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This document discusses recent trends in mathematical models and solitons used to analyze real-world problems. The focus is on the application of fractional calculus in modeling phenomena such as anomalous diffusion, non-Markovian processes, and random walk. The document provides summaries of 15 papers accepted for publication in a special issue, covering topics such as digestive system modeling, COVID-19 pandemic modeling, control systems for motors, symmetry in pine wilt disease, soliton solutions for various equations, and epidemic modeling. The papers employ various mathematical techniques and provide insights into different aspects of real-world problems. [Extracted from the article]

In this study, the nonlinear (2+1)-dimensional generalized breaking soliton system has been studied by introducing an appropriate travelling wave transformation. The model under consideration is integrable with constant coefficients and demonstrates Riemann wave propagation characteristics, it principally transformed into a third-order nonlinear ODE. The reliable and robust method, namely the modified exponential function method, is applied to the nonlinear system for the first time. The main goal is to investigate and obtain some explicitly exact travelling waves, periodic waves, and soliton solutions. The obtained solutions are in the form of exponential functions, trigonometric hyperbolic functions, and combined structures of the trigonometric hyperbolic with logarithmic functions. Furthermore, the obtained solutions are new and significant in revealing the pertinent features of the physical phenomenon. The results have been expressed in several graphs, including two- and three-dimensional plots, for the best visual assessment of the physical significance and dynamic characteristics. The most potent and efficacious tools are the computer software packages we utilize to derive solutions and graphs. [ABSTRACT FROM AUTHOR]

Fractional nonlinear evolution equations concerning conformable fractional derivative are effective models to interpret intricate physical phenomena in the real world. The space-time fractional foam drainage equation and the coupled mKdV equations with conformable fractional derivative are important model equations for shallow water waves, the waves of flow of liquid between bubbles, the capillary waves, the waves of foam density, the electro-hydro-dynamical model, the ion acoustic plasma waves etc. In this study, we extract the functional and further general exact wave solutions comprising the rational, trigonometric, exponential and hyperbolic functions to the above stated models taking the advantage of the auxiliary equation method with the assistance of the fractional complex transformation. The method is convenient, reliable and delivers fresh and useful solutions to fractional differential equations ascend in physical and engineering sciences. We depict figures of the obtained solutions in order to illustrate the inner structure associated to the phenomena. [ABSTRACT FROM AUTHOR]

This paper concerns with the integrability of variable coefficient fifth order nonlinear Schrödinger's equation describing the dynamics of attosecond pulses in inhomogeneous fibers. Variable coefficients incorporate varying dispersion and nonlinearity which are of physical significance in considering the nonuniform boundaries of fibers as well as the inhomogeneities of the media. The well-known exp(−φ(s))-expansion method is used to retrieve singular and periodic solitons with the aid of symbolic computation. The structures of the obtained solutions are discussed along with their existence criteria. Moreover, the modulation instability analysis is carried out to identify the instability regions. A dispersion relation is extracted between wave number and frequency. The optimal value of the frequency is found for the occurrence of the instability. A detailed discussion of the results is also given along with graphics. [ABSTRACT FROM AUTHOR]

The time-fractional generalized biological population model and the (2, 2, 2) Zakharov–Kuznetsov (ZK) equation are significant modeling equations to analyse biological population, ion-acoustic waves in plasma, electromagnetic waves, viscoelasticity waves, material science, probability and statistics, signal processing, etc. The new generalized G ′ / G -expansion method is consistent, computer algebra friendly, worthwhile through yielding closed-form general soliton solutions in terms of trigonometric, rational and hyperbolic functions associated to subjective parameters. For the definite values of the parameters, some well-established and advanced solutions are accessible from the general solution. The solutions have been analysed by means of diagrams to understand the intricate internal structures. It can be asserted that the method can be used to compute solitary wave solutions to other fractional nonlinear differential equations by means of fractional complex transformation. [ABSTRACT FROM AUTHOR]

Bernoulli sub‐equation function method is applied to obtain exact solutions of Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation. As a result of this, exact traveling‐wave and some new oscillating solutions to CDGSK are obtained. It may be observed that Bernoulli sub‐equation function method employed here is very effective and reliable to get explicit solutions for this nonlinear partial differential equation. Profiles of all constructed solutions are graphically illustrated entirely as well. [ABSTRACT FROM AUTHOR]

This paper studies (2 + 1) -dimensional stochastic chiral nonlinear Schrödinger equation dynamically. A number of novel solutions such as periodic, singular, dark and bright solitons solutions are retrieved. The extraction of these solutions is helped by three efficient and robust integration tools such as the Kudryashov's R function method, Jacobi elliptic function method (JEFM) and the modified auxiliary equations (MAE) method. The parametric constraints for the existence of such solutions are also listed. Moreover, 3D plots of few obtained solutions are presented in this paper. [ABSTRACT FROM AUTHOR]

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported. [ABSTRACT FROM AUTHOR]

This research paper focuses on the application of the tanh function method to find the soliton solutions of the (2+1)-dimensional nonlinear electrical transmission line model. Materials used to form a transmitting line are very important to transmit electric charge. In this sense, we find some new voltage behaviors such as dark, trigonometric, and complex function solutions. Choosing some suitable values of parameters, we present some various surfaces of results obtained in this paper. These results play an important role in telecommunications lines used to stand for wave propagations. [ABSTRACT FROM AUTHOR]

This paper focuses on the instability modulation and new travelling wave solutions of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar equation via the tanh function method. Dark, mixed dark–bright, complex solitons and periodic wave solutions are archived. Strain conditions for the validity of results are also reported. Instability modulation properties of the governing model are also extracted. Various wave simulations in 2D, 3D and contour graphs under the strain conditions are presented. [ABSTRACT FROM AUTHOR]

In this work, we investigate the perturbed optical solitons to the Gerdjikov-Ivanov equation consisting of group velocity dispersion and quintic nonlinearity coefficients, which communicate the propagation of pulses in nonlinear optics. The various kinds of solitons in single and combined forms like dark, singular, dark-singular, bright-dark are derived by Fan-extended sub equation method. Moreover, the singular periodic, triangular type solutions are also obtained. And, we also discuss the stability analysis of the studied nonlinear model with the concept of linear stability, we prove that the governing model is stable. Parametric conditions on physical parameters to ensure the existence criteria of optical solitons are also listed. We also plot 3D profiles for the physical behavior of the obtained solutions by selecting the suitable values of the parameters. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the various dimensional nonlinear Fokas and Breaking soliton equations via a powerful analytical method, namely, sine-Gordon expansion method. Many new solutions such as complex combined dark-bright soliton solutions, singular and hyperbolic functions are derived. Choosing the suitable values of these parameters, various novel simulations are also plotted. Such results explain the wave behavior of the governing models, physically. [ABSTRACT FROM AUTHOR]

In this paper, we apply an analytical method, namely, the sine-Gordon expansion method and extract some complex optical soliton solutions to the (2+1)-dimensional extended shallow water wave model, which describes the evolution of shallow water wave propagation. We obtain some complex mixed-dark and bright soliton solutions to this nonlinear model. Considering some suitable values of parameters, we plot the various dimensional simulations of every results found in this manuscript. We observe that our result may be useful in detecting some complex wave behaviors. [ABSTRACT FROM AUTHOR]

In this paper, we study the space-time fractional perturbed nonlinear Schrödinger equation under the Kerr law nonlinearity by using the extended sinh-Gordon equation expansion method. The perturbed nonlinear Schrödinger equation is a nonlinear model which arises in nano-fibers. Some family of optical solitons and singular periodic wave solutions are successfully revealed. The parametric conditions for the existence of valid solitons are stated. Under the choice of suitable values of the parameters, the 3-dimensional and 2-dimensional graphs to some of the reported solutions are plotted. [ABSTRACT FROM AUTHOR]

This work deals with finding new complex solitons to the perturbed nonlinear Schrödinger model with the help of an analytical method. Using several computation programs, we gain entirely new solitons to the considering model. Under the choice of suitable values of the parameters, density graphs to the reported solutions, such as complex, hyperbolic, trigonometric and exponential, are successfully presented. Simulations in various dimensions are also plotted by using package programs. [ABSTRACT FROM AUTHOR]

This study uses two mathematical approaches in constructing dark, bright, kink-type and singular soliton solutions to the Heisenberg ferromagnetic spin chain equation. The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation describes nonlinear dynamics of magnets. The acquired results in this study may help in explaining some physical meanings of some nonlinear physical models arising in electromagnetic waves. For instance, the hyperbolic tangent arises in the calculation of magnetic moment and rapidity of special relativity, the hyperbolic cotangent arises in the Langevin function for magnetic polarization. The 2-, 3-dimensional and the contour plots of all the acquired solutions are presented. [ABSTRACT FROM AUTHOR]

This study applies the powerful sine-Gordon expansion method in acquiring some new solutions to the strain wave equation in micro-structured solids arising in mathematical physics. The strain wave equation in micro-structured solids is used in modelling wave propagation in micro-structured materials. We successfully acquire some new solitary wave solutions with hyperbolic function structure. By taking some suitable values of parameters involved in the solutions, we plot the 2D, 3D and contour graphs of all the obtained solutions in this study. [ABSTRACT FROM AUTHOR]

This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin-Bona-Mahony and the coupled Klein-Gordon equations. The modified Benjamin-Bona-Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein-Gordon equation describes the quantized version of the relativistic energy-momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and use the new modified exp ( − Ω ( ξ ) ) -expansion method, (MEM). We apply the new MEM to nonlinear long–short-wave interaction systems (NLSWIS). Among our findings are sets of solutions including, but not limited to, new hyperbolic, complex, and dark soliton solutions. Not only is MEM shown to be highly adaptable for partial differential equations with strong nonlinearities, but also, it turns out to be highly efficient, despite its ease. [ABSTRACT FROM AUTHOR]

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions. [ABSTRACT FROM AUTHOR]

The modified exp - Ω ζ -expansion function method is applied to the Wu–Zhang system with conformable time-fractional derivative to construct new analytical solutions. We have obtained some soliton-type solutions such as dark, singular and combined soliton solutions. We have seen that all solutions have provided the mentioned equation system. For the suitable value of the solutions, the 2D–3D and contour surfaces have been plotted. [ABSTRACT FROM AUTHOR]

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized higher-order nonlinear Schrödinger equation analytically. We use two integral schemes for conducting this study. Dark, bright, combined dark-bright optical, singular soliton, soliton-like and trigonometric function solutions are successfully constructed. We give the constraint conditions for the existence of valid solutions. The 2D, 3D and the contour graphs for the dark and bright solitons are plotted. [ABSTRACT FROM AUTHOR]

The dynamical systems of soliton propagation through optical fibers for trans-continental and trans-oceanic distances is one of the most interesting areas of study. Optical solitons are restrained electromagnetic waves that stretch in nonlinear dispersive media and allow the intensity to remain unchanged due to the balance between dispersion and nonlinearity effects. In this study, we successfully acquire dark, bright, combined dark-bright, singular and combined singular soliton solutions to the decoupled nonlinear Schrödinger equation arising in dual-core optical fibers by using the extended sinh-Gordon equation expansion method. The constraint conditions for the existence of valid soliton solutions are given. We discuss how change in parameters affect the solitons transmission. We present the 2D, 3D and the contour graphs to some of the reported solutions. [ABSTRACT FROM AUTHOR]

This study constructs bright and singular optical solitons the (2+1)-dimensional NLSE and the Hirota equation by utilizing the new sine-Gordon expansion method. The parametric conditions for the existence of valid solitons are given. We give the variations of solitary and periodic wave solutions that depend on the parameters. We present the 2D, 3D and contour graphs to represent the kind of solutions secured in this study. [ABSTRACT FROM AUTHOR]

This study investigates the longitudinal wave equation in a magneto-electro-elastic circular rod by using the extended sinh-Gordon equation expansion method. Topological, non-topological and singular soliton solutions are extracted. To illustrate the physical appearance of the obtained solutions, 2D, 3D and the contour graphs to some of the obtained solutions are plotted. The reported results may be useful in explaining the physical meaning of the studied models and other nonlinear physical models arising in nonlinear sciences. [ABSTRACT FROM AUTHOR]

Abstract In this study, the applications of the extended sinh-Gordon equation expansion method to a nonlinear Schrödinger equation that describes the nonlinear spin dynamics of (2 + 1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit is presented. We successfully construct dark, bright, combined dark-bright, singular and combined singular soliton solutions to this equation. We give the parametric conditions for the existence of valid soliton to each of the obtained solutions. We plot the 2D and 3D graphics to some of the obtained solutions. The reported results of this study may be helpful in explaining the physical meaning of the studied model. Highlights • The application of the extended ShGEEM to the Heisenberg ferromagnetic spin chain equation is presented. • Dark, bright and other solition solutions are reported. • The physical meanings of the obtained solutions are stated. • The interesting 2D and 3D graphics to some of the obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

This study reveals the dark, bright, combined dark–bright, singular optical solitons and other solutions to the conformable space–time fractional Fokas–Lenells equation. We use two integral schemes in reaching such solutions. The constraints conditions which guarantee the existence of valid solutions are stated. By choosing the suitable values of the parameters involved, we plot the 2-dimensional and 3-dimensional surfaces to some of the obtained solutions. [ABSTRACT FROM AUTHOR]