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The space-time fractional non-linear Phi-4 equation is a significant equation to describes the fission and fusion process that ensued in chemical kinematics, solid-state physics, astrophysical fusion plasma, plasma physics, and electromagnetic interactions, etc. The Phi-4 non-linear partial differential equation is reshaped utilizing the three different fractional-order derivatives and constructed transformations corresponding to every fractional-order derivative to convert the partial differential equation into an ordinary differential equation. A new extended direct algebraic equation method was successfully applied to extract the solitons solutions. The solitons solutions are developed with the exponential, trigonometric, rational, and hyperbolic functions including different unknown constant parameters. The graphical interpretations of obtained solutions are also depicted by allocating the feasible values to unknown constant parameters. The proposed scheme is an effective and functional scientific technique to investigate different fractional systems and models in engineering and physics referenced to real physical problems. RECEIVED

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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.

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<abstract><p>The study aims to explore the nonlinear Landau-Ginzburg-Higgs equation, which describes nonlinear waves with long-range and weak scattering interactions between tropical tropospheres and mid-latitude, as well as the exchange of mid-latitude Rossby and equatorial waves. We use the recently enhanced rising procedure to extract the important, applicable and further general solitary wave solutions to the formerly stated nonlinear wave model via the complex travelling wave transformation. Exact travelling wave solutions obtained include a singular wave, a periodic wave, bright, dark and kink-type wave peakon solutions using the generalized projective Riccati equation. The obtained findings are represented as trigonometric and hyperbolic functions. Graphical comparisons are provided for Landau-Ginzburg-Higgs equation model solutions, which are presented diagrammatically by adjusting the values of the embedded parameters in the Wolfram Mathematica program. The propagating behaviours of the obtained results display in 3-D, 2-D and contour visualization to investigate the impact of different involved parameters. The velocity of soliton has a stimulating effect on getting the desired aspects according to requirement. The sensitivity analysis is demonstrated for the designed dynamical structural system's wave profiles, where the soliton wave velocity and wave number parameters regulate the water wave singularity. This study shows that the method utilized is effective and may be used to find appropriate closed-form solitary solitons to a variety of nonlinear evolution equations (NLEEs).</p></abstract>

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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 H1 and the rational function solution H34 from the obtained solutions, respectively.