Year: 2026
Type: preprint
Abstract: For any prescribed positive integer $n$, we establish the existence of a family of stationary solutions to the two dimensional capillary-gravity water wave equations in finite depth, whose stream functions exhibit $n$ well-separated vertically aligned vortex spikes with alternating signs. The construction is based on a nonlinear gluing scheme for a singularly perturbed semilinear elliptic equation coupled to the Bernoulli equation on the free surface. Our results extend the single-spike construction to arbitrary finite multi-spike configurations and reveal a mechanism through which exponentially localized vortex spikes can coexist with an almost flat free surface.
Source: arXiv (Cornell University)
Authors: Fan, Zhiqiang, Wang, Yuchen, Zheng, Youquan
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223907323