Dry dam-break over parabolic bathymetry

Motivated by dry dambreak flows, a perturbative framework for solving the 1D shallow water equations (SWE) over parabolic bathymetry, together with the inviscid Burgers' equation, is developed. Expressed in Riemann variables, the solutions are expanded as analytic power series in the bathymetric curvature $\omega^2$ and summed in closed form in terms of trigonometric functions. The solutions, termed generalised rarefaction waves (GRWs), exhibit finite-time singularities and are shown to describe the asymptotic behaviour of the fluid near dry (vacuum) points. The GRWs bear direct relevance to wave run-up at a shore. A repulsive parabolic hill bathymetry is also considered, yielding solutions in terms of hyperbolic functions. The connection to the nonlinear Schr\"odinger/Gross-Pitaevskii equation that models Bose-Einstein condensates confined in a harmonic potential is discussed. The analytical results agree with direct numerical simulations of the Burgers' equation and of the SWE in different scenarios.