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The Klein–Fock–Gordon equation (KFGE), defined as the equation of relativistic wave related to NLEEs, has numerous implications for energy particle physics and is useful as a model for several types of matter, with deviation in the basic stuffs of particles and in crystals. In this work, the Sardar subequation method (SSM) is used for finding the solution of this KFGE. The advantage of SSM is that it provides many different kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright solitons. The results show that the SSM is very reliable, simple and can be functionalized to other nonlinear equations. It is verified that all the attained solutions are stable by modulation instability process. To enhance the physical description of solutions, some 3D, contour and 2D graphs are plotted by taking precise values of parameters using Maple 18.

The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov’s method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems.

This paper focuses on the use of the Sardar sub-equation method to obtain optical solitons for a nonlinear model that incorporates Kudryashov’s quintuple power law and dual-form nonlocal nonlinearity. The model studies the propagation of pulses in optical fibers and the proposed technique is used to investigate various types of solitons including bright, dark, singular, periodic, combined dark-bright, and combined dark-singular solitons. The proposed technique is highly efficient and can also be utilized to solve higher-order nonlinear partial differential equations in fields such as fluid mechanics, plasma physics, and fiber optics. The paper includes a comparative analysis of the model and presents graphical representations for some of the obtained solutions.

The nonlinear Schrödinger equation is used to model various phenomena, such as solitons self-focusing effects and rogue waves. In the ocean engineering, the modified nonlinear Schrödinger equation investigates the behavior of water waves, considering the complex interaction of dispersion nonlinearity, and dissipation effects. By introducing fractional derivatives to the model, the M-fractional conformable modified nonlinear Schrödinger equation allows for the investigation of fractional order effects, which can study more accurately the behavior of wave propagation in real-world ocean engineering. The novelty of our research lies in the application of of the M-fractional conformable derivative on the governed equation which represents an advancement in the existing work, which have used nonlinear Schrödinger equations without fractional derivatives. Two powerful techniques: the Jacobi elliptic function method and unified solver method are applied to attain solutions to the M-fractional modified nonlinear Schrödinger equation. The several results, including dark, bright, singular, periodic, and dark-bright soliton solutions are obtained which provide valuable insights into the complex behavior of water waves in ocean engineering. Additionally, 3D and contour graphs have been provided to visually illustrate the impact of the fractional order. We also illustrate these solutions at different values of the fractional order which explain how variations in this parameter affect wave propagation. These findings will contribute to the advancement of ocean engineering techniques, enhancing our ability to design and implement effective solutions for coastal protection, offshore structures, and marine renewable energy systems.

In this study, we explore different methods to investigate the nonlinear Helmholtz equation. First, to investigate the dynamic behavior of the system, we apply bifurcation analysis, visually depicted via phase portraits. Next, we introduce the periodic functions into the dynamical system to explore chaotic, which introduces complexity into the system and leads the emergence of chaotic dynamics. We graphically present these dynamics with the aid of 3D, 2D phase plots and time plots. We then examine the chaotic nature of the system in detail, revealing intricate patterns of instability. By changing the initial conditions, we study the multistability, which demonstrates how the system shows different stable states by taking selected parameters. We perform the sensitivity analysis to evaluate how small changes in the system’s parameters affect its overall behavior and provide a deeper understanding of its robustness and responsiveness to perturbations. Finally, we apply the Sardar subequation method to derive exact solutions for the governed equation. These solutions are graphically represented through 3D plots with projections, polar plots, and 2D plots of the real, absolute, and imaginary components of the solutions. This study offers a thorough investigation of the equation by employing bifurcation analysis, chaotic dynamics, multistability, sensitivity analysis, and the Sardar subequation method to uncover the complex behaviors that arise in nonlinear wave propagation. The model’s strength lies in accommodating wide restoring spatial symmetry and beam angles, while its complexity poses a challenge. The proposed methods effectively uncover intricate wave phenomena, making them highly applicable to nonlinear wave propagation and optical systems. The novelty lies in the dynamic analysis of the nonlinear Helmholtz equation through bifurcation, chaos, multistability, and sensitivity studies.

This study explores optical solitons in the Biswas–Arshed equation within birefringent fibers. Employing the unified solver method, the 1 φ ( η ) , φ ′ ( η ) φ ( η ) method, and new Kudryashov’s method, we extract various optical soliton solutions, encompassing dark, singular, bright, and periodic forms. These solutions deepen our understanding of dynamic phenomena in birefringent fibers, showcasing their potential practical applications. The results, effectively visualized in 3D and 2D plots, reveal intricate patterns. Our research underscores the efficacy and simplicity of these approaches in obtaining optical solitons for diverse nonlinear evolution equations. The novelty lies in the advanced methodologies applied to investigate the Biswas–Arshed equation, yielding a diverse array of soliton solutions and their practical implications. This study not only presents a variety of solutions but also highlights their applicability across disciplines and real-world scenarios. Consequently, our research significantly contributes to advancing our understanding of optical solitons in birefringent fibers, offering a methodological breakthrough in engineering and applied physics.

In this study, we explore different methods to investigate the nonlinear Helmholtz equation. First, to investigate the dynamic behavior of the system, we apply bifurcation analysis, visually depicted via phase portraits. Next, we introduce the periodic functions into the dynamical system to explore chaotic, which introduces complexity into the system and leads the emergence of chaotic dynamics. We graphically present these dynamics with the aid of 3D, 2D phase plots and time plots. We then examine the chaotic nature of the system in detail, revealing intricate patterns of instability. By changing the initial conditions, we study the multistability, which demonstrates how the system shows different stable states by taking selected parameters. We perform the sensitivity analysis to evaluate how small changes in the system’s parameters affect its overall behavior and provide a deeper understanding of its robustness and responsiveness to perturbations. Finally, we apply the Sardar subequation method to derive exact solutions for the governed equation. These solutions are graphically represented through 3D plots with projections, polar plots, and 2D plots of the real, absolute, and imaginary components of the solutions. This study offers a thorough investigation of the equation by employing bifurcation analysis, chaotic dynamics, multistability, sensitivity analysis, and the Sardar subequation method to uncover the complex behaviors that arise in nonlinear wave propagation. The model’s strength lies in accommodating wide restoring spatial symmetry and beam angles, while its complexity poses a challenge. The proposed methods effectively uncover intricate wave phenomena, making them highly applicable to nonlinear wave propagation and optical systems. The novelty lies in the dynamic analysis of the nonlinear Helmholtz equation through bifurcation, chaos, multistability, and sensitivity studies.

The newly developed (3+1)-dimensional nonlinear Kudryashov’s equation offers a framework for studying the behavior of propagating modulated envelope signals and is particularly useful in the fields, where the propagation of wave-like phenomena is essential such as in the field of fluid dynamics, it helps to analyze phenomena like water waves and fluid flow behavior. In plasma physics, it helps to understand plasma instabilities and fusion device behavior. To obtain soliton solutions for the (3+1)-dimensional nonlinear Kudryashov’s equation, the ϕ 6 -model expansion method is employed to derive various forms of solitons, each with its own unique characteristics and properties. This analytical technique is known for its effectiveness in constructing soliton solutions for nonlinear equations and also provides constraints and conditions for the existence of these solitons. Furthermore, the study effectively emphasizes the physical significance of the proposed equation by presenting insightful graphical representations of the constructed soliton solutions. By considering a higher level of nonlinearity, this latest version of the equation presents a significant advancement in the understanding of soliton dynamics and provides a more comprehensive framework for studying wave phenomena. Our study delves into the fascinating realm of wave propagation and soliton dynamics, making a significant and original contribution to this field.

This study examines the (3 + 1) dimensional extended quantum Zakharov–Kuznetsov equation in weakly nonlinear ion-acoustic phenomena and quantum electron-positron-ion magneto plasma. For accomplishing this goal, two distinct mathematical approaches namely new mapping method and new Kudryashov’s method are fashioned. These solutions encompass dark, bright, singular, periodic singular, and some other rational solutions. Graphical depictions of some obtained solutions by 2D, 3D and contour plots meticulously crafted within figures offered profound insights into the deep physical appearances of the structures under examination. Our outcomes highlight that the proposed methods assist as an efficient and inclusive approaches to discover the solitons for the present model. The comparison with previous papers highlights that the methods employed in our study are being utilized for the first time within the context of the extended quantum Zakharov–Kuznetsov equation which underscores the novelty of our paper. By retaining these two methods, we not only boost our consideration of the dynamical behavior of these kind of models but also provide a useful tool for finding specific soliton solutions of nonlinear evolution equations.

The time-regularized long-wave equation is pivotal in understanding diverse wave dynamics, such as shallow water waves, pressure waves in liquids and gas bubbles, ion-acoustic waves in plasma, and nonlinear transverse waves in magnetohydrodynamics. The analysis is initiated by deriving the infinitesimal generators of the Lie group symmetries, followed by constructing a commutator table and adjoint table to study the algebraic structure of the equation’s symmetries. Using this symmetries, the time-regularized long-wave equation is systematically reduced and solved for invariant solutions associated with each symmetry. These solutions give insight into the inherent properties of the system and wave behavior. The extended hyperbolic function method is employed to derive exact solutions to the simplified versions of the time-regularized long-wave equation. This method enhances the solution process by generating soliton solutions including dark, singular, bright and periodic-singular solutions. To illustrate their behavior of obtained solutions, graphical representations are given by using different values of parameters. A comprehensive bifurcation analysis is also conducted to explore the qualitative changes in the dynamics of system, while the Hamiltonian structure of the equation is constructed to investigate its conservation properties. The study also investigates the phase portraits of the system to offer a visual interpretation of the solution trajectories in phase space. Finally, the modulation instability of the time-regularized long-wave equation through linear stability analysis is analyzed to provide a clearer understanding of the conditions that lead to the onset of instability in wave propagation.