This manuscript aims to explore localized waves for the nonlinear partial differential equation referred to as the $(1+1)$-dimensional generalized Kundu–Eckhaus equation with an additional dispersion term that describes the propagation of the ultra-short femtosecond pulses in an optical fiber. This research delves deep into the characteristics, behaviors, and localized waves of the $(1+1)$-dimensional generalized Kundu–Eckhaus equation. We utilize the multivariate generalized exponential rational integral function method (MGERIFM) to derive localized waves, examining their properties, including propagation behaviors and interactions. Motivated by the generalized exponential rational integral function method, it proves to be a powerful tool for finding solutions involving the exponential, trigonometric, and hyperbolic functions. The solutions we found using the MGERIF method have important applications in different scientific domains, including nonlinear optics, plasma physics, fluid dynamics, mathematical physics, and condensed matter physics. We apply the three-dimensional (3D) and contour plots to illuminate the physical significance of the derived solution, exploring the various parameter choices. The proposed approaches are significant and applicable to various nonlinear evolutionary equations used to model nonlinear physical systems in the field of nonlinear sciences.

The primary aim of this research article is to investigate the soliton dynamics of the M-truncated derivative nonlinear Kodama equation, which is useful for optical solitons on nonlinear media, shallow water waves over complex media, nonlocal internal waves, and fractional viscoelastic wave propagation. We utilized two recently developed analytical techniques, the generalized Arnous method and the generalized Kudryashov method. First, the nonlinear Kodama equation is transformed into a nonlinear ordinary differential equation using the homogeneous balance principle and a traveling wave transformation. Next, various types of soliton solutions are constructed through the application of these effective methods. Finally, to visualize the behavior of the obtained solutions, three-dimensional, two-dimensional, and contour plots are generated using Maple (2023) mathematical software.

In this study, we investigate a nonlinear Schrödinger equation relevant to the evolution of optical beams in weakly nonlocal media. Utilizing the modified F-expansion method, we construct a variety of novel soliton solutions, including dark, bright, and wave solitons. These solutions are illustrated through comprehensive graphical simulations, including 2D contour plots and 3D surface profiles, to highlight their structural dynamics and propagation behavior. The effects of the temporal parameter on soliton formation and evolution are thoroughly analyzed, demonstrating its role in modulating soliton shape and stability. To further explore the system’s dynamics, chaos and sensitivity theories are employed, revealing the presence of complex chaotic behavior under perturbations. The outcomes underscore the versatility and richness of the present model in describing nonlinear wave phenomena. This work contributes to the theoretical understanding of soliton dynamics in weakly nonlocal nonlinear optical systems and supports advancements in photonic technologies. This study reports a novel soliton structure for the weak nonlocal cubic–quantic NLSE and also details the comprehensive chaotic and sensitivity analysis that represents the unexplored dynamical behavior of the model. This study further demonstrates how the underlying nonlinear structures, along with the novel solitons and chaotic dynamics, reflect key symmetry properties of the weakly nonlocal cubic–quintic Schrödinger model. These results enhanced the theoretical framework of the nonlocal nonlinear optics and offer potential implications in photonic waveguides, pulse shape, and optical communication systems.

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations.

This study examines the nonlinear dynamical behavior of a Van der Waals system in the viscosity–capillarity regularization form. The solitary wave solutions of the proposed model are investigated using advanced analytical techniques, including the generalized Arnous method, the modified generalized Riccati equation mapping method, and the modified F-expansion approach. Additionally, we use mathematical simulations to enhance our comprehension of wave propagation. Moreover, a machine learning algorithm known as the multilayer perceptron regressor neural network was adopted to predict the performance results of our soliton solutions. Another important aspect of this study is the exploration of the chaos of the studied model by introducing a perturbed system. Chaotic analysis is supported by different techniques, such as return maps, power spectra, a bifurcation diagram, and a chaotic attractor. This multifaceted investigation not only emphasizes the rich dynamical pattern of the studied model but also presents a robust mathematical framework for studying nonlinear systems. The studied model also presents a robust mathematical framework for studying nonlinear systems. This study offers novel insights into nonlinear dynamics and wave phenomena by assessing the effectiveness of modern methodologies and clarifying the distinctive characteristics of a system’s nonlinear dynamics.

This paper develops a functional operator-theoretic framework for nonlinear Erdelyi–Kober (EK) fractional dynamical systems formulated in Banach spaces endowed with the compact-open topology. Within this setting, sufficient conditions for existence, uniqueness, and Ulam–Hyers stability of solutions are established using the Banach and Schaefer fixed-point theorems. The continuity, boundedness, and Lipschitz properties of the associated nonlinear operators are analyzed to ensure well-posedness of the fractional system. As a constructive complement to the theoretical results, a power series iterative method (PSIM) is employed to obtain an explicit fractional series representation of the solution in the case $0<\alpha <1$. The applicability of the theoretical framework is illustrated through a nonlinear fractional dynamical Belousov–Zhabotinsky system (DBZS), where the assumptions of the main theorems are verified and the solution is constructed via the proposed series scheme. The results provide a coherent link between abstract fixed-point analysis and a constructive semi-analytical representation of solutions for EK fractional systems.