Solutions to a unified dispersive wave equation

Non-hydrostatic dispersive model can better describe mass flows in high mountain slopes. Pudasaini (2023) derived a complex dispersive wave equation to model the non-hydrostatic motion of landslides. However, no explicit exact analytical solutions for the landslide depth and its velocity have yet been established for mass flows that include the non-hydrostatic, dispersion effects. Here, we construct a novel unified dispersive wave equation that combines both the flow depth and velocity into a single, well-structured non-linear partial differential equation, characterizing the complex mechanics and the entire dynamics of landslide. We present some analytical functions solving the new unified dispersive wave equation. These solutions establish intrinsic functional relationships between the flow velocity and depth, allowing for the estimation of the one when the other is known. This is the advantage of the constructed solution. To illustrate the structure of the solution, we present a simple demonstrative example. The solution reveals interesting structures as the wave of higher dispersion travels along the slope that amplifies in time. We also construct some simple analytical solutions for the dispersive wave equation for the landslide motion. The physical significance and mathematical consistency of the solutions are proven. We reveal that a small change in the boundary conditions can lead to large differences in the wave structure. This manifests that the wave dynamics is strongly controlled by its dispersion mechanism and the physical − mathematical boundary conditions. The results indicate the necessity of properly selecting the physical parameters while describing the dispersive wave phenomena in mass transport. We highlight the physical essence of the new models and their analytical solutions, and outline their application potential.