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The Delta and Omicron variants' system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). [ABSTRACT FROM AUTHOR]

The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields. [ABSTRACT FROM AUTHOR]

The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using the new extended direct algebraic method. As a result, the soliton patterns of the Riemann wave equation have been successfully illustrated, with exact solutions offered by the plane solution, trigonometry solution, mixed hyperbolic solution, mixed periodic and periodic solutions, shock solution, mixed singular solution, mixed trigonometric solution, mixed shock single solution, complex soliton shock solution, singular solution, and shock wave solutions. Graphical visualization is provided of the results with suitable values of the involved parameters by Mathematica. It was visualized that the velocity of the soliton and the wave number controls the behavior of the soliton. We are confident that our research will assist physicists in predicting new notions in mathematical physics. [ABSTRACT FROM AUTHOR]

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. [ABSTRACT FROM AUTHOR]

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ 6 -model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature. [ABSTRACT FROM AUTHOR]

The propagation of optical soliton profiles in plasma physics and atomic structures is represented by the (1 + 1) − dimensional Schrödinger dynamical equation, which is the subject of this study. New solitary wave profiles are discovered by using Nucci's scheme and a new extended direct algebraic method. The new extended direct algebraic approach provides an easy and general mechanism for covering 37 solitonic wave solutions, which roughly corresponds to all soliton families, and Nucci's direct reduction method is used to develop the first integral and the exact solution of partial differential equations. Thus, there are several new solitonic wave patterns that are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, a mixed trigonometric solution, a trigonometric solution, a shock solution, a mixed shock singular solution, a mixed singular solution, a complex solitary shock solution, a singular solution, and shock wave solutions. The first integral of the considered model and the exact solution are obtained by utilizing Nucci's scheme. We present 2-D, 3-D, and contour graphics of the results obtained to illustrate the pulse propagation characteristics while taking suitable values for the parameters involved, and we observed the influence of parameters on solitary waves. It is noticed that the wave number α and the soliton speed μ are responsible for controlling the amplitude and periodicity of the propagating wave solution. [ABSTRACT FROM AUTHOR]

This study examines the Chen–Lee–Liu dynamical equation, which represents the propagation of optical pulses in optical fibers and plasma. A new extended direct algebraic technique and Nucci's scheme are used to find new solitary wave profiles. The method covers thirty-seven solitonic wave profiles, including approximately all soliton families, in an efficient and generic manner. New solitonic wave patterns are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions. The exact fractional solution is obtained using Nucci's reduction approach. The impact of the fractional order parameter on the solution is considered using both mathematical expressions and graphical visualization. The fractional order parameter is responsible for controlling the singularity of the solution which is graphically displayed. A sensitivity analysis was used to predict the sensitivity of equations with respect to initial conditions. To demonstrate the pulse propagation characteristics, while taking suitable values for the parameters involved, 2-D, 3-D, and contour graphics of the outcomes achieved are presented. The influence of the fractional order ζ is shown graphically. A periodic-singular wave with lower amplitude and dark-singular behaviour is inferred from the graphical behaviour of the trigonometric function solution H 1 and the rational function solution H 34 from the obtained solutions, respectively. [ABSTRACT FROM AUTHOR]

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example, nonlinear sciences, marine engineering, fluid dynamics, scientific applications, and ocean plasma physics. The new extended algebraic method is applied to solve the model under consideration. Furthermore, the nonlinear model is converted into an ordinary differential equation through the next wave transformation. A well-known analytical approach is used to obtain more general solutions of different types with the help of Mathematica. Shock, singular, mixed-complex solitary-shock, mixed-singular, mixed-shock singular, mixed trigonometric, periodic, mixed-periodic, mixed-hyperbolic solutions are obtained. As a result, it is found that the energy-carrying capacity of liquid with gas bubbles and its propagation can be increased. The stability of the considered model is ensured by the modulation instability gain spectrum generated and proposed with acceptable constant values. Two-dimensional, three-dimensional, and contour surfaces are plotted to see the physical properties of the obtained solutions. [ABSTRACT FROM AUTHOR]