Homoclinic-shedding in the steady forced water-wave problem

Abstract: Recently, an infinite set of steady solitary-wave solution branches was discovered for inviscid and irrotational flow over a localised topographic depression. These solutions were restricted to critical flow, when the Froude number is unity. Several questions were raised, including i) what happens to the solution structure as the flow speed is continued into the supercritical regime? and ii) how does the decay rate of the topography affect the solution structure? In this article, we answer these questions by examining exact steady solutions of the fKdV model and numerical solutions of the fully nonlinear Euler system over i) trench, ii) Gaussian and iii) generalised Agnesi topographies. We uncover a highly non-trivial bifurcation structure consisting of spirals and closed loops, characterised by the phenomenon of `homoclinic shedding', where pairs of solitary waves are emitted up- and downstream as the amplitude is varied.