Convexity of Stokes waves of extreme form (Q1879438)

The authors establish existence of a periodic Stokes waves of extreme form which has a convex profile between consecutive stagnation points where there is a corner with contact angle of \(120^\circ\). The methods reduce the problem first to finding a holomorphic function with simultaneously prescribed Dirichlet and Neumann boundary conditions on a piecewise-convex, periodic, planar curve. The problem is then rewritten as an integral equation, following Nekrasov's theory. Finally, the integral equation is homotoped to a simpler form, where existence of a convex solution is explicitly known. The major difficulty lies in preserving convexity during the homotopy.