Non–existence of solitary water waves in three dimensions

Abstract

In the subject of free–surface water waves, solitary waves play an important role in the theory of two–dimensional fluid motions. These are steady solutions to the Euler equations that are localized, positively elevated above the mean fluid level and travelling at velocities with supercritical Froude number. They provide a stable mechanism in bodies of water for transport of mass, momentum and energy over long distances. In this paper, we prove that in the three– (or higher–) dimensional problem of surface water waves, there do not exist any localized steady positive solutions to the Euler equations.