Search

This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Soliton solutions of the Konopelchenko–Dubrovsky (KD) equation using four analytical methods are established. The KD system is used to study the portrays in physics with weak dispersion. The investigated results are obtained in different forms such as trigonometric, hyperbolic, and exponential functions. For the physical behavior of the concerned nonlinear system, some solutions are plotted graphically via assigning the certain values to the parameters. Mathematica software 11.11 is used to handle all results as well as figures. Hence, searched results have rewarding recompenses in nonlinear science.

The Phi-Four (also embodied as ∅4) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE. [ABSTRACT FROM AUTHOR]