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 nontrivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $Fr$ (a non-dimensional wave speed) plays the central role. So far, the best analytic result Fr < √2 was obtained by Starr (1947), while the numerical evidence of Longuet-Higgins & Fenton (1974) states Fr ≤ 1.294. On the other hand, as shown recently by Kozlov (2023), the upper bound $Fr < 1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy utilizing the flow force function and rigorously establish the improved upper bound $Fr < 1.37838$.