Integral and Asymptotic Properties of Solitary Waves in Deep Water

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 that the "excess mass" vanishes.

Existence, Nonexistence, and Asymptotics of Deep Water Solitary Waves with Localized Vorticity

“…This generalizes the recent work of Wheeler [35] on the irrotational case. In the twodimensional irrotational and vortex sheet cases, Sun [30] established similar decay rates (but not asymptotics) under analogous assumptions.…”

“…Formulas (1.12) and (1.13) agree in the limit where ω and ρ + vanish. In this case they recover the formula obtained by Wheeler in [35]; also see [20]. With nontrivial vorticity in the bulk, it becomes considerably more difficult to find clean expressions for p, as the definition of V is not in any way related to the location of the interface S.…”

“…For both capillary-gravity and gravity waves in two dimensions, Sun [30] showed that the decay η = O(1/|x ′ | 1+ε ) automatically improves to η = O(1/|x ′ | 2 ), and in this case ruled out the existence of waves of pure elevation or depression. Wheeler [35] obtained similar results in three dimensions, and also found leading-order expressions for the asymptotic form of the waves.…”

“…For two-dimensional irrotational solitary waves and vortex sheets in infinite depth, this was proved by Sun [30], and in the three-dimensional irrotational case it appears in [35]. With strong surface tension (σ > |c| 2 /4g), Sun is able to assume even weaker decay than (1.8).…”

“…The main idea behind this argument is that the asymptotic properties (3.7) can be determined via elliptic theory from the kinematic boundary condition, the equation satisfied by ϕ, and the decay assumption (1.8). In comparison to the irrotational case considered in [35], there are additional terms coming from the vortical contribution V that must be understood using the asymptotic information contained in Lemma 3.1 and Corollary 3.2.…”