Muhammad Abbas

This paper explores the nonlinear Zoomeron equation that is much used in the modeling of nonlinear wave propagation processes in several physical systems including fluid dynamics and nonlinear optics. The great nonlinearity of this equation renders search of analytical solutions to this equation to be a hard and significant problem. In order to solve this, the governing nonlinear partial differential equation is converted into an ordinary differential equation through the use of the modified Khater method (MKM) and its exact solutions are then built. In this method, a vast variety of new analytical solutions are discovered, including kink, anti-kink, singular periodic and dark solitary wave solutions. These solutions are further illustrated using the dynamical behavior by two-dimensional, three-dimensional and contour plots in mathematica. The results obtained prove that the Modified Khater method is a good and valid tool of analysis in solving nonlinear evolution equations. Additionally, the derived solutions add to the solution space of the Zoomeron model and give further insight into the complex nonlinear wave dynamics, which can be helpful in future theoretical and practical research.

AIDS-related cancers are a major complication in HIV-infected individuals, driven by chronic immune suppression and viral oncogenesis. To better understand the dynamic interplay among healthy cells, HIV-infected cells, and cancer cells, this study proposes a fractional-order HIV-1 model using the Caputo-Fabrizio derivative. The research problem centers on how fractional-order dynamics-representing memory effects-influence the stability and long-term behavior of this biological system, particularly the conditions under which cancer persists alongside HIV infection. The model’s well-posedness is established by proving the existence and uniqueness of solutions, and local stability of equilibria is analyzed, with biological interpretation of the disease-free and coexistence steady states. Notably, numerical simulations employing the three-step fractional Adams-Bashforth method reveal that the system exhibits chaotic dynamics as the fractional order decreases from 1, indicating heightened sensitivity and complex interactions among cell populations. Two- and three-dimensional phase portraits together with time-series analyses demonstrate that lower fractional orders intensify chaotic behavior, suggesting that memory effects critically modulate disease progression and cancer development in HIV-1. These findings underscore the importance of fractional modeling in capturing realistic, biologically plausible dynamics in viral oncogenesis.

In this research, a [Formula: see text]-model approximation method is employed to investigate the localized wave solutions for the Lakshmanan-Porsezian-Daniel equation with beta-derivative. This equation integrates the fundamental phenomena such as Space-Time Dispersion (STD), Group Velocity Dispersion (GVD) and parabolic-law-governed by nonlinear behavior. A variety of optical soliton waves are obtained by applying the [Formula: see text]-model expansion approach. These waves are expressed as Jacobi elliptic functions [Formula: see text] that based on the particular values of the parameter A, that can be converted into solutions of trigonometric or hyperbolic functions. This technique provides variety of solutions, including dark soliton solutions, hyperbolic solutions, periodic waves solutions, bright solitons, singular soliton solution and singular periodic waves solutions. To further explore the system's behavior, bifurcation analysis is done. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers deep understanding of the phase portraits, time series, chaotic behavior and sensitivity analysis of the equation to external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.

This research paper is used to examine the generalized third-order nonlinear Schrödinger equation that is important for the description of complex nonlinear wave propagation in optical fibers, plasma physics, and in fluid mechanics. By using the generalized auxiliary equation method and improved modified Sardar-sub equation method various analytical soliton solutions are obtained. The Periodic, bell, anti-bell, dark, W-type, kink, anti-kink, and M-shape solitons solutions of the problem are obtained by using proposed methods. The behavior of several soliton solutions is depicted graphically. The findings are important for applications in engineering and mathematical physics. A key component of planar dynamical system theory is the bifurcation of the dynamical system of the governing equation caused by the Galilean transformation. By emphasizing how susceptible the system is to its initial conditions and how uncertain its long-term evolution is, the study also investigates chaotic behavior. Lastly, sensitivity analysis is performed to examine the effects of altering system parameters on the stability and results of the solutions. By taking these things into account, this work aids in our comprehension of the proposed model’s dynamical characteristics and offers helpful insights into its use in nonlinear media and other fields of research. In contrast to previous research, this work is novel because it applies enhanced analytical methods to the generalized third-order nonlinear Schrödinger equation, resulting in a wider range of solution classes.

This paper explores a nonlinear telecommunication transmission line model with the incorporation of the M‐truncated fractional derivative, providing memory‐dependent effects that go beyond the classical nonlinear transmission line models. The Hamilton–Jacobi and Jacobian dynamical frameworks are, for the first time, utilized in this fractional nonlinear transmission line context to methodically derive fractional travelling wave solutions and define the system’s nonlinear stability. Fractional system dynamics are investigated by bifurcation analysis, Lyapunov exponents, time series behavior, initial condition sensitivity, identifying periodic, quasiperiodic, and chaotic regimes. A rich variety of wave structures like kink, antikink, bell‐shaped, singular, M‐shaped, and W‐shaped solutions are derived and visualized through 2D profiles, 3D surfaces, and contour plots. Stability is investigated by Hamiltonian criteria and linearized perturbation analysis, and it confirms long‐time dynamic robustness. The results show that nonlinear wave dynamics and stability characteristics are greatly improved in telecommunication transmission lines by fractional order processes, and thus a new fractional order nonlinear transmission line theory is formed. In this paper, it is shown how the memory effect leads to the change of the stability region and the faster emergence of chaos within the framework of fractional order systems.

Fractional calculus (FC) has become more popular during the past four decades due to its extensive applications in mathematics, physics, engineering, and statistics. B‐spline functions offer flexible and incredibly precise approximations because of their piecewise polynomial structure and smoothness at knots. For a class of third‐order conformable boundary value problems (BVPs), we develop approximation solutions using the quartic B‐spline method. The conformable fractional derivative (CFD) is utilized to formulate fractional problems. More specifically, singularities are used to modify the category of conformable Lane–Emden models. Three numerical examples are shown and examined to demonstrate the approach’s effectiveness. The numerical results are highly accurate and require less computational work, and they closely match the exact solutions. MSC2020 Classification: 39A12, 39B62, 33B10, 26A48, 26A51.

This study investigates the optical soliton solutions to the generalized third-order nonlinear Schrödinger equation involving the Caputo fractional derivative using new mapping method. The fractional generalized third-order nonlinear Schrödinger equation is frequently utilized in various fields, including mathematical physics, nonlinear optical phenomena, optical communication technologies and plasma physics. The obtained solutions have different solitons including bell shape, w-shape, anti bell, kink, dark and periodic wave solution. Bifurcation analysis is performed to further explore the behaviour of the system. For this analysis planar dynamical system is obtained by using Galilean transformation. This analysis offers valuable insights into the phase portraits, time series, chaotic behaviour and the sensitivity of the model to the external perturbations. The sensitivity and dynamics of optical solitons are thoroughly investigated that offers significant insights into their behavior within fractional models.