Quasi-periodic standing wave solutions of gravity-capillary water waves

Abstract

We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $x$) of a $2$-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

Revised version to appear in Memoirs of the American Mathematical Society, MEMO 891

References in corpus (3)

• KAM for autonomous quasi-linear perturbations of mKdV
• Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations
• Gravity capillary standing water waves

Cited by in corpus (8)

• Traveling quasi-periodic water waves with constant vorticity
• Full description of Benjamin-Feir instability of Stokes waves in deep water
• Spatially quasi-periodic water waves of infinite depth
• Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions
• On the stability of periodic multi-solitons of the KdV equation
• On the construction of Sobolev Almost periodic invariant tori for the 1d NLS
• Reducibility of relativistic Schrödinger equation with unbounded perturbations
• 2N-Dimensional Canonical Systems and Applications