What is it about?

Waves in deep water retain much of their mysteries, in spite of the large amount of work on them since the discovery of their equations of motion by Euler in the 18th century. The idealized case of inviscid irrotational deep-water gravity waves in one dimension is a much studied subject in mathematical physics. Beyond its conceptual simplicity, a large part of its appeal comes from the fact that in some sense it comes close to integrability, i.e. explicit solutions at all times could be obtained for an arbitrary initial data. In this paper, we give rather strong evidence that this may indeed be the case by exhibiting a set of new conservation laws beyond energy and momentum, a characteristic of such integrable systems.

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Why is it important?

This might lead to a proof of the Integrability of this system of partial integro-differential equations which has no immediate analogue so far, and presumably to the discovery of new mathematical structures.


Much work remains to explore the possibilities which are obvious from the article. For example push the calculation of the conservation laws to higher orders in perturbation in the amplitude of the surface displacement, find relations between these conservation laws, find new ones, etc.

André Neveu
Universite de Montpellier

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This page is a summary of: Perturbatively conserved higher nonlocal integral invariants of free-surface deep-water gravity waves, Physics of Fluids, March 2021, American Institute of Physics,
DOI: 10.1063/5.0039868.
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