Bifurcation analysis and novel wave patterns to Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation with truncated M-fractional derivative

Abstract The Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation (ZKBBME) is a crucial mathematical model used in fractional quantum mechanics, optical fiber signal processing, ion-acoustic waves in plasma, water waves driven by gravity, turbulent flow, fluid flow waves, and for describing many other real-world phenomena. This article employs the modified exp-function method and exp $$(-\Phi (\psi ))$$ -expansion method, along with a truncated M-fractional wave transformation, to investigate new rational, trigonometric, hyperbolic, and exponential function solutions. Assigning specific parameter values generates diverse wave shapes most significantly, a new combined wave type called the compacton-kink and a class of peakon waves, which has not yet been documented in previous research of this model. 2-dimensional, 3-dimensional, contour, density, and polar plots illustrate the physical properties of soliton solutions, demonstrating the method’s suitability for analyzing a range of nonlinear fractional models with truncated M-fractional derivative (TMFD). Furthermore, utilizing the Galilean transformation to transform the equation into a planar dynamical system, bifurcation theory is applied to investigate its bifurcation and equilibrium points. The findings show that the TMFD framework captures intricate nonlinear wave dynamics and considerably enriches the ZKBBME solution space. These results advance our knowledge of wave structures in engineering and applied physics models controlled by fractional-order nonlinear partial differential equations (FNLPDEs).