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This research article examines the correctness of two new analytical methods for solving the internal solitary waves of shallow seas. To get the computational solutions to the positive (2 + 1)-dimensional Gardner–Kadomtsev–Petviashvili model, the extended simplest equation and modified Kudryashov methods are used. Numerous new traveling wave solutions in various forms are developed in order to assess the starting conditions required for the variational iteration technique, one of the most accurate semi-analytical methods. The semi-analytical solutions are used to demonstrate the precision of the solutions obtained and the analytical methods employed. The dynamical behavior of internal solitary waves in shallow waters is shown using many three-dimensional drawings. The performance of the used schemes demonstrates their efficacy and power, as well as their capacity to handle a large number of nonlinear evolution equations.

This article proposes and analyzes a fractional-order susceptible, infectious, susceptible (SIS) epidemic model with saturated treatment and disease transmission by employing four recent analytical techniques along with a novel fractional operator. This model is computationally handled by extended simplest equation method, sech–tanh expansion method, modified Khater method, and modified Kudryashov method. The results’ stable characterization is investigated through the Hamiltonian system’s properties. The analytical solutions are demonstrated through several numerical simulations.

This article investigates the dynamical and physical behavior of the second positive member in a new, utterly integrable hierarchy. This investigation depends on constructing novel analytical and approximate solutions to the Qiao model. The model’s name is after the researcher who derived the mathematical formula of it in 2007. This model possesses a Lax representation and bi-Hamiltonian structure. This study employs the unified and variational iteration (VI) method to obtain analytical and numerical solutions to the considered model. The obtained analytical solutions are used to calculate the necessary conditions for applying the suggested numerical method that makes checking the obtained solutions’ accuracy a valuable option. The obtained solutions are sketched in different techniques to explain more physical and dynamics details of the Qiao model and show the matching between obtained solutions.

Sardar subequation approach, which is one of the strong methods for solving nonlinear evolution equations, is applied to conformable resonant Schrödinger’s equation. In this technique, if we choose the special values of parameters, then we can acquire the travelling wave solutions. We conclude that these solutions are the solutions obtained by the first integral method, the trial equation method, and the functional variable method. Several new traveling wave solutions are obtained including generalized hyperbolic and trigonometric functions. The new derivation is of conformable derivation introduced by Atangana recently. Solutions are illustrated with some figures.

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ( ℂ ) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ( ℂ ) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics. [ABSTRACT FROM AUTHOR]

This study comprehensively explores the (3 + 1) -dimensional Mikhailov–Novikov–Wang (ℕ) integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified () , Khater II ( hat.II) method, and He's variational iteration (ℍ ) method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation's pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics. [ABSTRACT FROM AUTHOR]

This study introduces innovative analytical solutions for the (2 + 1) -dimensional nonlinear Jaulent–Miodek () equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of waves. Within the realm of nonlinear fluid dynamics, the equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the (2 + 1) -dimensional equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field. [ABSTRACT FROM AUTHOR]

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He's Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This study aims to tackle the generalized coupled nonlinear Schrödinger ( ℂ – ℕ ) equations, with a focus on understanding their physical significance and stability, especially in the realm of plasma physics. These equations are crucial for grasping the complex dynamics of wave interactions within plasma systems, which are fundamental for phenomena like wave-particle interactions, turbulence, and magnetic confinement. We employ analytical methods such as the generalized rational ( ℝ at) and Khater II ( hat.II) techniques, along with characterizing the system using Hamiltonian principles, to carefully examine the stability of solutions. The relevance of this model extends across various plasma phenomena, including electromagnetic wave propagation, Langmuir wave dynamics, and plasma instabilities. By applying these analytical techniques, we derive solutions and investigate their stability using Hamiltonian dynamics, providing valuable insights into the fundamental behavior of nonlinear plasma waves. Our findings reveal the existence of stable solutions under specific conditions, thus advancing our understanding of plasma dynamics significantly. This research carries significant implications for fields such as plasma physics, astrophysics, and fusion research, where a deep understanding of plasma wave stability and dynamics is crucial. Essentially, our study represents a scholarly effort to offer fresh perspectives on the behavior of ℂ – ℕ equations within plasma systems, contributing to the academic discourse on plasma wave phenomena. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt ( ) model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model's stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines. [ABSTRACT FROM AUTHOR]

This study aims to solve the nonlinear Pochhammer–Chree (ℕℙℂ) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II (핂hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the ℕℙℂ equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the ℕℙℂ equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics. [ABSTRACT FROM AUTHOR]

In this study, we investigate the complex dynamics of the (1+1)-dimensional generalized Burgers–Fisher (GBF) model, a nonlinear partial differential equation that encapsulates the interplay between wave propagation, diffusion, and reaction processes. Our work employs a combination of the modified Khater (MKhat) method, the unified (UF) method, and He’s variational iteration (HVI) scheme to derive and validate analytical and numerical solutions. We present a comprehensive analysis of solitary wave, shock wave, and diffusion-driven phenomena within the GBF framework. The novelty of our study lies in the integration of these methods to provide deeper insights into the model’s physical implications, specifically highlighting the interactions between nonlinear advection, diffusion, and reaction mechanisms. This approach not only enhances the accuracy and applicability of the derived solutions, but also contributes to the advancement of nonlinear wave theory and related interdisciplinary fields. [ABSTRACT FROM AUTHOR]

This paper delves into the intricacies of the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov ( ℕ ℕ ) model, a nonlinear partial differential equation governing ion-acoustic wave propagation in plasma. By employing advanced analytical and numerical approaches, the study explores innovative solitary wave solutions, particularly focusing on the dynamics of isochoric flow. Isochoric flow analysis is crucial for unraveling the complex behaviors exhibited by incompressible fluids like elastomers and bio-elastomers, which maintain a constant density. The derivation of the (2 + 1)-dimensional ℕ ℕ equation stems from fluid equations governing plasma dynamics. This model serves as a valuable tool for simulating experimental observations of plasma waves. The computational methodology applied in this research demonstrates a commendable level of precision and consistency, yielding novel solitary wave solutions previously unreported in the ℕ ℕ model. These results underscore the study's importance and novelty. The outcomes not only contribute to our understanding of incompressible fluid dynamics, but also lay the groundwork for future investigations in this domain. The revealed solitary wave solutions have the potential to inform the development of more accurate models for predicting fluid dynamics, thereby advancing the field. [ABSTRACT FROM AUTHOR]