Two nonlinear evolution equations, namely the Kadomtsev-Petviashvili (KP) equation which describes the dynamics of soliton and nonlinear wave in the field of fluid dynamics, plasma physics and the Oskolkov equation which describes the dynamics of an incompressible visco-elastic Kelvin-Voigt fluid are investigated. We deliberate exact traveling wave solutions, specially kink wave, cusp wave, periodic breather waves and periodic wave solutions of the models applying the modified simple equation method. The solutions can be expressed explicitly. The dynamics of obtained wave solutions are analyzed and illustrated in figures by selecting appropriate parameters. The modified simple equation method is reliable treatment for searching essential nonlinear waves that enrich variety of dynamic models arises in engineering fields.

This work explores diverse novel soliton solutions of two fractional nonlinear models, namely the truncated time M-fractional Chafee-Infante (tM-fCI) and truncated time M-fractional Landau-Ginzburg-Higgs (tM-fLGH) models. The several soliton waves of time M-fractional Chafee-Infante model describe the stability of waves in a dispersive fashion, homogeneous medium and gas diffusion, and the solitary waves of time M-fractional Landau-Ginzburg-Higgs model are used to characterize the drift cyclotron movement for coherent ion-cyclotrons in a geometrically chaotic plasma. A confirmed unified technique exploits soliton solutions of considered fractional models. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. Keeping special values of the constraint, this inquisition achieved kink shape, the collision of kink type and lump wave, the collision of lump and bell type, periodic lump wave, bell shape, some periodic soliton waves for time M-fractional Chafee-Infante and periodic lump, and some diverse periodic and solitary waves for time M-fractional Landau-Ginzburg-Higgs model successfully. The required solutions in this work have many constructive descriptions, and corporal behaviors have been incorporated through some abundant 3D figures with density plots. We compare the m-fractional derivative with the beta fractional derivative and the classical form of these models in two-dimensional plots. Comparisons with others’ results are given likewise.

This work focuses on the study of the paraxial wave model with a space–time fractional form. This model has more importance for describing light propagation in nonlinear optical fibers and telecommunication lines. The main aim of this work is to observe the effect of the fractional parameters and compare the truncated $$M$$M -fraction with the beta-fraction, conformable fraction form, and classical form of the PW model. For this observation, we applied the Simplest equation technique to acquire analytical solutions to the space–time (spatiotemporal) $$M$$M -fractional paraxial wave model. We are able to acquire several new optical soliton solutions, including periodic waves, kink-type waves, rogue-type waves, and several novel periodic waves, by providing the appropriate fractional parametric values. These solutions have significance for shedding light on a number of physical phenomena in the realms of optical fiber and communication sciences. The diverse values of fractional parameters and the three-dimensional and contour plot graphs of certain chosen solutions are depicted, which are the most accurate physical characterizations of the outcomes. We also sketch the comparative graph of diverse fractional forms and the classical form of the paraxial wave equation in two-dimensional plots. Consequently, our findings represent an important breakthrough in this complex area and help further develop our comprehension of the behavior of solitons.

In this article, we study the soliton solutions with a time-dependent variable coefficient to the Kolmogorov-Petrovsky-Piskunov (KPP) model. At first, this model was used as the genetics model for the spread of an advantageous gene through a population, but it has also been used as a number of physics, biological, and chemical models. The enhanced modified simple equation technique applies to get the time-dependent variable coefficient soliton solutions from the KPP model. The obtained solutions provide diverse, exact solutions for the different functions of the time-dependent variable coefficient. For the special value of the constants, we get the kink, anti-kink shape, the interaction of kink, anti-kink, and singularities, the interaction of instanton and kink shape, instanton shape, kink, and bell interaction, anti-kink and bell interaction, kink and singular solitons, anti-kink and singular solitons, the interaction of kink and singular, and the interaction of anti-kink and singular solutions to diverse nature wave functions as time-dependent variable coefficients. The presented phenomena are clarified in three-dimension, contour, and two-dimension plots. The obtained wave patterns are powerfully exaggerated by the variable coefficient wave transformation and connected variable parameters. The effect of second-order and third-order nonlinear dispersive coefficients is also explored in 2D plots.