Extended Lie Method for Mixed Fractional Derivatives, Unconventional Invariants and Reduction, Conservation Laws and Acoustic Waves Propagated via Nonlinear Dispersive Equation

This study primarily aims to investigate the application of the Lie symmetry method and conservation law theories in the analysis of mixed fractional partial differential equations where both Riemann–Liouville time-fractional and integer-order x-derivatives are present simultaneously. Specifically, the focus is on the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation. The fractionally modified equation is subjected to invariant analysis using the prolongation formula for mixed derivatives \(\partial _{t}^{\alpha }(u_{x})\) and \(\partial _{t}^{\alpha }(u_{xxx})\) for the first time. Through the introduction of a novel reduction method, we utilize the Lie symmetry technique to convert the (2+1) dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation into a fractional ordinary differential equation. It’s worth noting that this transformation is carried out without employing the Erdélyi–Kober fractional differential operator. Following this, we introduce a comprehensive expression for deriving conservation laws, involving the notion of nonlinear self-adjointness. Further, two different versatile techniques, the extended Kudryashov method and the Sardar subequation method have been used to extract a wide array of fresh sets of solitary wave solutions encompassing variations like kink, bright, singular kink, and periodic soliton solutions. To provide an intuitive grasp and investigate the ramifications of the fractional derivative parameter on these solitary wave solutions, we conduct a visual exploration employing both 3D and 2D plots.