Smooth stationary water waves with multiple vortex spikes

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.