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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.

This study investigates the nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) model, which is widely applied to phenomena such as population dynamics, combustion processes, wave propagation in excitable media, and other systems characterized by reaction–diffusion behavior. The primary objective is to examine the intricate interplay between diffusion and nonlinear growth, crucial for understanding processes where spatial spreading is coupled with local growth or decay, such as species invasion, disease transmission, and chemical reactions. The problem addressed is the challenge of obtaining accurate, analytical solutions to the nonlinear Fisher–KPP equation, which often resists conventional solution techniques due to its complex, coupled dynamics. This study employs two advanced analytical methods — the modified Khater (MKhat) and unified (UF) techniques — to derive new, exact solutions to the Fisher–KPP model. These solutions provide deeper insights into the underlying dynamics of the reaction–diffusion systems, showcasing the balance between diffusion-driven spreading and nonlinear growth effects. Numerical validation of the obtained solutions is carried out using He’s variational iteration (HVI) scheme, ensuring the reliability and accuracy of the analytical results. This combination of exact and numerical solutions strengthens the study’s findings and offers robust tools for future research. The expected results include a set of novel, exact analytical solutions that contribute significantly to the understanding of the diffusion and nonlinear growth interplay in reaction–diffusion systems. These solutions have practical applications in forecasting and modeling processes such as population growth, epidemic spread, combustion dynamics, and chemical reaction rates. The study’s innovative use of MKhat and UF methods introduces new approaches for solving nonlinear differential equations, expanding the toolkit available to researchers in applied mathematics and related fields. In conclusion, this research provides a comprehensive exploration of the nonlinear Fisher–KPP model, offering both theoretical advancements and practical applications. The findings have wide-ranging interdisciplinary implications, benefiting fields such as biology, physics, chemistry, and engineering, where reaction–diffusion systems play a central role.

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model’s applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order.

This research paper studies the optical soliton wave solutions of the model of sub-10-fs-pulse propagation by the implementation of the modified Khater method. This model describes the dynamics of light pulses that represent a higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term. The validity of this model depends on one primary hypothesis, which is the carrier wavelength of the soliton is much shorter than the spatial width. This means that the amplitude of the soliton frequency must be less than the carrier frequency. The shorter femtosecond pulses ($<$100 fs) are desired to increase the bit rate of pulse propagation. The losing of distribution in such short-wavelength pulses through waveguides is a negligible loss. Our solitary analytical wave solutions are approved with the waveguide made of highly nonlinear optical materials.

This research paper extracts novel analytical and semi-analytical wave solutions of the longitudinal wave equation in a magneto-electro-elastic circular rod by using the modified Khater method as one of the most novel and general computational methods and the Adomian decomposition method as a semi-analytical method. The longitudinal waves in metallic thin films are explained for the first time by Nilsson and Lindau, who used the visual evidence. They noted subdued anomalies in the ports of thin ($\cong$100 Å) Ag layers deposited on amorphous silica for p-polarized light at frequencies padlock to the dynamic plasma frequency. These properties are studied by our two suggested methods and are explained by sketching some of our obtained solutions. Moreover, the stability property is tested for our obtained solutions by using the features of the Hamiltonian system. The performance of our used methods shows the power and effectiveness of these methods and their ability to apply on many different forms of nonlinear partial differential equations.

The primary aim of this paper is to investigate the dynamic behavior of generalized nonlinear fractional Tzitzéica-type equations and to derive optical soliton solutions. To achieve this goal, we employ the modified Khater method, focusing on obtaining solitary wave solutions for generalized fractional Tzitzéica-type (TT) equations. Through this approach, we unveil novel solutions for both Tzitzéica and Tzitzéica–Dodd–Bullough (TDB) equations expressed in terms of fractional derivatives. The significance of employing the modified Khater method lies in its ability to yield a diverse array of soliton solutions. These solutions encompass dark, bright, singular, periodic, kink, singular kink, and combined dark–bright solitons. The derived solutions are visually represented through two-dimensional (2D) and three-dimensional (3D) graphs. Our findings underscore that the proposed method serves as a comprehensive and efficient approach to explore exact solitary wave solutions for generalized fractional TT evolution equations. By employing the modified Khater method, we not only enhance our understanding of the dynamic behavior of these equations, but also provide a versatile tool for obtaining precise soliton solutions in the realm of nonlinear fractional evolution equations.

In this research, the analytical and numerical solutions of the fractional nonlinear space-time Phi-four model are investigated by employing two systematic schemes and the B-spline schemes. A new fractional operator definition is applied to this model to convert the model from its fractional formula to an integer-order nonlinear ordinary differential equation. The considered model is of major interest for studying the nuclear interaction, elementary particles in a condensed medium, and propagation of dislocations in crystals. Explicit wave solutions are obtained.