Lie Symmetries, exact solutions, and stability analysis of the (2+1)-dimensional modified Kadomtsev–Petviashvili equation in nonlinear wave dynamics and systems

Abstract

This paper analyzes the group invariant solutions of the modified Kadomtsev–Petviashvili (mKP) equation using the Lie symmetry approach. Due to its cubic nonlinearity, the mKP system has rich physical and mechanical relevance. Nonlinear waves propagate at nonuniform velocities, which can be described by nonlinear evolution equations (NLEEs) and their solutions using arbitrary functions. Integrating a nonlinear evolution equation displays the various aspects of natural occurrences. The Lie symmetry transformation method is responsible for reducing the number of independent variables in the system and forming ordinary differential equations (ODEs). These reduced ODEs are solved with some imposed parametric restrictions. These solutions are more general than previously established results due to the existing arbitrary functions. The derived solutions are mainly kink waves, anti-kink waves, negatons, positons and elastic multisolitons. Further, using bifurcation theory, stability analysis of the results has been shown to make the research more worthy. We use phase portraits to explore the remarkable characteristics of the exact wave solutions. These portraits confirm the existence of certain families of homoclinic and periodic orbits around the equilibrium points. Finally, a brief discussion on the solutions of mKP has been made, showing the richness of our results due to several arbitrary constants and functions.