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A fundamental question in the study of water waves is the existence andstability of solitary waves. Solitary waves have been proved to exist and havebeen studied in many interesting situations, and often arise from the balanceof different forces/factors influencing the fluid dynamics, e.g. gravity,surface tension or the fluid bottom. However, the existence of solitary waveshas remained an open problem in two of the simplest cases, namely for eitherpure gravity waves or pure capillary waves in infinite depth. In this articlewe settle both of these questions in two space dimensions. Precisely, weconsider incompressible, irrotational, infinite depth water wave equation,either with gravity and without surface tension, or without gravity but withsurface tension. In both of these cases we prove that there are no solitarywaves.

No solitary waves in 2-d gravity and capillary waves in deep water