On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

Abstract

We study the bifurcation of periodic travelling waves of the capillary–gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary–gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov–Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and, therefore, positive and convex; the threshold value for this to happen turns out to be $$T = \frac{4}{\pi ^2}$$, not the critical Bond number $$\frac{1}{3}$$), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in $$L^2(\mathbb {S})$$ (but not in $$L^\infty $$) towards one of the two constant solution curves.