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Higher-order nonlinear partial differential equations, such as the eighth-order Kac-Wakimoto model, are useful for studying wave turbulence in fluids, where energy transfers across a range of wave numbers. This phenomenon is observed in oceanographic research involving sea surface and internal waves, where intricate multi-dimensional interactions play a crucial role. In this work, we use the improved modified extended tanh function method for the first time to extract the exact solutions of the eighth-order (3+1)-dimensional Kac-Wakimoto equation, which describes the dynamics of fields and the structure of solutions in various physical and mathematical contexts. The proposed method is simple and quick to execute, and it offers more innovative solutions than other methods. As a consequence, through the donation of suitable assumptions for the parameters, some new solutions for dark and singular soliton, as well as Jacobi elliptic, exponential, hyperbolic, and singular periodic forms, are developed. Furthermore, to enhance understanding, graphical representations of certain solutions are included.

In this study, we introduce the new (3+1)-dimensional $ \beta $-fractional Boussinseq-Kadomtsev-Petviashvili (KP) equation that describes the wave propagation in fluid dynamics and other physical contexts. By using the modified extended direct algebraic method, we investigate diverse wave solutions for the proposed fractional model. The acquired solutions, include (dark, bright) soliton, hyperbolic, rational, exponential, Jacobi elliptic function, and Weierstrass elliptic doubly periodic solutions. The primary objective is to investigate the influence of fractional derivatives on the characteristics and dynamics of wave solutions. Graphical illustrations are presented to demonstrate the distinct changes in the amplitude, shape, and propagation patterns of the soliton solutions as the fractional derivative parameters are varied.

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated.

This study is focusing on the integrable (3+1)-dimensional equation that combines the potential Kadomtsev-Petviashvili (pKP) equation with B-type Kadomtsev-Petviashvili (BKP) equation, also known as the pKP-BKP equation. The idea of combining integrable equations has the potential to produce a variety of unexpected outcomes such as resonance of solitons. This article provides a wide range of alternative exact solutions for the pKP-BKP equation in three dimensional form, including dark solitons, singular solitons, singular periodic solutions, Jacobi elliptic function (JEF) solutions, rational solutions and exponential solution. The improved modified extended (IME) tanh function method is employed to investigate these solutions. All of the obtained solutions for the investigated model are presented using the Wolfram Mathematica program. To further help in understanding the solutions' physical characteristics and dynamic structure, the article provides visual representations of some derived solutions using 2D representation in addition to the 3D graphs via symbolic computation. This article aims to use a potent strategy using a powerful scheme to derive different solutions with various structures. Additionally, the results greatly improve and enhance the literature's solutions to a combined pKP-BKP equation and allow deep understanding of the nonlinear dynamic system through different exact solutions.

Abstract One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.

We have introduced various novel soliton waves and other analytic wave solutions for nonlinear Schrödinger equation with cubic, quintic, septic, and nonic nonlinearities. The modified extended direct algebraic method governs the transmission of various solitons with different effects. The combination of this system enables the obtaining of analytical soliton solutions with some unique behaviors, including bright, dark, and mixed dark-bright soliton solutions; singular soliton solutions; singular periodic, exponential, rational wave solutions; and Jacobi elliptic function solutions. These results realize the stability of the nonlinear waves' propagation in a high-nonlinear-dispersion medium that is illustrated using 2D and 3D visuals and contour graphical diagrams of the output solutions. This research focused on determining exact soliton solutions under certain parameter conditions and evaluating the stability and reliability of the soliton solutions based on the used modified extended direct algebraic method. This will be useful for many various domains in technology and physics, such as biology, optics, and plasma physical science. At the end, we use modulation instability analysis to assess the stability of the wave solutions obtained.

We investigated the dynamics of highly dispersive nonlinear gap solitons in optical fibers with dispersive reflectivity, utilizing a conformable fractional derivative model. The modified extended direct algebraic method was employed to obtain various soliton solutions, including bright solitons and singular solitons, as well as hyperbolic and trigonometric solutions. The key findings demonstrated that the fractional derivative parameter (α) can effectively control the wave propagation, causing a shift in the wave signal while maintaining the same amplitude. This is a novel contribution, as the ability to control soliton properties through the conformable derivative is explored for the first time in this work. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. This research provides insights into the potential applications of fractional calculus in the design and optimization of photonic devices for optical communication systems.

Abstract As delays are common, persistent, and ingrained in daily life, it is imperative to take them into account. In this work, we explore the averaging principle for impulsive Atangana–Baleanu fractional stochastic delay differential equations driven by Lévy noise. The link between the averaged equation solutions and the equivalent solutions of the original equations is shown in the sense of mean square. To achieve the intended outcomes, fractional calculus, semigroup properties, and stochastic analysis theory are used. We also provide an example to demonstrate the practicality and relevance of our research.

This study tackles a nonlinear Schrödinger’s equation in three dimensions, using three dispersive components of fourth order, often resulting in pure-quartic solitons. Pure-quartic bullets are known to deviate from conventional solitons due to their balance of fourth-order dispersion and nonlinearity. As bright and dark optically modulated bullets, we derive many solutions. Experiments on bullet transmission using optical nanofibers can benefit from the solutions found. Our creative solutions are produced by implementing the well-known scheme which is the improved modified extended tanh-function scheme. We find novel kinds of solutions (dark, bright, singular solitons, exponential, singular periodic, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic functions) that make their originality for the problem at hand evident by using the previously described approach. Contour plots and 2D and 3D visualizations in Wolfram Mathematica software are used to show how the well-furnished results propagate for various values of the necessary free parameters. The results demonstrate the computational processes’ precise, well-informed, and efficient nature. Through their integration with representational computations, they may be applied to increasingly complex phenomena. This work constitutes a major advancement in our comprehension of the intricate and erratic behavior of this mathematical model.

An extended version of (3+1)-dimensional non-linear Schrödinger equation that has a cubic–quintic nonlinear component under the stochastic effects is examined in this investigation. Several stochastic exact solutions of this model is acquired through the application of the improved modified extended tanh-function scheme (IMETFS). This method offers a practical and effective approach to finding precise solutions to several kinds of nonlinear partial differential equations. In addition, these solutions include stochastic soliton solutions (bright, singular, combo dark-singular), and exact solution such as singular periodic, Jacobi elliptic function, Weierstrass elliptic doubly periodic solution, rational, and exponential functions. Since it is the first study of its sort to examine multiplicative white noise’s impacts in this particular setting, it offers fresh insights and innovative research approaches for the field’s future studies. The work adds much to our understanding of soliton theory and how it relates to optical fiber technology while illuminating hitherto unknown facets of multiplicative white noise. To illustrate the impact of the noise, a few recovered solutions with varying noise strengths are given graphically as examples.

Stochastic optical solitons are a fascinating phenomenon in nonlinear optics where soliton-like behavior emerges in systems affected by stochastic noise. This study investigates the influence of Brownian motion on wave propagation in optical fibers. The propagation is modeled using a stochastic nonlinear Schrödinger equation incorporating quintuple power-law nonlinearity and nonlinear chromatic dispersion. To explore this, the improved modified extended tanh (IMET) scheme, leveraging the extended Riccati equation, is employed. This technique facilitates the extraction of various stochastic solutions, including bright, dark, and singular solitons. Furthermore, solutions in the shapes of exponential, singular periodic, and Weierstrass elliptic forms are investigated. The study looks at how the strength of noise impacts various solutions, and Matlab software is used to create 2D and 3D graphs that show the results. It has been noted that when noise intensity rises, signal level falls and surface flattens.

In the domains of fluid mechanics, hydrodynamics, and marine engineering, Bogoyavlensky–Konopelchenko equations are of great interest to mathematicians and physicists as a means of illuminating the diverse dynamics of non-linear wave events. In this study, to pique readers' interest, we investigate the soliton solutions of a dynamical model, which is the mathematical physics equivalent of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation (GBKE). Utilizing the improved modified extended tanh-function scheme (IMETFS), we generate several innovative solutions. Utilizing the previously described approach, we find new types of solutions that have never been found before to demonstrate their originality for the problem at hand, such as dark, singular soliton, exponential, hyperbolic, singular periodic, Jacobi elliptic function (JEF), and rational solutions. The results show that the computational procedures are clear, informed, and effective. By integrating them with representational calculations, they may be used for more intricate phenomena. The efficacy of our method indicates that it may be utilized to tackle other non-linear challenges in many domains, particularly in soliton theory, since the examined model appears in many applications. Utilizing the computer algebra system, Wolfram Mathematica®, the propagation of the well-furnished results is visualized through contour plots, 2D and 3D visualizations for different values of the required free parameters. All of the research's conclusions are necessary to comprehend the behavior and physical significance of the examined equation, highlighting how crucial it is to examine various non-linear wave phenomena in the field of engineering mathematics and physical sciences.

The rise in everyday internet users worldwide can be ascribed to advancements made to the internet worldwide. Attacks by malware cause data loss, hardware destruction, and large financial losses. As a result, this research creates a novel mathematical compartmental dynamical model that, under both fixed and dynamic size network assumptions, can be utilized to precisely predict malware outbreaks. The modeling processes divided the network into seven distinct states and considered the diverse ways that users of the internet interacted with potentially hazardous links. The Hilfer-Katugampola fractional operator was employed to obtain the intended varied states of networks. The existence, uniqueness, equilibrium, and stability of the provided models are thoroughly examined. Then the best control plan is implemented for every network topology. The controllers' objectives are to minimize the costs of data loss due to infections, malware tracing, and public awareness improvements. Ultimately, we confirm the theoretical results by observing the spread of malware using numerical simulations. Results demonstrated the effectiveness of the enforced control measures in maintaining the necessary goals of malware infection management.

Sufficient conditions for controllability of impulsive nonlinear integro-differential equations with ψ-Hilfer fractional derivative are established. The result are obtained by using fractional calculus and Schaefer’s fixed point theorem. Finally, a numerical examples are then provided to demonstrate the outcomes.

This study examines the analytic wave solutions of a fourth-order (2+1)-dimensional nonlinear Schrödinger equation with higher-order odd and even terms using the modified Sardar sub-equation method. The modified Sardar sub-equation method has simple calculations, high accuracy, low computational effort and provides variety of solution forms. The results show a wide range of solutions including bright, dark and singular solitons as well as hyperbolic, periodic, singular periodic, and exponential solutions. Assigning suitable values to free parameters allows for the graphic representation of the dynamical wave structures of some analytical solutions using two- and three-dimensional graphics.

This manuscript explores a new class of Hilfer fractional stochastic differential system, as driven by the Wiener process and Rosenblatt process through the application of non-instantaneous impulsive effects and Poisson jumps. Existence of a mild solution to the considered system is proved. Sufficient conditions for the controllability of the proposed control system are established. To prove our main results, we utilize fractional calculus, stochastic analysis, semigroup theory, and the Sadovskii fixed point theorem. In addition, to illustrate the theoretical findings, we present an example.

In this paper, we investigate soliton solutions and other exact solutions of the highly dispersive perturbed nonlinear Schrödinger equation having Kudryashov's arbitrary form with sextic-power law of refractive index and generalized non-local laws. Studying is conducted by applying the modified extended direct algebraic method, which offers several types of solutions. The extracted solutions including (combo bright-dark, singular, dark, bright) solitons, exponential solutions, rational solutions and singular periodic solutions. The extracted solutions confirmed the effectiveness and strength of the current technology. To illustrate the properties of some solutions, graphic representations of those solutions are given. This study contributes to the understanding of nonlinear wave phenomena and showcases the applicability of the modified extended direct algebraic method in obtaining exact solutions for complex nonlinear equations. The optical solitons produced in respect to this form have never been explored by the proposed technique before, and the results have never been published.

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of stochastic Hilfer-Katugampola fractional differential inclusions (SHKFDIs) with non-instantaneous impulsive (NIIs) that is strengthened by Brownian motion (BM) and the Clarke sub-differential. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using stochastic analysis, the Clarke sub-differential's characteristics, and the multi-valued fixed point theorem. Subsequently, by employing Balder's theorem, the existence of optimal control pairs for the considered system is investigated. Eventually, an example is provided to validate the obtained results.

Because it is used in communication and information technology, the propagation of pulses in optical fibers has become a significant research subject in recent years. Mathematical models have been created to describe the behaviors of optical solitons in optical fibres, such as the Kaup-Newell equation, the nonlinear Schrödinger equation with various types of nonlinearity, and others. A model among these is the newly generalized nonlinear Schrödinger equation with triple refractive index and non-local nonlinearity. To secure optical solitons for a particular case of this novel model, we used the improved modified extended tanh function approach. Different solutions have been acquired such as bright solitons, dark solitons, singular solitons, singular periodic solutions, trigonometric solutions, and hyperbolic solutions. Some of the discovered solutions are shown in 2D and 3D graphs for better comprehension and demonstration.

In this paper, we consider the nonlinear Schrödinger equation (NLSE) with nonlocal nonlinearity having nonlinear chromatic dispersion and Kudryashov’s generalized quintuple-power. For the suggested model, dark solitons, and singular solitons are investigated using the modified extended mapping approach. Furthermore, rational solutions, exponential solutions, hyperbolic wave solutions, singular periodic solutions, and Jacobi elliptic function solutions are shown. We impose certain limits on the parameters in order to guarantee the existence of the obtained soliton solutions. Furthermore, a few chosen solutions are also displayed graphically to illustrate the physical characteristics of the precise solutions. Furthermore, we use the idea of modulation instability analysis to discuss the stability of the derived wave solutions.

Global Navigation Satellite System (GNSS) provides ellipsoidal heights, whereas engineering projects need orthometric heights of points. Thus, to take advantage of GNSS in engineering work, the relation between orthometric and ellipsoid height must be determined as a function of position by geoid modeling. The Red Sea coastal cities have various natural potentials that render them economically promising cities that are attractive to numerous investors and visitors alike. Over the past years, there has been an increased governmental awareness of setting up large projects in these cities. With the increasing utilization of GNSS in a number of these projects, there is an imperative need to develop a local geoid model in this region. This study is an attempt to use a geometric method for the study area in order to develop a local geoid model and a local digital elevation model. In addition, the performance of EGM2008, SRTM, and ASTER is evaluated. In order to develop the local geoid model, the polynomial regression method and Artificial Neural network (ANN) technique are employed. Performance evaluation of the studied methods is based on a compilation of various statistical parameters and goodness-of-fit measures, followed by a comparison of methods outputs. The study indicates significant improvements in the local geoid model when using both methods. Additionally, the model resulting from the ANN appears more reliable compared to the calculated model using polynomial regression. In this study, the accuracy of EGM2008 is about ±0.466 m. Moreover, regarding the global digital elevation model, ASTER outperforms SRTM by about ±0.905 m.

In this paper, we explore the approximative controllability of fractional stochastic differential inclusions (SDIs) of Sobolev-type with fractional derivatives in Atangana-Baleanu (AB) sense and Poisson jumps. Our findings are supported by the fixed point theorem, multi-valued map theory, compact semigroup theory and stochastic analysis principles. In the later part, an illustration is provided to clarify the established outcomes.

In this work, we consider a concatenation model that is a concatenated version of the familiar nonlinear Schrödinger’s equation, Lakshmanan–Porsezian–Daniel equation and the Sasa–Satsuma equation. With the help of the improved modified extended tanh-function method (IMETFM), new families of solutions are extracted for the proposed model. Bright solitons, dark solitons, combo bright-dark solitons, singular solitons, periodic solutions, exponential solutions, rational solutions and Jacobi elliptic solutions are obtained. The IMETFM enables to recover a wide spectrum of solutions to the governing model. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature.

In this paper, the modified extended direct algebraic technique is used for determining new and various traveling wave solutions to the coupled nonlinear Schrödinger equation (NLSE) with Kudryashov's law of nonlinearity. This model describes the propagation of optical solitons through magneto-optic wave-guides. Various solutions such as dark solitons, singular solitons, hyperbolic solutions, singular periodic solutions, rational solutions, Jacobi elliptic solutions and Weierstrass elliptic function solution are raised. To demonstrate the physical behavior of the obtained solutions, 3D and 2D graphs are introduced.

In this paper, the improved modified extended Tanh function technique is used to find analytical solutions to the cubic-quartic Lakshmanan-Porsezian-Daniel model which describes the pulse propagation in optical fiber. Using the traveling wave theory, the nonlinear partial differential form of the cubic-quartic Lakshmanan-Porsezian-Daniel equation is transformed into an ordinary differential equation. Then applying the proposed methods bright, dark, bright-dark, periodic, singular, and Jacobian elliptic solutions were derived as analytical solutions. Besides, hyperbolic, rational, trigonometric, and exponential functions were also obtained. In order to understand the obtained solutions, graphical simulations were depicted by 3D surfaces and 2D graphs.

In this work, we investigate the fourth order nonlinear Schrödinger's equation with cubic–quintic–septic–nonic nonlinearity that is frequently encountered in modern physics study. Additionally, it is an important equation for researching problems with light pulse transmission in nonlinear dispersive optical medium. Studying is conducted by applying the improved modified extended tanh-function scheme. Various types of solutions, such as bright solitons, hyperbolic solutions, singular solitons and dark solitons solutions are extracted for the investigated model. The extracted solutions confirmed the efficacy and strength of the current technique. To illustrate the physical characteristics of the established solutions, many selected solutions are visually depicted.

The higher order nonlinear Schrödinger equation (NLSE) describes ultrashort pulse propagation in optical fibres. In this paper, we discuss analytically the non-linear schrödinger equation with higher order dispersion terms up to sixth order and sextic power law of nonlinearity with the help of the improved modified extended tanh-function scheme. New families of solutions, such as dark solitons, singular solitons, rational solutions, periodic solutions and exponential type solutions, have been extracted. Extracted solutions will be useful in the development of the communication systems. Furthermore, some solutions are graphically depicted to demonstrate their physical nature by using distinct values of the parameters.

Optical soliton solutions in birefringent fibers and other analytic solutions for Sasa-Satsuma equation with Kerr law nonlinearity are investigated using extended F-expansion method. Bright, dark, singular, combo bright-singular, combo dark-singular and combo singular soliton solutions are extracted. Also, periodic, rational and Jacobi elliptic functions solutions are derived. Finally, we offer some 3D, 2D and contour plots to illustrate the nature of the provided solutions.

In this study, we investigate the soliton solutions for stochastic (3+1) dimensional nonlinear Schrödinger equation (NLSE). Our Study mainly depends on applying the modified extended mapping method to construct various and novel stochastic solutions for the proposed model. These solutions including (dark, bright, and singular) solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, and Jacobi elliptic function (JEF) solutions are raised. For a variety of nonlinear stochastic partial differential equations, this method offers a practical and effective method for determining exact stochastic solutions. The stochastic noise effects forcing the wave to behave differently from deterministic cases prognosticated. To illustrate the physical characteristics of the established stochastic solutions, some selected solutions are presented graphically. We used Mathematica (11.3) packages to find the coefficients and Matlab (R2015a) packages to plot the graphs.

In this manuscript, the improved modified extended tanh integration technique is implemented to investigate the exact solutions of stochastic Ginzburg–Landau model driven by Wiener process which appeared in various fields of chemistry, mathematics and physics. Many types of stochastic wave solutions are raised such as hyperbolic stochastic solutions, trigonometric stochastic solutions, Weierstrass elliptic stochastic solutions and Jacobi elliptic stochastic solutions. Moreover, 3D and 2D graphical representations are depicted to show the effect of the multiplicative noise on the propagated waves. The results reveal that as the noise level increases, the wave degrades. The wave is distorted when σ⩾1. In addition, the 3D graphical representation show that the propagated wave is flattered when the noise strength is increased. These solutions may be applicable in various applications in applied science. The proposed method is efficient, direct and powerful.

Throughout this article, the improved modified extended tanh function method is presented for deriving fresh travelling wave alternative solutions in different forms of nonlinear Schrödinger equation (NLSE) with higher-order dispersive and nonlinear effects that describes the propagation of ultra-short optical solitons in an optical fiber medium. For this model, we have a variety of new solutions, including solitons (bright, dark and singular), exponential, singular periodic and Jacobi elliptic solutions. Graphical representations of selected solutions are provided to demonstrate the physical properties of the raised solutions.

In this article, stochastic solutions of Radhakrishnan–Kundu–Lakshmanan equation (RKLE) with multiplicative noise are investigated. Studying is conducted with the aid of the modified extended direct algebraic integration technique which depended on the extended Riccati equation. To aid in our evaluation, a graphic representation of noise’s impact on the retrieved solutions is presented using MATLAB software packages.

New exact solutions for the perturbation higher order nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity are investigated using the ϕ6 expansion method. This type of NLSE models the propagation of femtosecond pulses in fiber optics which has many applications such as infrared time-resolved spectroscopy, ultrahigh-bit-rate optical communication systems and ultrafast optics. The application of ϕ6 expansion technique provided many different types of solutions. Dark solitons, bright solitons, singular solitons, hyperbolic solutions, periodic and singular periodic solutions are obtained. The graphical representations of some selected solutions are provided to present both the dynamical behavior of solutions and the effect of changing the nonlinearity degree on their dynamical behavior. This study concludes that the nonlinearity degree controls the dynamical behavior of the light pulse.

The stochastic complex Ginzburg–Landau model with parabolic law nonlinearity and chromatic dispersion is investigated in this research study. In many fields of science, including fluid dynamics and optical fibers, this model is utilized to represent the wave nature. In this study, the improved modified extended tanh approach is used. This method provides a wide variety of solutions, including dark, singular and bright solitons. Furthermore, exponential, singular periodic and Weierstrass elliptic solutions are provided. For a variety of nonlinear partial differential equations, this method offers a practical and effective method for determining exact solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths.

Cubic-quartic optical solitons and other solutions for Biswas-Milovic equation with dual-power law nonlinearity are investigated using improved modified extended tanh-function method. Bright soliton solutions, dark soliton solutions, singular soliton solutions, singular periodic solutions, Jacobi elliptic solutions and hyperbolic solutions are extracted. Furthermore, some selected solutions are described graphically to demonstrate the physical nature of obtained solutions.

In our study we implement an improved modified extended tanh-function (IMETF) method for constructing new solitons and other types of solutions with NLSE having Kudryashov’s generalized nonlinear refractive index. Bright, dark and singular optical soliton solutions have been recovered for the model by this approach. Moreover, 3D and contour graphs are plotted for more illustration.

Optical solitons is one of the fastest growing areas of research in the field of nonlinear fiber optics. In the current work, the modified extended direct algebraic method is implemented to obtain optical solitons for the nonlinear Schrödinger’s equation with group velocity dispersion and second order spatiotemporal dispersion that describe the propagation of optical solitons in nonlinear media. Bright soliton solutions, dark soliton solutions and singular soliton solutions are obtained. Also, periodic solutions, exponential solutions, Jacobi elliptic solutions and Weierstrass elliptic solutions are extracted. Under specific parameter values, 2D, 3D and contour graphs are introduced to visualize the model and demonstrate their accurate physical behaviors.

This paper is an wide account of bright solitons and periodic solutions for twin-core couplers and multiple-core couplers with polynomial law of nonlinearity and nonlinear perturbation terms using the Jacobi elliptic function expansion method. The restrictions on the parameters are considered to guarantee the existence of the obtained solitons. Moreover, graphs of some solutions are presented to show their features.

In the present article, the improved modified extended tanh function method is implemented to investigate the strain wave model which governs the wave propagation in micro-structured solids. Solitary and other wave solutions are provided for this model such as singular and bright solitary type solutions. In addition Kink, periodic, Jacobi elliptic, hyperbolic, exponential and Weierstrass elliptic type solutions are constructed. To show the physical characteristics of the raised solutions, graphical illustration of some solutions is presented.

This work predicts the dynamics of the COVID-19 under widespread vaccination to anticipate the virus's current and future waves. We focused on establishing two population-based models for predictions: the fractional-order model and the fractional-order stochastic model. Based on dose efficacy, which is one of the main imposed assumptions in our study, some vaccinated people will probably be exposed to infection by the same viral wave. We validated the generated models by applying them to the current viral wave in Egypt. We assumed that the Egyptian current wave began on 10th September 2021. Using current actual data and varying our models’ fractional orders, we generate different predicted wave scenarios. The numerical solution of our models is obtained using the fractional Euler method and the fractional Euler Maruyama method. At the end, we compared the current predicted wave under a high vaccination rate with the previous viral wave. Through this comparison, the vaccination control effect is quantified.

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation.

This context focuses on examining the manner in which localized waves travel through a non-linear medium that adheres to a dual-power law. This medium encompasses a range of disruptions like inter modal dispersion, self frequency shift, and self-steepening effects. This investigation entails employing an improved modified extended tanh function (IMETF) technique to obtain various novel optical soliton solutions which include bright, dark, singular and combo singular-dark solitons. Furthermore, we extract singular periodic, exponential, rational, and Weierstrass elliptic solutions. To showcase their physical characteristics, we provide graphical representations in 3D and 2D of some of the derived solutions.

In this paper, the modified simple equation method and the extended simplest equation method are applied to secure optical solitons and other solutions for Radhakrishnan-Kundu Lakshmanan equation. Bright solitons, dark solitons and singular solitons are obtained. Also, periodic solutions, singular periodic solutions, rational-type solutions and hyperbolic solutions are reported. Moreover, for the physical illustration, some of the obtained solutions are represented graphically.

In this work, the improved modified extended tanh scheme is implemented to extract exact travelling wave solutions for perturbed nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index and dual form of generalized nonlocal nonlinearity. Various types of solutions are extracted such as bright solitons, singular solitons, dark solitons, singular periodic wave solutions, periodic wave solutions, Jacobi elliptic functions, plane wave and hyperbolic wave solutions. Moreover, 3D and contour plots of some solutions are illustrated to show the physical nature of obtained solutions.

In this article, the improved modified extended tanh-function method is used to construct new structure of highly dispersive optical solitons for nonlinear Schrödinger equation in birefringent fibers with Kerr law and non-local law of refractive index. Bright solitons, dark solitons, bright–dark combo solitons and singular solitons are obtained. Also, periodic solutions, singular periodic solutions, hyperbolic solutions, rational-type solutions and exponential solutions are reported. After giving suitable values to parameters, the novel structures of solutions are depicted. The physical surfaces of some gained solutions in different forms are shown graphically. The results show the superiority of executed method.

In this paper, a four-dimensional competition model, driven by the Riesz–Caputo operator, is established. Then, the presented model’s uniqueness, existence, and stability are discussed. After that, the model is applied to describe the profit competition between Egyptian banks. Using the Hamiltonian definition of optimal control, we set a control strategy for banks’ profits during crises as a pre-step measure to counteract their negative impact. Finally, the model is solved numerically using an extended Adams–Bashford scheme. The valid data between the end of 2016 and 2020 are used for numerical simulations, while the data between the end of 2021 and 2022 are used for prediction measurements. From the results, it can be seen that the COVID-19 crisis resulted in a sudden decrease in banks’ profits during 2020, and if there were a control system enabled, it could have compensated for this decrease.

Symmetry in systems arises as a result of natural design and provides a pivotal mechanism for crucial system properties. In the field of control theory, scattered research has been carried out concerning the control of group-theoretic symmetric systems. In this manuscript, the principles of stochastic analysis, the fixed-point theorem, fractional calculus, and multivalued map theory are implemented to investigate the null boundary controllability (NBC) of stochastic evolution inclusion (SEI) with the Hilfer fractional derivative (HFD) and the Clarke subdifferential. Moreover, an example is depicted to show the effect of the obtained results.

In this paper, the modified extended mapping method is applied to obtain solitary wave solutions and other solutions for the Gilson–Pickering equation which describes the wave propagation in plasma physics. Many types of solutions are obtained such as shock wave, singular waves, periodic waves, singular periodic waves, Jacobi elliptic functions, plane waves, hyperbolic waves, Weierstrass elliptic solutions and exponential type solutions. Moreover, 3D graphs of some solutions are introduced for the physical illustration.

Abstract We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main results, we use fractional calculus, stochastic analysis, fractional power of operators and Sadovskii’s fixed point theorem. At the end, an example is also given to show the applicability of our obtained theory.

We introduce the conformable fractional (CF) noninstantaneous impulsive stochastic evolution equations with fractional Brownian motion (fBm) and Poisson jumps. The approximate controllability for the considered problem was investigated. Principles and concepts from fractional calculus, stochastic analysis, and the fixed-point theorem were used to support the main results. An example is applied to show the established results.