Long time dynamics of space periodic water waves

TL;DR

This paper reviews recent progress in understanding the long-term behavior of space-periodic water waves, including bifurcation phenomena, well-posedness, and stability, using advanced mathematical theories for Hamiltonian PDEs.

Contribution

It synthesizes recent developments employing novel KAM, Birkhoff normal form, and symplectic perturbation techniques for analyzing water wave dynamics.

Findings

Bifurcation of quasi-periodic solutions identified

Long-time well-posedness established

Modulational instability of Stokes waves analyzed

Abstract

We review recent advances regarding the long-time dynamics of space-periodic water waves, focusing on 1) bifurcation of quasi-periodic solutions, both standing and traveling; 2) long-time well-posedness results; 3) modulational instability of Stokes waves. These results rely on unconventional approaches to KAM and Birkhoff normal form theories for Hamiltonian quasi-linear PDEs and symplectic Kato perturbation theory for separated eigenvalues of reversible and Hamiltonian operators.