Lie symmetry and memory-driven dynamics of a (2+1)-dimensional time-fractional nonlinear wave equation in memory media

In the present work, we focus on discussing detailed analysis and important features of a (2+1)-dimensional nonlinear time-fractional diffusion-wave equation. The equation models wave phenomena in materials where memory effects play an important role, for example porous mediums and viscoelastic structures. We use the Lie symmetry methods together with the traveling wave reductions to obtain the exact solutions. Some of the solutions are expressed using special functions, like the Mittag-Leffler function and the Lambert W function. These functions describe the role of nonlinearity and fractional-order temporal damping on wave propagation. The graphical representation of the solutions suggest that the parameter α more or less dictates the behavior of the wave by affecting its concentration in space, decay rate, and speed at which it propagates. Further, the system correspond to distinct physical behaviors depending on different values of α, for example, sub-diffusive (0 < α < 1), diffusive (α=1), or wave-like motion with memory effects (1 < α ≤ 2). From these observation, it becomes clear that the dynamics of the system strongly gets affected when fractional memory combines with nonlinearity. This effect may have possible applications in modeling transportation processes as well as viscoelastic wave propagation arising in biological systems and porous materials.