Stability of Solitary Capillary-Gravity Water Waves in Three Dimensions
Changfeng Gui, Shanfa Lai, Yong Liu, Juncheng Wei, Wen Yang
TL;DR
The paper tackles the conditional orbital stability of fully localized three-dimensional capillary-gravity solitary waves in finite depth, constructed non-variationally via a Lyapunov–Schmidt reduction. By marrying Grillakis–Shatah–Strauss with Mielke's conditional stability framework and exploiting the Hamiltonian structure given by the Dirichlet-to-Neumann operator, it derives a reduced augmented potential and a mixed local–nonlocal linearized operator whose spectral properties drive the stability analysis. In the small-amplitude, high-surface-tension KP-I regime, the authors perform a precise spectral analysis: the rescaled operator converges to a KP-I–type limit with a single negative eigenvalue and a two-dimensional neutral spectrum from translations, while the remainder is positive; they then establish convexity $d''(c)>0$ and construct modulation to obtain conditional orbital stability of the solitary waves, up to horizontal translations. This work extends stability theory to non-variational 3D capillary-gravity waves, linking KP-I lump dynamics with finite-depth water waves and offering a rigorous stability framework for reduction-based solitary 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.