– Alam, M.-R., Liu, W and Haller, G. “Closed Loop Separation Control: An Analytic Approach”, Physics of Fluids, 18 (4): Art. No. 043601 APR 2006. (PDF)
It is known that when a boat advances near critical speed in a wide canal, solitons can be radiated upstream in a transient manner. Real canals are not flat, therefore we considered a canal with a randomly rough bed. Governing equations were shown to be in the form of forced-damped KdV equation and numerical evaluations showed less transient behavior in forces and moments. However, striking results emerged when we considered an array of ships advancing in the canal. When a caravan of ships moves over a flat (with no bathymetry undulation) canal, bow and stern waves unsteadily interact resulting in irregular and extreme forces and moments from time to time. But surprisingly, if roughness is taken into account and in a relatively short transient time, the bow wave of a front ship upon encountering and interacting with the stern wave of a back ship forms a smooth steady pattern: another miracle of nonlinear equations.
If you are wondering where on the planet we may get an array of ships moving in a canal, here are a couple of examples: The left photo depicts early steam ships in the Suez canal (about 1880) and the right photo is a picture supplied by Stella Elliott (BBC) USS AMERICA (CV 66) during its transit through the Suez canal.
Right: Ship caravan over a flat bottom, Left: Ship Caravan over a randomly rough bottom (for more information see the references)