Mathematical Formulation of Non-Linear Surface Stress Localization and Catenary Soliton Geometry over Bounded Free Fluid Boundaries

The spatial localization and non-linear geometric asymmetry of finite-amplitude waves on a free fluid surface have long remained fundamental questions in non-linear fluid mechanics. The non-linear decomposition established within this framework projects the fluid velocity potential into orthogonal couplings of longitudinal and transverse components, thereby resolving the spatial localization of three-dimensional short-crested waves. Through a high-order Stokes perturbation expansion, the structural profile asymmetry between sharp crests and flattened troughs is analytically derived. Furthermore, the transverse boundary energy dissipation and membrane tension balance, which arise from the concentration of the stress tensor principal axes at the wave crest singular points, are shown to reduce the two-dimensional non-linear Schrödinger (2D-NLS) equation to a non-linear ordinary differential equation. The exact solution is uniquely determined as a hyperbolic secant function, representing the inverse catenary geometric profile. Finally, the stationarity and structural stability of this solitary wave configuration are verified through its exotic Fourier invariance in multi-dimensional wavenumber hyperspaces, establishing a complete mathematical closure for localized non-linear fluid structures.