Abundant dynamical structure of solutions to truncated M-fractional modified Korteweg–de Vries model: Effects of dispersion, nonlinearity and fractionality

Abstract

This work explores various wave pattern dynamics due to fractional derivative, dispersive and nonlinearity effects for the nonlinear time M-fractional modified Korteweg–de Vries (tM-fMKDV) model. To reconnoiter such dynamics, the unified and new form of modified Kudryashov’s techniques execute to integrate the nonlinear tM-fMKDV model for achieving diverse solitonic and travelling wave envelopes. As a result, trigonometric, hyperbolic and rational solutions have been found via a unified technique, and constants base wave solutions have been derived via the novel Kudryashov’s technique. The dynamical behaviors of the obtained solutions in the pattern of periodic waves, different types of periodic rogue waves, kink waves, and different types of double periodic waves have been illustrated with 3D and density plots for arbitrary choice of the permitted parameters. Analyzing the effect, anyone can observe that an increase in dispersion coefficient causes strictly shock to slopy shocked, an increase of nonlinearity reduces the wavelength and reduces fractionality causing smoothly banding wave shape to strictly banding. As a result, our findings demonstrate that the proposed schemes are highly effective, efficient, and accurate in capturing the characteristics of waves. Compared to other approaches, the solutions obtained from our tM-fMKDV models are more abundant.

Keywords

• Unified Scheme
• New form of modified Kudryashov’s scheme
• Time M-fractional modified Korteweg–de Vries
• Rogue wave
• Kink wave
• Periodic wave