Integral and asymptotic properties of solitary waves in deep water

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.