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This study focuses on solving the Caudrey–Dodd–Gibbon (CDG ) equation using the Khater II (KII ) method and the Variational Iteration ( V I ) method. The CDG equation is a pivotal mathematical model in nonlinear wave dynamics, essential for understanding the evolution, interaction, and preservation of wave forms in dispersive media. Its applications span various fields, including fluid dynamics, nonlinear optics, and plasma physics, where it plays a crucial role in analyzing solitons and complex wave interactions. In this research, we meticulously implement the KII and V I methods to derive solutions for this nonlinear partial differential equation. Our findings reveal new aspects of the equation’s behavior, offering deeper insights into nonlinear wave phenomena. The significance of this study lies in its contribution to advancing the understanding of these phenomena and their practical applications in the academic realm. The results underscore the effectiveness of the employed methods, their innovative contributions, and their relevance to applied mathematics. [ABSTRACT FROM AUTHOR]

The current study examines the (2 + 1)-dimensional fractional Chaffee–Infante (FCI) model, which is a nonlinear evolution equation that characterizes the processes of pattern generation, reaction-diffusion, and nonlinear wave propagation. The construction of analytical solutions involves the use of analytical methods, namely the Khater III and improved Kudryashov schemes. The He’s Variational Iteration method is employed as a numerical approach to validate the accuracy of the obtained solutions. The main objective of this study is to get novel analytical and numerical solutions for the FCI model, with the intention of gaining a deeper understanding of the system’s dynamics and its possible implications in the fields of fluid mechanics, plasma physics, and optical fiber communications. The study makes a valuable contribution to the area of nonlinear science via the use of innovative analytical and numerical methodologies in the FCI model. This research enhances our comprehension of pattern creation, reaction–diffusion phenomena, and the propagation of nonlinear waves in diverse physical scenarios. [ABSTRACT FROM AUTHOR]

The primary aim of this study is to solve the complex Akbota equation using the Khater II (Khat II) and unified (UF) methods as analytical techniques, while validating the accuracy of the constructed solutions through the Adomian decomposition method as a numerical scheme. The complex Akbota equation is a significant nonlinear evolution equation with crucial applications in fluid dynamics, optical fibers, and quantum field theory. It shares characteristics with other well-known nonlinear equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, highlighting its importance in modeling diverse physical phenomena. The Khater II and UF methods enable the construction of exact analytical solutions, while the Adomian decomposition method ensures the numerical accuracy of these solutions, demonstrating their applicability to real-world scenarios. The study's findings reveal precise solutions that align closely with numerical results, underscoring the robustness of the proposed methods. This research contributes to the field by providing novel analytical solutions and verifying their accuracy, thereby enhancing our understanding of the complex Akbota equation's physical behavior. The significance lies in the potential applications of these solutions in various scientific and engineering domains. Conclusively, the study offers new insights and methodologies for tackling complex nonlinear equations, advancing both theoretical and practical knowledge in applied mathematics and physics. [ABSTRACT FROM AUTHOR]

This study undertakes a comprehensive analytical and numerical investigation of the nonlinear Kairat model, a significant evolution equation that governs a wide range of physical phenomena, including shallow water waves, plasma physics, and optical fibers. The Kairat model effectively describes the propagation of nonlinear waves in shallow water, capturing the intricate interplay between nonlinearity and dispersion. It exhibits similarities with well-known nonlinear evolution equations such as the Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, thereby offering insights into their common underlying dynamics. To achieve the objectives of this research, we employ the modified Khater (MKhat) and unified (UF) methodologies to derive exact solutions for the Kairat model. Furthermore, the trigonometric-quantic-B-spline (TQBS) scheme is utilized as a numerical technique to verify the accuracy of these derived solutions and validate their applicability within the domain of shallow water wave propagation. This investigation yields a collection of innovative and precise analytical solutions, elucidating the complex nonlinear behavior of the Kairat model and its effectiveness in capturing the dynamics of shallow water waves. Moreover, these analytical solutions are corroborated through numerical simulations conducted using the TQBS scheme, ensuring their reliability and practical significance in understanding and predicting shallow water wave phenomena. The significance of this endeavor lies in its contribution to a deeper understanding of the dynamics of the Kairat model and its potential applications in fields such as coastal engineering, oceanography, and related disciplines. The integration of analytical and numerical techniques offers new perspectives and methodologies for exploring nonlinear evolution equations, potentially benefiting researchers in applied mathematics, physics, and engineering. In summary, this comprehensive analytical and numerical investigation provides novel insights, precise solutions, and a robust foundation for further exploration of the physical implications and applications of the Kairat model in the context of shallow water wave propagation. [ABSTRACT FROM AUTHOR]

This study employs advanced computational methodologies to investigate the nonlinear complex fractional generalized Zakharov equations, herein referred to as NCFGZ . The primary objective is to elucidate precise and groundbreaking solitary wave solutions. These equations find wide–ranging applications in diverse domains, including plasma physics, nonlinear optics, and wave dynamics. The pursuit of highly accurate solitary wave solutions is of paramount significance, as it contributes to both theoretical comprehension and practical real-world applications. By harnessing cutting-edge computational techniques along with conformable fractional derivative (CFD) and leveraging the capabilities of Mathematica 13.1 software, this research significantly advances theoretical knowledge and provides valuable practical insights. The computational approaches employed in this research outperform conventional methods in terms of their efficacy, demonstrating a high degree of precision and efficiency in solving complex fractional differential equations. These findings underscore the pivotal role that precise solitary wave solutions play in forecasting and comprehending the behavior of nonlinear systems governed by the generalized Zakharov, or G Z , equations. Furthermore, this study yields profound implications for the symmetry theorem of solitons, enhancing our understanding of symmetry properties within the context of G Z equations and their broader influence on wave phenomena. [ABSTRACT FROM AUTHOR]

This study is centered on addressing the complexities of the nonlinear time-fractional Harry Dym equation through a combined application of analytical and numerical methodologies. The principal objective is to establish a comprehensive framework for solving this intricate fractional partial differential equation. The introduction of the Khater II method as an analytical technique presents a novel approach specifically designed to handle such equations. In tandem, the utilization of numerical schemes-cubic-B-spline, quantic-B-spline, and septic-B-spline methods-further refines the accuracy and efficiency of the solutions. The outcomes underscore the efficacy of these proposed methods in effectively resolving the challenges posed by the Harry Dym equation. Emphasis is placed on the dual facets of analytical insights and numerical precision. The significance of this study lies in its contribution to advancing comprehension of nonlinear time-fractional equations while offering practical tools for their resolution. The conclusions drawn from this research provide valuable insights into the practical applicability of the Khater II method and B-spline schemes in navigating complex mathematical models. The novelty of this work lies in the amalgamation of an innovative analytical approach with established numerical techniques, signifying a noteworthy contribution to this field of study. [ABSTRACT FROM AUTHOR]

This study presents a new approach for accurately obtaining solitary wave solutions in the Boiti–Leon–Pempinelli (BLP) system, which was developed by Italian mathematicians Luca Boiti, Franco Leon, and Guido Pempinelli in the late 1980 s. The BLP system is very significant in physics and is employed in several fields. It is used to simulate the behavior of nonlinear media in electromagnetic and acoustic wave propagation, as well as to understand the features of Bose–Einstein condensates. This research thoroughly examines the acquired solutions, with a particular focus on crucial factors such as the maximum amplitude, duration, and velocity of the waves. The accurate solitary wave solutions play a crucial role in enhancing the comprehension of these physical phenomena, establishing the foundation for possible practical uses. The suggested modified exponential function approach outperforms previous numerical strategies in terms of efficacy and dependability, demonstrating the significant potential for future study in this field. The results of our investigation are graphically shown using meticulously designed graphs, providing valuable insights into the behavior of various solutions inside the (x, t) domain. This visual depiction facilitates the identification of key characteristics such as velocity, magnitude, and form, hence improving the clarity of our study findings. [ABSTRACT FROM AUTHOR]

This research delves into the Schäfer–Wayne equation, a theoretical model that describes the nonlinear propagation of ultra-short pulses in silica optical fibers. The significance of this study lies in our utilization of three distinct analytical techniques to obtain a range of solutions. Additionally, we employ the multiplier approach to calculate conservation laws and identify the Lie point symmetries of the equation. A novel aspect of our investigation is the exploration of periodic, quasiperiodic, and weakly chaotic behaviors of nonlinear optical pulses for the Schäfer–Wayne equation, a topic explored here for the first time. To elaborate on our findings, we provide detailed descriptions of specific solutions. This work contributes to our understanding of the nonlinear dynamics of optical pulses and extends beyond previous efforts in the literature by introducing new analytical methods and uncovering unexplored behaviors within the context of the Schäfer–Wayne equation. [ABSTRACT FROM AUTHOR]

Understanding the intricate Korteweg–de Vries ( c K d V ) equation is paramount for comprehending various nonlinear wave phenomena, owing to its capacity to depict the propagation of nonlinear waves in media characterized by complex-valued dispersive and nonlinear coefficients. The practical ramifications of the c K d V equation are extensive, spanning disciplines such as optics, plasma physics, and other realms dealing with intricate media. This investigation employs the Khater II method and a generalized rational approach as analytical methodologies to generate novel exact solutions for the c K d V equation. The Khater II method is employed to transform the nonlinear partial differential equation into a nonlinear ordinary differential equation, thus facilitating a systematic resolution. Moreover, the generalized rational technique utilizes a rational ansatz function to derive diverse forms of solutions. Application of these methodologies leads to the discovery of fresh exact solutions, encompassing solitary and periodic wave solutions for the c K d V equation. These solutions are expressed in rational forms featuring arbitrary functions, thereby expanding the repertoire of known solutions for the model. The efficacy of the analytical methodologies employed becomes evident through the discovery of these novel exact solutions, thereby enriching our understanding of the physical interpretations and wave characteristics associated with the c K d V equation. The derived solutions augment the existing body of knowledge pertaining to the model. This research enhances our comprehension of the c K d V equation, thereby advancing nonlinear wave analysis with potential applications in physics, optics, plasma science, and allied engineering domains. Future investigations may explore the extension of these methodologies to address other nonlinear wave equations. [ABSTRACT FROM AUTHOR]

This research employs advanced computational methodologies to analyze solitary wave solutions associated with the fractional nonlinear Chaffee–Infante ( C I ) equation, extending classical diffusion models with broad applications in materials science, fluid dynamics, and signal processing. The study makes notable contributions to the modeling of anomalous diffusion in porous materials, the comprehension of nonlinear dynamics, and the analysis of wave behavior incorporating memory and non-local effects. The research enhances our understanding of practical applications and provides valuable insights into complex wave dynamics within fluid dynamics, nonlinear optics, and plasma physics. The computational techniques utilized in this investigation, specifically the extended unified ( E U ) and trigonometric–quantic–B-spline (TQBS ) approaches, demonstrate superior effectiveness in comparison to existing methods, promising heightened accuracy and efficiency in solving fractional partial differential equations. The numerical scheme, Trigonometric–Quantic–B–spline, in tandem with analytical solutions, serves to validate the model and adapt it to real-world complexities. This integrated approach not only quantifies accuracy but also supports parameter estimation, ensuring the model's applicability in diverse engineering and scientific scenarios. Graphical representations of both analytical and numerical solutions are presented, offering a visual elucidation of the model's characteristics and validating solution accuracy. The incorporation of the Trigonometric–Quantic–B–spline scheme provides a robust foundation for further exploration and applications in varied scientific and engineering domains. [ABSTRACT FROM AUTHOR]

This study delves into investigating the stability of solutions pertaining to the generalized Davey–Stewartson ( D S ) equation by employing the generalized rational method and projective Riccati equations method within the Hamiltonian system framework. The central focus lies in unraveling the physical significance of this model within the realm of plasma physics, shedding light on its intricate connections with analogous equations. The generalized D S equation, a pivotal nonlinear evolution model, serves as a vital tool in comprehending the evolution of coupled waves across various physical systems, notably within the domains of fluid dynamics and plasma physics. Its significance lies in its capacity to elucidate the intricate interplay among different wave components, offering a holistic perspective on wave interactions within plasma mediums. Through the meticulous application of rigorous analytical methods, this investigation uncovers pivotal insights into the stability of solutions, thereby enriching our understanding of mathematical models in plasma physics. The findings presented in this research contribute significantly to advancing the understanding of nonlinear dynamics, with profound implications for future exploration in plasma physics and related fields. In essence, this study underscores the analytical prowess of the generalized rational and projective Riccati equations methods, accentuating their relevance and applicability to the generalized D S equation and, by extension, their role in enhancing our comprehension of coupled wave phenomena within diverse plasma systems. [ABSTRACT FROM AUTHOR]

This study presents two novel analytical methodologies for the derivation of solitary wave solutions within the framework of the Benney-Luke ( B L ) model. The B L equation, characterized as a nonlinear evolution equation, plays a pivotal role in the description of long-wavelength, weakly nonlinear, and dispersive waves in shallow water. It holds particular importance in comprehending the behaviors of water waves in shallow channels and across the surfaces of shallow water bodies. By encompassing nonlinearity, dispersion, and dissipation, this equation encapsulates fundamental factors that significantly impact water wave dynamics, providing invaluable insights into attributes such as wave amplitude, frequency, and propagation speed. The soliton wave solutions obtained are rigorously validated using a contemporary numerical scheme, demonstrating a remarkable congruence with the analytical results. Further analysis of these solutions is carried out through comprehensive graphical representations, effectively highlighting the distinctive and innovative characteristics inherent in the B L model. This investigation makes a substantial contribution to the understanding of the intricate dynamics within the B L equation, thereby expanding the repertoire of available analytical and numerical techniques for the study of similar nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This study undertakes the challenging task of unraveling the intricacies embedded in the integrable Kuralay equation, a cornerstone in the field of mathematical physics. Specifically, our focus extends to the integrable dynamics governing space curves, encompassing both the Kuralay-I equation ( K - IE ) and the Kuralay-II equation ( K - IIE ). Our methodological approach leverages innovative analytical techniques, namely the Bernoulli sub-equation function (BSF ) method and the Khater II ( K h I I ) method. For the derivation of numerical solutions, we employ the exponential cubic-B-spline (ECBS ) scheme. The outcomes of our investigation signify a triumphant resolution of the Kuralay equation, thereby enriching our comprehension of its dynamic behavior. The profound impact of this study reverberates throughout the domain of mathematical physics, particularly in illuminating the intricate dynamics inherent in integrable systems. The distinctive aspect of our research lies in the application of the BSF and K h I I methods for analytical solutions, coupled with the utilization of the ECBS scheme for numerical solutions. These methodologies, hitherto underexplored in the context of Kuralay equations, impart a unique quality to our investigation. In conclusion, this research constitutes a significant contribution to the progression of mathematical physics. It not only furnishes valuable insights, numerical solutions, and innovative methodologies for addressing the complex Kuralay equation but also advances our collective understanding of this foundational problem. [ABSTRACT FROM AUTHOR]

This paper examines the complex characteristics of the modified Benjamin–Bona–Mahony equation ( m BBM ) and the Klein–Gordon ( K G ) equation in the field of mathematical physics. The m BBM equation is a basic model used to describe surface water waves, especially in shallow water situations. It provides valuable information on wave propagation, stability, and the formation of solitons. The applications of this instrument are wide-ranging, including fields such as oceanography, where it plays a crucial role in comprehending wave behavior. On the other hand, the K G equation is of utmost importance in quantum field theory since it sheds light on the dynamics and interactions of scalar fields such as mesons. Within the field of particle physics, it offers substantial insights into basic concepts, acting as a fundamental basis for comprehending particle behavior. The primary goal of our work is to develop strong analytical techniques for solving these problems. In order to tackle these issues, we use two novel methodologies: the extended simple equation technique and the generalized Kudryashov method. Furthermore, we validate our results by using the extended cubic–B–spline approach for numerical computations. The work effectively solves these intricate equations, resulting in encouraging results. The presented approaches demonstrate their effectiveness, providing significant advances to mathematical physics. This work has inherent worth by presenting innovative analytical methods and perspectives, particularly designed to solve complex nonlinear equations such as the m BBM and K G equations. The finding has significant ramifications that extend across several scientific fields, offering novel approaches to tackle complex issues in mathematical physics. To summarize, our paper introduces innovative analytical techniques designed to solve nonlinear equations in the field of mathematical physics, with a particular emphasis on the m BBM and K G equations. [ABSTRACT FROM AUTHOR]

Nonlinear dispersive waves influenced by higher–order dispersion effects play a pivotal role across various scientific disciplines. While numerical simulations offer valuable approximations, analytical solutions provide a comprehensive mathematical characterization. In this study, exact solitary wave solutions for the high-order Dispersive Extended Nonlinear Schrödinger (DENLS ) equation were systematically derived using two recent computational techniques. The DENLS model incorporates third- and fifth-order dispersion terms, extending beyond the standard nonlinear Schrödinger equation and rendering it non–integrable. This model delineates the mathematical and physical characteristics of nonlinear dispersive waves, with applications spanning optics and plasma physics. The derived solutions significantly advance our understanding of high-order nonlinear wave behaviors in such systems. Multiple solution types, including periodic, rational, and hyperbolic solitary waveforms, were obtained employing the Khater II method and generalized rational methods. Notably, the periodic solution unveiled the emergence of secondary peaks, highlighting the profound impact of higher-order dispersion on the envelope structure. Additionally, the localized rational and hyperbolic solutions depicted robust nonlinear excitation. Validation of the solutions was conducted through consistency checks, stability analyses, examination of conserved quantities such as momentum, and comparisons with numerical simulations. Both the Khater II and generalized rational methodologies proved effective in deriving closed-form representations of waves governed by the non–integrable high-order DENLS model. The analytical wave–forms presented herein contribute to an enriched mathematical depiction of nonlinear dispersive phenomena, facilitating quantitative parameter extraction. This study enhances our understanding of complex nonlinear wave propagation in domains characterized by higher–order chromatic effects, such as ultrafast fiber optics, plasma physics, and Bose–Einstein condensates. The formulations introduced in this study advance theoretical tools with broad applicability in the realm of nonlinear sciences. [ABSTRACT FROM AUTHOR]

This study delves into the perturbed Radhakrishnan–Kundu–Lakshmanan ( (p RKL) ) model, a pivotal component within nonlinear optics and communication engineering. In addressing this challenge, we employed the Bernoulli sub-equation function method and the Khater II method as analytical approaches, complemented by numerical solutions authenticated through the exponential cubic B–spline (ECBS) method. Our investigation yielded innovative solitary wave solutions, depicted elegantly through three-dimensional, two-dimensional, and contour plots. To ascertain the precision and dependability of these outcomes, we conducted a comparative analysis between analytical and numerical findings, visually presenting them via two-dimensional graphs. The implications of our discoveries are significant, offering insights into perturbations of optical solitons within nonlinear optical fibers. Our research effectively concludes that the proposed methodologies successfully solve the p RKL model, yielding highly accurate solutions. Notably, our study introduces novelty to nonlinear optics by applying the Bernoulli sub-equation function (BSE) method, the Khater II (Kh II) method, and the exponential cubic B-spline (ECBS) method to the p RKL model, with a specific focus on solitary wave solutions. [ABSTRACT FROM AUTHOR]

This study investigates the modified nonlinear time fractional Harry Dym equation, incorporating the conformable fractional derivative. Functioning as a mathematical framework for examining nonlinear phenomena in shallow water waves, particularly solitons, this model elucidates the intricate effects of dispersion and nonlinear steepening on wave dynamics. Employing a blend of analytical and numerical methodologies, the research aims to decipher the physical implications of the equation and its interconnectedness with other nonlinear evolution equations. The model delineates the evolution of a nonlinear wave in 1 + 1 dimensions (one spatial dimension x and time t ). The proposed methodology encompasses the G ′ G , 1 G expansion method, an analytical technique, alongside three numerical schemes utilizing B-spline methods. These methodologies facilitate the exploration of the equation’s behavior and enable precise computations of its solutions. The principal findings underscore the effective application of the proposed methodologies in resolving the modified nonlinear time fractional Harry Dym equation, furnishing valuable insights into its dynamics and significantly contributing to its physical interpretation. The significance of these discoveries lies in their contribution to the broader comprehension of nonlinear evolution equations and their pertinence across various scientific and engineering domains. This study provides novel insights into the modified nonlinear time fractional Harry Dym equation through a combined analytical and numerical approach. It advances the field of nonlinear dynamics and carries implications for analyzing analogous nonlinear evolution equations. These findings deepen our understanding of the equation’s physical interpretation and lay the groundwork for future explorations in related domains. [ABSTRACT FROM AUTHOR]

In our investigation, we've utilized an advanced computational technique, the modified exp - ϕ (ζ) -expansion method, to derive precise solitary wave solutions within the (2+1)–dimensional Ablowitz–Kaup–Newell–Segur (AKNS ) equation. These solutions, categorized by peak amplitudes, wave speeds, and widths, underscore the efficacy and reliability of our approach compared to conventional numerical methods. The significance of our study extends to practical applications of the (2+1)–dimensional AKNS equation in modeling various physical phenomena, from nonlinear optics to water waves and plasma physics. Our research, by precisely determining solitary wave solutions, not only deepens our comprehension of these phenomena but also establishes a robust basis for practical applications. [ABSTRACT FROM AUTHOR]

This study explores analytical and approximate explicit solutions for the generalized Hietarinta-type equation, denoted as ( G H ), in the context of (2+1) dimensions. This investigation utilizes contemporary computational and numerical techniques, including the Khater II technique, He’s variational iteration method, and the septic–B–spline scheme. The G H equation serves as a mathematical representation of wave propagation on water surfaces, accounting for both gravity and surface tension effects. In fluid dynamics, different wavelengths of waves are associated with distinct phase velocities and frequencies. This research introduces innovative solitary wave solutions, visually depicted through figures, and rigorously assesses their computational accuracy using state-of-the-art numerical methods. The stability of these solutions is scrutinized through the characteristics of the Hamiltonian system. The solutions are reintroduced into the original G H model using Mathematica 13.1 software to validate the obtained results. This study holds significant importance in mathematical modeling and fluid dynamics because it explores the G H equation in (2+1) dimensions. The G H equation is a vital tool for modeling wave propagation on water surfaces, encompassing the interplay of gravity and surface tension, a phenomenon with broad practical applications. Additionally, the study contributes to the field by conducting a stability analysis of these solutions, providing insights into their dynamic behavior and long-term evolution. [ABSTRACT FROM AUTHOR]

This study investigates the integrableMikhailov-Novikov-Wang (MNW) equation in (1+1)- dimensional space-time, utilizing rigorous analytical methodologies including the Unified (UF) and Khater II (Khat.II) methods, alongside numerical solutions. Situated within the established mathematical framework of physics, the MNW equation holds significant relevance in elucidating various physical phenomena, encompassing solitons, nonlinear wave propagation, quantum mechanics, and field theories, contingent upon the specific context and parameters selected for analysis. The primary objective of this study is to conduct a comprehensive analysis of the MNW equation and propose effective resolution techniques. This is achieved through the amalgamation of diverse analytical and numerical methodologies, facilitating an in-depth exploration of the equation's characteristics. Noteworthy findings indicate that the UF, Khat.II methods, in conjunction with He's variational iteration method, yield robust and highly accurate solutions. The significance of these results lies in their potential to address complex nonlinear equations, providing researchers and practitioners with versatile tools for analysis and interpretation. This research contributes a novel perspective by skillfully integrating these analytical methodologies, thereby advancing the field of mathematical physics and nonlinear differential equations. It is imperative to note that this study is purely theoretical in nature and does not involve specific subjects or participants. The approach adopted is grounded in meticulous numerical experimentation and analytical scrutiny, ensuring rigor and reliability in the findings presented. [ABSTRACT FROM AUTHOR]

This research provides the first known solution to the nonlinear Hamiltonian amplitude equation using the combined Bernoulli sub-equation function and trigonometric-quintic-B-spline methods. While previous works have analyzed this model through limited linear approximations, an exact solution accounting for nonlinearity remains lacking. We address this gap by employing innovative techniques to reveal and scrutinize solitary wave solutions, elucidating complex behaviors inaccessible through linear methods. Our findings show the unequivocal effectiveness of our approach, comprising the first successful solution to this equation. This breakthrough fundamentally advances the comprehension of nonlinear physical systems across domains encompassing wave propagation, quantum mechanics, and optical phenomena. [ABSTRACT FROM AUTHOR]

The primary objective of this investigation revolves around scrutinizing the generalized Kawahara equation, a consequential nonlinear evolution equation prevalent in diverse physical domains such as shallow water waves, ion-acoustic waves within plasmas, and nonlinear acoustics. This equation manifests intricate dynamical attributes, notably featuring the emergence of solitary waves, shock waves, and chaotic solutions. In our pursuit to elucidate the nuanced behavior inherent in the generalized Kawahara equation, we deploy two robust analytical methodologies: the Khater II method and the variational iteration method. These methodologies offer adept and precise avenues for approximating solutions to nonlinear partial differential equations. Through our analytical scrutiny, a diverse spectrum of solutions comes to light, encompassing solitary wave solutions, periodic solutions, and chaotic solutions. Our exploration delves into the impact wielded by diverse parameters on solution behaviors, thereby furnishing invaluable insights into the foundational physical mechanisms at play. The findings derived from our inquiry contribute significantly to comprehending the intricate dynamics of the generalized Kawahara equation, concurrently establishing connections with other nonlinear evolution equations. This study augments the existing reservoir of knowledge concerning nonlinear wave phenomena and sets the stage for further investigations into the convoluted behaviors exhibited by these intricate systems. [ABSTRACT FROM AUTHOR]

This research employs recent and precise computational techniques to identify new and accurate solitary wave solutions of the modified Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. The KdV-KP equation is a nonlinear partial differential equation that characterizes the evolution of waves in diverse physical systems, including nonlinear optics, fluid dynamics, and plasma physics. The equation generalizes the KdV and KP equations, which are fundamental models in their respective fields. Physically, the modified KdV-KP equation describes wave propagation where the effects of nonlinearity and dispersion can produce intriguing and complicated wave phenomena such as wave turbulence and solitary waves. The equation has numerous real-world applications, including modeling shallow water waves such as tsunamis and river waves in fluid dynamics, describing the behavior of plasma waves in fusion reactors and astrophysical plasmas, and studying the propagation of light in nonlinear media such as optical fibers in nonlinear optics. One of the most significant applications of the equation is in the study of solitons, which are self-sustaining solitary waves that preserve their shape and velocity even after colliding with other solitons. Solitons are used in various applications such as optical communications, where they transmit information over long distances without distortion, and in fluid dynamics, where they model long-lasting waves in the ocean. The study's importance lies in its impact on the utilization of the modified KdV-KP equation in diverse fields of physics, including fluid dynamics and plasma physics. The effectiveness of the proposed techniques is demonstrated by comparing them with other computational methods, indicating their superiority. [ABSTRACT FROM AUTHOR]

This study addresses the challenge of solving the nonlinear Gilson–Pickering G P equation, employing the unified method for analytical investigation and verifying the results through the He's variational iteration method. The primary aim is to offer a robust solution for this complex equation. The research methodology integrates the unified method for analytical purposes and validates the outcomes through numerical approaches. The study's significance lies in successfully resolving the G P equation, showcasing its applicability in various scientific contexts. This research's implications are substantial, as it introduces an effective solution method with broad interdisciplinary relevance. In conclusion, the study underscores the compatibility of the proposed unified method with numerical solutions, contributing a novel perspective to mathematical modeling. This work pertains to the field of mathematical sciences and is theoretical, involving no specific subjects or participants. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "On some novel bright, dark and optical solitons to the cubic-quintic nonlinear non-paraxial pulse propagation model" published in the journal Optical & Quantum Electronics. The authors apologize for a typesetting error in the equations in the original publication. The corrected equations are provided for four different cases, along with corresponding graphs. The document includes mathematical formulas and does not provide any additional information beyond the correction. [Extracted from the article]

This study explores the impact of the integrability criterion on nonlinear Schrödinger (NLS ) equations with mixed derivatives, specifically focusing on the Rangwala–Rao ( R R ) equation formulated by A. Rangwala in 1990. The objective of this investigation is to enhance our understanding of the dispersion effect by examining novel soliton wave solutions and their interactions. By doing so, we aim to gain insights into the behavior of the slowly changing envelope of the electric field and pulse propagation in optical fibers. To achieve this, we employ recent computational approaches, including the Sardar Sub-equation and modified rational methods, to identify distinctive solitary wave solutions for the analyzed model. The significance of studying the Rangwala–Rao equation lies in its potential contribution to the development of more efficient optical fiber communication systems. The presented numerical solutions in this paper showcase the dynamic nature of optical fiber pulse propagation, thereby highlighting the originality of this work compared to existing scholarly contributions. [ABSTRACT FROM AUTHOR]

This study introduces recent computational advancements targeting the discovery of precise and novel solitary wave solutions in the (1 + 1)-dimensional Mikhailov-Novikov-Wang (MNW ) integrable equation. Solitary wave solutions are of paramount importance for comprehending complex nonlinear phenomena and have practical implications in diverse scientific fields such as fluid dynamics, nonlinear optics, and plasma physics. The proposed computational techniques encompass powerful numerical methods, including the Khater II (KHA - II) method, modified Rational method (MRE ) method, and He’s variational iteration (HVI ) method. These techniques offer robust and accurate approaches for computing solitary wave solutions by leveraging the inherent integrability properties of the MNW equation. Notably, these approaches demonstrate substantial improvements in efficiency, accuracy, and the ability to explore a broader range of solitary wave solutions compared to traditional computational methods. The efficacy of these techniques is effectively demonstrated through extensive numerical simulations and comprehensive comparisons with other computational approaches. The results highlight the superior performance of the proposed methods in identifying new solitary wave solutions, precisely characterizing their properties, and providing deeper insights into the model’s behavior. Overall, these computational techniques significantly contribute to advancing our understanding of the underlying dynamics of the MNW equation and hold promising potential for applications across multiple scientific disciplines. [ABSTRACT FROM AUTHOR]

The generalized modified Equal–Width (GMEW) equation is often used to show how a one-dimensional wave moves through a medium that is not linear and has dispersion processes. In this article, we’ll use two very precise, cutting-edge analytical and numerical methods to find the exact traveling wave solutions for the model we’re looking at. These discoveries are really new, and they could immediately change how people train in engineering and physics. Now that a numerical approach has been described, we can roughly evaluate the replies’ accuracy. Analytical and quantitative data were shown using contour plots and two- and three-dimensional graphs. Our method of symbolic computing shows that it has the potential to be a powerful mathematical tool. It can be used to solve a wide range of nonlinear wave problems. Our findings are the outcome of our topic investigation. [ABSTRACT FROM AUTHOR]

The purpose of this study is to employ the Sine–Cosine expansion approach to produce some new sort of soliton solutions for the cubic–quintic nonlinear Helmholtz problem. The nonlinear complex model compensates for backward scattering effects that are overlooked in the more popular nonlinear Schrödinger equation. As a result, a number of novel traveling wave structures have been discovered. We also investigate the stability of solitary wave solutions for the governing model. Furthermore, the modulation instability is discussed by employing the standard linear-stability analysis. The 3D, contour and 2D graphs are visualized for several fascinating exact solutions to comprehend their behaviour. [ABSTRACT FROM AUTHOR]

In this study, the solitary wave solutions of the complex nonlinear perturbed Fokas–Lenells (pFL) equation by implementing the Khater II method. The nonlinear pFL model's soliton wave solutions describe the propagation of light pulses in the femtosecond or sub–picosecond range. These waves are used in recent electronic communications such as Twitter comments, Facebook communication, and Internet blogs. The model perturbation terms are also used to describe the soliton cooling of the observed phenomena. In contrast, the stochastic perturbation terms analyze the related Langevin equations to get the mean free velocity of the soliton. The accuracy of the obtained computational solutions is checked by employing the Adomian decomposition method. The quasi-monochromatic soliton wave solutions are represented in some distinct graphs. The implemented schemes' performance shows their powerful, effective, direct ability to handle many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

Using the extended truncated expansion approach and three precise B-spline numerical methods, this paper examines the structure of the analytical and numerical solutions to the combined mKdV equation and KdV equation. This model represents the propagation of weakly nonlinear long waves in a KdV-typed medium by changing the coefficients of dispersion and nonlinear coefficients. Additionally, it also describes the flow below a pressure surface in a fluid. The numerical schemes of ExCBS, SBS, and TQBS are used. The handled model may describe many different phenomena, including the propagation of a thermal pulse through a single sodium fluoride crystal and the motion of confined particles under a harmonic force in a one-dimensional nonlinear lattice. The link between fast and slow soliton, which causes a phase shift, is shown in several instances. The energy density along the route of rapid and slow colliding solitons is shown to be very small and relatively large, respectively, thanks to the phase shift shown by the contour map. If the analytical and numerical solutions vary by the same amount, then the spline-connected and distribution graphs will be consistent. [ABSTRACT FROM AUTHOR]

This study gives more attention to soliton wave solutions of the nanoparticle hybrid system through two recent (generalized rational (GRat) and Kudryashov (GKud)) schemes. Because of the hybrid model's nanoscale size, the nanoparticles in these new solitonic liquid wave structures exhibit unique electrical and optical characteristics crucial to several applications. The constructed results are numerically simulated in polar, contour, two, and three-dimensional graphs to demonstrate the dynamic behavior of an inviscid, incompressible, and non-rotating flow of fluid of constant depth h. Moreover, the obtained solutions' accuracy is also checked by implementing two precise numerical (quintic-B-spline (QBS) and septic-B-spline (SBS)) schemes. The matching between computational and approximate solutions is explained through some distinct graphs in two-dimensional and distribution graphs. The novelty of our results is also elucidated by comparing our results with those constructed previously. [ABSTRACT FROM AUTHOR]

In this work, we study the exact traveling wave solutions of (2 + 1) -dimensional Chiral nonlinear Schrödinger equation with the aid of generalized auxiliary equation method. The aforementioned model is used as a governing equation to discuss the wave in quantum field theory. The suggested technique is direct, effective, powerful, and offers constraint conditions to ensure the existence of solutions. The solutions obtained are bright solitons, dark solitons, singular solitons, mixed solitons, periodic waves, exponential, rational, and complex solutions that are relevant in various applications of applied science. Finally, some solutions are depicted in two and three dimensional to better understand the behavior of the considered model. [ABSTRACT FROM AUTHOR]

In this article, the modified Khater method is utilized to extract new analytical solutions to the cubic-quintic nonlinear non-paraxial pulse propagation model in a planar waveguide with Kerr nonlinearity, fifth-order nonlinearity, and spatiality dispersion due to non-paraxial effects. As a result, a variety of some new exact solutions are observed. Moreover, the three-dimensional shapes are visualized with the aid of the latest scientific tools. [ABSTRACT FROM AUTHOR]

The accuracy of computational solutions for the ionic current obtained using the microtubule (MTU) equation is analyzed. The MTU equation has various applications for the ionic current in biological non-linear dispatch line and can be used to describe the ionic transport throughout the intracellular environment. The extended simple equation (ESE) and generalized exp - ϕ (ξ) expansion (GEE) schemes result in many distinct solutions. The aim of this work is to investigate the accuracy of these solutions and thereby the applied computational schemes. The process depends on extracting the initial and boundary conditions from the obtained analytical solutions followed by the application of the well-known Adomian decomposition (AD) and homotopy perturbation (HP) methods. Microtubules are one of the main components of the cytoskeleton, carrying out many operations including intracellular transmit, and DNA division, intracellular transmission, and DNA division, thereby motivating their study using the described model. The results demonstrate that the proposed methods are very powerful, innovative, facile, suitable, and convenient for solving many such nonlinear models. [ABSTRACT FROM AUTHOR]

In this survey, the ionic currents along the microtubules equation handled by applying two different techniques. It describes the ionic transport throughout the intracellular environment. This phenomenon explains the behavior of many applications in a biological nonlinear dispatch line for ionic currents. Using two different analytical techniques on this equation allows us to get new forms of analytical solitary traveling wave solutions. The obtained solutions support many researchers concerned with the discussion of the physical properties of the ionic currents along microtubules. Microtubules are one of the main components of the cytoskeleton, and function in many operations, comprehensive constitutional backing, intracellular transmit, and DNA division. The thing which makes us applies this property in many applications for this model. The earned results demonstrate that the proposed methods are significantly powerful, evangelist, relaxing, suitable, and convenient for solving many nonlinear models. [ABSTRACT FROM AUTHOR]

This research paper is connected with one of the most popular models in shallow water wave applications. The computational wave solutions of the Benjamin-Bona-Mahony-Peregrine (BBMP) equation with power-law nonlinearity are investigated via three new analytical schemes. This model is considered a natural extension of the well-known mathematical model called the Korteweg–de Vries (KdV) equation. The KdV has been used to study the shallow water wave behavior of long wavelengths and small amplitude. In contrast, the BBMP equation has been employed for the same purpose, but it concentrates on studying this kind of wave, especially in ocean beaches. Applying the modified auxiliary equation method (MAE) method, the direct algebraic (DA) method, and generalized Kudryashov (GK) method to the BBMP equation after harnessing suitable wave transformation gives plentiful unprecedented solitary wave solutions. Many novel solutions have been constructed through the suggested methods that have been built along with their 2D, 3D, and contour graphical configurations for clarity and exactitude. [ABSTRACT FROM AUTHOR]

This study performs two recent methods on the nonlinear long-short wave interaction system (NLSWIS) to obtain novel formulas of solitary solutions representing an optical field that does not alter by multiplication a sensitive balance between the different terms impacts in the medium. Elastic medium is defined as the medium that can adjust in the figure due to a deforming strength and come back shape to its original form when the force eliminates. The wave is produced by vibrations that are a consequence of acoustic power. Also known as the sound wave, or acoustic wave. For examples of elastic mediums are air, water and so on. The improved G ′ G expansion method and novel G ′ G expansion method are implemented, which are considered as two general methods in this field. Applying these methods to many nonlinear partial differential equations gives diverse and different forms and configuration solutions that allow users of those two methods a large number of solutions that provide broader scope to use. [ABSTRACT FROM AUTHOR]

In this manuscript, the computational solutions of the nonlinear Klein–Gordon–Zakharov (KGZ) model are scrutinized through a new generalized analytical scheme. This mathematical model describes the evolution of strong Langmuir turbulence in plasma physics. Many distinctive solutions are obtained, such as linear, rational, hyperbolic, parabolic, and so on. 2D, 3D, contour, stream plots are plotted to demonstrate further physical and dynamical attitudes of the investigated model. The power, usefulness, and accuracy of the compensated schemes are revealed and tested. The capabilities for managing a class of nonlinear evolution equations of the new generalized method is assessed. Moreover, the stability property of the obtained solutions is checked by using the Hamiltonian system's characteristics. [ABSTRACT FROM AUTHOR]

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches. [ABSTRACT FROM AUTHOR]

This paper aims to determine some novel solitary wave solutions of the Chaffee–Infante equation, which have not yet been presented for this equation. This equation arises in various branches of science and technology, such as plasma physics, coastal engineering, fluid dynamics, signal processing through optical fibers, ion-acoustic waves in plasma, the sound waves, and the electromagnetic waves field. The (2 + 1) -dimensional Chaffee–Infante equation describes the dynamical behavior of gas diffusion in a homogeneous medium. Soliton solutions are obtained for this equation using several computational schemes. Many physical significances are explained by sketching some two-dimensional and three-dimensional diagrams for the acquired solutions in three different types. These figures give us a better understanding of the behavior of these solutions. Moreover, the stability property is investigated based on the Hamiltonian system's characterizations. The methods provide efficient way for the solving other equations that occur in other branches of science. [ABSTRACT FROM AUTHOR]

This research paper illustrates a novel structure of the optical soliton wave solutions of nonlinear Schrödinger (NLS) equation with the higher-order (a model for the sub-10-fs-pulse propagation) by using the modified Khater method. Moreover, this research is studying the stability properties of obtained solutions to show their ability to apply in the model's applications. The NLS equation with the higher-order describes the quantum aspects of a quantum-mechanical system. Some sketches are plotted to show more physical properties of the real, imaginary, and absolute obtained solutions. All obtained solutions are verified of its accuracy by putting them back into the original equation. [ABSTRACT FROM AUTHOR]

We use an analytical scheme to construct distinct novel solutions of two well-known fractional complex models (the fractional Korteweg–de Vries equation (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation). A new fractional definition is used to covert the fractional formula of these equations into integer-order ordinary differential equations. We obtain solitons, rational functions, the trigonometric functions, the hyperbolic functions, and many other explicit wave solutions. We illustrate physical explanations of these solutions by different types of sketches. [ABSTRACT FROM AUTHOR]

Novel explicit wave solutions are constructed for the Kudryashov–Sinelshchikov (KS) equation through liquid–gas bubbles mix under the thermodynamic conditions. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types of this model that is used to describe the phenomena of pressure waves through liquid–gas bubbles mix under the thermodynamic conditions. The stability property of the obtained solutions is tested to show the ability of our obtained solutions through the physical experiments. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical behavior of this model. [ABSTRACT FROM AUTHOR]

This research investigates the dynamical behavior of the transmission of nerve impulses of a nervous system (the neuron) by studying the computational solutions of the FitzHugh–Nagumo equation that is used as a model of the transmission of nerve impulses. For achieving our goal, we employ two recent computational schemes (the extended simplest equation method and Sinh–Cosh expansion method) to evaluate some novel computational solutions of these models. Moreover, we study the stability property of the obtained solutions to show the applicability of them in life. For more explanation of this transmission, some sketches are given for the analytical obtained solutions. A comparison between our results and that obtained in previous work is also represented and discussed in detail to show the novelty for our solutions. The performance of the two used methods shows power, practical and their ability to apply to other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe–Cr–X (X = M o , C u ) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software. [ABSTRACT FROM AUTHOR]

This research uses the extended exp- (− φ (ϑ)) -expansion and the Jacobi elliptical function methods to obtain a fashionable explicit format for solutions to the fragmented biological population and the same width models that depict popular logistics because of deaths or births. In mathematical terminology, the linear, hyperbolic, and trigonometric equation solutions that have been found describe several innovative aspects from the two models. Sketching these solutions in different types is used to give them more details. The accuracy and performance of the method adopted show their ability to be applied to various nonlinear developmental equations. [ABSTRACT FROM AUTHOR]