2014/11/20 by David M. Ambrose, Walter A. Strauss, Ambrose, David M. +3 · 1 citation
Earth and Planetary Sciences · #Analysis of PDEs (math.AP) #Coastal and Marine Dynamics #FOS: Mathematics #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.1411.5569
openalex publication_date 2014/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the global bifurcation problem for spatially periodic traveling\nwaves for two-dimensional gravity-capillary vortex sheets. The two fluids have\narbitrary constant, non-negative densities (not both zero), the gravity\nparameter can be positive, negative, or zero, and the surface tension parameter\nis positive. Thus, included in the parameter set are the cases of pure\ncapillary water waves and gravity-capillary water waves. Our choice of\ncoordinates allows for the possibility that the fluid interface is not a graph\nover the horizontal. We use a technical reformulation which converts the\ntraveling wave equations into a system of the form "identity plus compact."\nRabinowitz' global bifurcation theorem is applied and the final conclusion is\nthe existence of either a closed loop of solutions, or an unbounded set of\nnontrivial traveling wave solutions which contains waves which may move\narbitrarily fast, become arbitrarily long, form singularities in the vorticity\nor curvature, or whose interfaces self-intersect.\n