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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 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 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]

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 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]

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 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]

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]

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]

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 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 work, we investigate unprecedented optical closed form of solutions for the generalized higher-order nonlinear Schrödinger equation. This vital model in the optical fiber. Schrödinger equation describes the promulgation of the light-wave in an optical fiber. To achieve the desired objective of the search we apply a generalized Kudryashov method. This method is considered as one of the recent methods that developed in last decades. In this discussion, we discuss what differentiates this method from other methods in the field of finding precise solutions to non-linear partial differential equations and that debate takes place using two axes. The first is comparing the same way with the other ways ahead of this area while the second axis is comparing the solutions obtained using this method with solutions found using other methods and how these solutions converge and how much able to be utilized in that modern method (Generalized Kudryashov method). [ABSTRACT FROM AUTHOR]

In this research, we examine exact and solitary traveling solutions of nonlinear complex Kundu-Eckhaus equation with variable coefficients. Nonlinear complex Kundu-Eckhaus equation model has vital role in description of the proliferation of the pulses of the ultra-short femto second in an optical fiber, weakly nonlinear fragmented water waves, nonlinear optics and the quantum field theory. We implement a new auxiliary equation method and novel G′G-expansion technique on this model to obtain a new exact and solitary traveling wave solutions. The propose of this research is applying new method on this models to discover new form of soliton solutions that this makes the study of the electric and magnetic field of these pulses and the extent of their effect in the optical fiber more interesting. [ABSTRACT FROM AUTHOR]