Soliton dynamics via a novel method and dynamical analysis of the nonlinear fractional mKdV-KP equation

The main purpose of this work is to apply the new extended direct algebraic (nEDA) method to achieve exact traveling wave solutions for the nonlinear time-fractional modified Korteweg–de Vries Kadomtsev Petviashvili (mKdV-KP) equation. The model incorporates a fractional term utilizing the beta derivative, aiming to attain soliton solutions applicable to distinct physical processes such as ion-acoustic waves and plasma physics. The novelty of this research work lies in its application to the time-fractional mKdV-KP equation, along with the analysis of bifurcation, chaos, Lyapunov exponent analysis, sensitivity analysis and multi-stability to study the dynamical behavior of the system. The nEDA method is applied to achieve distinct soliton solutions for the governing model for the first time. These solutions are further examined by utilizing graphical simulations in the form of two-dimensional and three-dimensional surface plots yielded by MATHEMATICA- $11$. The dynamical characteristics and physical structure of the solutions are investigated, presenting their relevance and accuracy. The solutions are applied to real-world systems like biological processes, diffusion, solid-state physics and plasma phenomena. Notably, the analysis of bifurcation, chaos and sensitivity provides key insights into the stability and behavior of the model, emphasizing the novelty of the results. While the investigation presents a theoretical framework, further experimental verification and practical implementation of the soliton solutions are needed to understand their applicability in real-world systems better. Future work could explore more complex fractional models to gain deeper insights into the nonlinear dynamics of various systems. This paper applies an nEDA method for the first time to the time-fractional mKdV-KP equation, providing new exact soliton solutions such as bright-dark, lump-kink and periodic.