Stability of Solitary Capillary-Gravity Water Waves in Three Dimensions

TL;DR

This paper proves the conditional orbital stability of localized three-dimensional capillary-gravity water waves with strong surface tension, using spectral analysis and Hamiltonian structure considerations.

Contribution

It introduces a novel stability analysis for fully localized solitary waves in 3D water waves, extending the Grillakis-Shatah-Strauss framework to this context.

Findings

Established conditional orbital stability of 3D solitary waves

Developed spectral analysis techniques for linearized dynamics

Extended stability framework to non-energy minimizer waves

Abstract

This paper establishes the conditional orbital stability of fully localized solitary waves for the three-dimensional capillary-gravity water wave problem in finite depth under strong surface tension. The waves, constructed via a non-variational Lyapunov-Schmidt reduction in [26], are not energy minimizers and thus require a direct stability analysis. We adapt the Grillakis-Shatah-Strauss framework within Mielke's approach to handle the mismatch between well-posedness and energy spaces. The proof relies on spectral analysis of the linearized dynamics and careful treatment of the Hamiltonian structure defined by the energy and momentum functionals.