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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 Fr (a non-dimensional wave speed) plays the central role. So far, the best analytical result Fr < √2 was obtained by Starr (1947 J. Mar. Res., vol. 6, pp. 175–193), while the numerical ev
