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The Euler’s equations describe the motion of inviscid fluid. In the case of shallow water, when a perturbative asymtotic expansion of the Euler’s equations is taken (to a certain order of smallness of the scale parameters), relations to certain integrable equations emerge. Some recent results concerning the use of integrable equation in modeling the motion of shallow water waves are reviewed in this contribution.
Ivanov, R. (2007). Water Waves and Integrability. Philosophical Transactions of the Royal Society A, vol. 365, pg. 2267—2280. doi:10.1098/rsta.2007.2007