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The space–time fractional Landau-Ginzburg-Higgs equation and coupled Boussinesq-Burger equation describe the behavior of nonlinear waves in the tropical and mid-latitude troposphere, exhibiting weak scattering, extended connections, arising from the interactions between equatorial and mid-latitude Rossby waves, fluid flow in dynamic systems, and depicting wave propagation in shallow water. The improved Bernoulli sub-equation function method has been used to achieve new and wide-ranging closed-form solitary wave solutions to the mentioned nonlinear fractional partial differential equations through beta-derivative. A wave transformation is applied to renovate the fractional-order equation into an ordinary differential equation. Some standard wave shapes of multiple soliton type, single soliton, kink shape, double soliton shape type, triple soliton shape, anti-kink shape, and other types of solitons have been established. The more updated software Python is used to display the solutions by using 3D and contour plotlines to describe the physical significances of attained solutions more clearly. The findings of this study are straightforward, adaptable, and quicker to simulate. It has been notable that the improved Bernoulli sub-equation function method is practical, effective, and offers more sophisticated solutions that can help to generate a large number of wave solutions for various models.

The time-fractional unstable nonlinear Schrödinger (NLS) equations capture the time evolution of disturbances within media, tailored for describing phenomena in unstable media to help model and understand the intricate dynamics of systems prone to instability, the behavior of disturbances in complex and unstable environments and many more. In this study, we investigate the exact and analytical solutions of time-fractional unstable nonlinear Schrödinger (NLS) equations with the time-fractional β-time derivative, a powerful technique known as the sine-Gordon expansion (SGE) method is used. Non-locality behaviors that cannot be modeled using classical calculus are included in this equation. It discusses the relationships between waves of different frequencies or the same frequency but with distinct polarizations having a diverse set of key applications in nonlinear optics. We begin by using a fixed-point argument to prove the existence and uniqueness of solutions to the time-fractional problem under consideration. The partial differential equation of fractional order is transformed into an ordinary differential equation using the complex transformation. Numerous exact solutions are achieved with all the arbitrary parameters. The specific values of the obtained solutions' associated parameters are analyzed graphically with 3D plots as multiple soliton types of different wave solutions with the help of symbolic software. Understanding that the features of the solutions are far from intuitive because it is based on the parameter selection from the figures is difficult. The main objective of this paper is to investigate some fresh and further general solitary wave solutions to the suggested equation that properly explain the above-stated phenomena.

Nonlinear fractional evolution equations play a crucial role in characterizing assorted complex nonlinear phenomena observed in different scientific fields, including plasma physics, quantum mechanics, elastic media, nonlinear optics, surface water waves, nonlinear dynamics, molecular biology, and some other domains. In this study, we investigate diverse solitary wave solutions to the space–time fractional modified Equal Width equation, focusing on their significance for wave propagation behaviors in plasma and optical fiber systems. Two effective approaches, namely the improved Bernoulli sub-equation function and the new generalized (G′/G)-expansion methods are exploited. A fractional wave transformation technique is employed to convert the fractional-order equation into an ordinary differential equation. The solitary wave solutions obtained in terms of exponential, hyperbolic, rational, and trigonometric functions, have wide applications in various nonlinear phenomena. The 3D, spherical, contour, and vector plots are presented to elucidate the physical implications of the obtained solutions for specific parameter values. The soliton structures reveal various wave types, including bell-shaped, multi-soliton, kink, single soliton, anti-bell-shaped, periodic soliton, compacton, and other types for definite values of the free parameters. We compare the attained solutions with the solutions available in the literature to demonstrate the feasibility of the solutions. It is observed that both approaches are efficient, straightforward, and significant for investigating diverse nonlinear models that modulate complex phenomena.

The general time fractional Burger- Fisher (TF-BF) and the space–time regularized long-wave (STF-RLW) equations are considered as examples of gravitational water waves in cold plasma as well as so many areas. The above equations are used in nonlinear science and engineering to study long waves in seas and harbors that travel in just one direction. First, the two equations are transformed to ODEs by applying a fractional complex transform along with characteristics of confirmable fractional derivative (CFD). Then, the extended tanh-function (ET-F) approach is investigated to find a variety of analytical solutions with different geometrical wave structures the mentioned models. The results are in the form of kink, one-, two-, multiple-solitons solutions, and other types sketched in 2D, 3D, and contour patterns.

Nonlinear fractional partial differential equations (NLFPDEs) are well suited for describing a broad range of factors in engineering and science, including plasma physics, optical fiber, acoustics, finance, turbulence, mechanical engineering, control theory, nonlinear biological systems, and so on. Through this memorandum, a novel technique is used for solving the class of NLFPDEs. We opted to construct a traveling wave solution to the nonlinear space-time fractional Phi-4 and Allen-Cahn (AC) equations, which are often used as frameworks for a variety of phenomena such as nano fluids, reaction–diffusion model and used to investigate the phase separation process in a variety of components, order–disorder interchange in the bass system and, so on. The subsidiary new generalized (G'/G)-expansion technique for the mentioned equations is used to obtain new precise solutions with the help of conformable derivatives. This strategy is utilized in this article to obtain some new, appealing, and more general outcomes that are simple, versatile, and faster to simulate. Using the proposed technique, some dynamical wave shapes of compaction types, periodic compaction type, single soliton type, kink types, singular kink types, and some other types are accomplished and we used Mathematica to describe the solutions through 3D, contour, 3D list point plot, and vector plots to describe the physical sketch more clearly.

We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

The nonlinear Riemann wave equations (RWEs) and the Landau-Ginsburg-Higgs (LGH) equation are related to plasma electrostatic waves, ion-cyclotron wave electrostatic potential, superconductivity, and drift coherent ion-cyclotron waves in centrifugally inhomogeneous plasma. In this article, the interactions between the maximum order linear and nonlinear factors are balanced to compute realistic soliton solutions to the formerly stated equations in terms of hyperbolic functions. The linear and nonlinear effects rheostat the structure of the wave profiles, which vary in response to changes in the subjective parameters combined with the solutions. The established solutions to the aforementioned models using the extended tanh scheme are descriptive, typical, and consistent, and include standard soliton shapes such as bright soliton, dark soliton, compacton, peakon, periodic, and others that can be used to analyze in ion-acoustic and magneto-sound waves in plasma, homogeneous, and stationary media, particularly in the propagation of tidal and tsunami waves.