Rectification of a deep water model for surface gravity waves

Vincent Duchêne (1) , Benjamin Melinand (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this work we discuss an approximate model for the propagation of deep irrotational water waves; specifically the model obtained when retaining only quadratic nonlinearities in the water waves system under the Zakharov-Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite-regularity spaces, explaining spurious oscillations reported in numerical simulations for instance in GN07, and thus agreeing with the conclusion of ABN14 although not on the proposed instability mechanism. We show that the system can be "rectified". Indeed, through the introduction of suitable regularizing operators we can recover well-posedness properties without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. Our study is supported with detailed and reproducible numerical simulations.

Mots clés: Water waves; asymptotic model; ill-posedness; regularization; numerical experiments

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Duchêne, Benjamin Melinand. Rectification of a deep water model for surface gravity waves. 2022. ⟨hal-03598747v1⟩