Integral and asymptotic properties of solitary waves in deep water

TL;DR

This paper analyzes the asymptotic behavior of solitary water waves in deep water, revealing their dipole-like decay, relating dipole moments to kinetic energy, and ruling out certain wave types.

Contribution

It establishes the dipole asymptotics for velocity potential, links dipole moments to kinetic energy, and rules out pure elevation or depression waves in deep water.

Findings

Velocity potential behaves like a dipole at infinity.

Dipole moment is related to the kinetic energy.

Pure elevation or depression waves are ruled out.

Abstract

We consider two- and three-dimensional gravity and gravity-capillary solitary water waves in infinite depth. Assuming algebraic decay rates for the free surface and velocity potential, we show that the velocity potential necessarily behaves like a dipole at infinity and obtain a related asymptotic formula for the free surface. We then prove an identity relating the "dipole moment" to the kinetic energy. This implies that the leading-order terms in the asymptotics are nonvanishing and in particular that the angular momentum is infinite. Lastly we prove a related integral identity which rules out waves of pure elevation or pure depression.