of nonlinear equations encountered in the engineering field. The visualization of our solution is further enhanced by 3D and 2D graphs.

In this study, the dynamical behavior of a nonlinear DNA double-chain model, which depicts the double-helix formation of deoxyribonucleic acid (DNA), is analyzed. The proposed model is essential for the transfer of genetic information and the propagation of energy along biomolecular chains. The hydrogen bonds that connect the polynucleotide strands are depicted by modeling the two DNA chains as elastic rods joined by a flexible membrane, which gives rise to a nonlinear evolution equation that describes the transverse displacements occurring within the molecule. By employing the multivariate generalized exponential differential function method and the generalized Arnous technique, a number of wave solutions are derived. The acquired solutions include bright solitons, kink soliton, lump soliton, W-shaped wave profiles, and periodic wave patterns in suitable parametric values. The acquired wave structures are graphically represented using 3D surface graphs, density plots, and contour plots, providing a clear visualization of the wave behaviors. These analytical results provide a theoretical framework for describing localized excitations and vibrational dynamics in DNA chains, offering analytical insight into internal nonlinear interactions within the biomolecular system. Moreover, we have studied the stability of the proposed model using linear stability analysis. The reported results extend existing studies on nonlinear DNA models and yield a more comprehensive insight into the processes of energy localization and transfer in biomolecular systems, while also showcasing the reliability and applicability of the solution methods utilized.

This work addresses the main inducements: the modified extended direct algebraic approach and residual neural network (MEDARNN) structures to obtain the exact analytical solutions for the (2+1)-dimensional modified Veronese web (MVW) equation. This is an integrable nonlinear partial differential equation (NLPDE) and is used to admit a differential covering with non-removable physical parameters. We present for the first time a novel analytical technique called the MEDARNN method, which combines models of residual neural networks (RNNs) with the auxiliary equation to provide precise solutions for NLPDEs. In addition, by establishing particular activation functions within the '2-2-2-1' and '2-2-3-1' NNs models, the introduction of a novel activation function derived from the auxiliary equations solutions is a significant innovation of this approach, creating a new mathematical connection between deep learning and differential equation theory. The different forms of solutions, such as shock, complex solitary-shock, shock-singular and periodic-singular forms, exhibit both single and coupled wave structures. For the physical interpretation of some of the constructed solutions are draw in the form of three-dimensional (3D), two-dimensional (2D) and corresponding contours by choosing the different values of parameters. This paper offers a new methodological approach for dealing with NLPDEs that is widely applicable in engineering and science domains.

In this paper, we investigated the existence of mild solutions and the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions driven by a higher-order Atangana-Baleanu fractional derivative in the Caputo sense. The system is formulated in an infinite-dimensional framework and incorporates Brownian motion processes, impulsive effects, and non-smooth multi-valued nonlinearities described via Clarke's generalized sub-differential. By employing methods from fractional evolution theory, stochastic analysis, and multi-valued fixedpoint theory, we established sufficient conditions for solvability and approximate controllability. The results extended classical controllability frameworks to systems exhibiting memory, randomness, and impulsive dynamics. An illustrative example is provided to demonstrate the applicability of the theoretical findings.

In numerous branches of physics, the nonlinear cubic Schr & ouml;dinger equation serves as a fundamental model with profound applications, particularly in optics, plasma physics, fluid dynamics, and quantum mechanics. In this study, four analytical techniques have been employed to explore and derive soliton solutions of the complex Schr & ouml;dinger equation. The obtained solutions, expressed via hyperbolic, trigonometric, exponential, and rational functions, exhibit rich wave structures and dynamics. To elucidate the physical significance and practical applicability of the proposed model, several graphical simulations have been performed by assigning appropriate values to the key parameters, thereby illustrating the influence of these parameters on wave propagation characteristics. The proposed methods efficiently generate diverse traveling solutions in physics and applied sciences.

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.