Babenko's equation for periodic gravity waves on water of finite depth: derivation and numerical solution

The nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko's equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $2 \pi$-periodic Hilbert transform $\mathcal{C}$ used in the equation for deep water, the equation obtained here contains a certain operator $\mathcal{B}_r$, which is the sum of $\mathcal{C}$ and a compact operator whose dependence on the parameter involves on the depth of water. Numerical computations are based on an equivalent form of Babenko's equation derived by virtue of the spectral decomposition of the operator $\mathcal{B}_r \D / \D t$. Bifurcation curves and wave profiles of the extreme form are obtained numerically.