Can you hear the shape of the river-bed? Stability and bifurcation of fully nonlinear hydraulic fall solutions to the forced water-wave problem

Determining the size and shape of an obstacle on a river-bed or a ocean-floor based on measurements of the wave-profile on the surface is a fundamental problem in nonlinear wave studies and is important in understanding a number of physical phenomena, including tsunami detection and river-bank erosion. One theoretical approach is to prescribe a free-surface profile and then solve the governing equations for the bottom topography profile. In this talk, we propose a new time-dependent 'inverse method' for establishing the size of the forcing based solely on measurements of the unsteady wave-profile far upstream of the obstacle. We solve the fully nonlinear steady and time-dependent Euler equations numerically using a novel numerical framework based on the finite-element Galerkin method and by probing the stability of hydraulic-fall solutions we are able to identify a new invariant object of the Euler equations that consists of a time-dependent wave-pulse at the centre of the forcing that emits waves far upstream. The measurement of the spatial and temporal features of these emitted waves allows us to identify the size of the forcing and forms the basis of our new time-dependent 'inverse' approach. In addition, we also establish a connection between the hydraulic-fall solutions and the recently uncovered solitary-wave solutions at Fr=1 in Keeler et al. (2017), thus partially answering an open question posed by Forbes et al. (2021).