An improved upper bound for the Froude number of irrotational solitary water waves

A classical and central problem in the theory of water waves is to classify parameter regimes for which non-trivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $ \textit{Fr}$ (a non-dimensional wave speed) plays the central role. So far, the best analytical result $ \textit{Fr} \lt \sqrt {2}$ was obtained by Starr (1947 J. Mar. Res. , vol. 6 , pp. 175–193), while the numerical evidence of Longuet-Higgins & Fenton (1974 Proc. A , vol. 340 , pp. 471–493) states $ \textit{Fr} \leq 1.294$ . On the other hand, as shown recently by Kozlov (2023 On the first bifurcation of Stokes waves), the hypothetical upper bound $ \textit{Fr} \lt 1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy and rigorously establish the improved upper bound $ \textit{Fr} \lt 1.3451$ , which is the first rigorous improvement of Starr’s bound. In this process, we establish several new inequalities for the relative horizontal velocity, which are of separate interest and for which we delicately make use of the bound on the slope of the surface profile established by Amick (1987 Arch. Ration. Mech. Anal. , vol. 99 , pp. 91–114). As an application we show that the velocity at the bottom below the crest of any solitary wave does not exceed $47\,\%$ of the propagation speed.