No solitary waves exist on 2D deep water

TL;DR

This paper proves that under certain conditions, no solitary wave solutions exist for two-dimensional deep water flows governed by gravity, using a nonlinear pseudodifferential equation and Pohozaev identity.

Contribution

It establishes a nonexistence result for solitary waves in 2D deep water flows, a significant theoretical insight in fluid dynamics.

Findings

No solutions decay faster than linearly at infinity.

Solitary waves cannot exist in the specified 2D deep water setting.

The proof employs a Pohozaev identity for the governing equation.

Abstract

The solitary wave problem at the free surface of a two-dimensional, infinitely-deep and irrotational flow of water, under the influence of gravity, is formulated as a nonlinear pseudodifferential equation. A Pohozaev identity is used to show that it admits no solutions which asymptotically vanish faster than linearly.