Professor Triantafyllos Akylas on: Exponential asymptotics: applications to water waves
Abstract: This Lecture will focus on applications of the wavenumber exponential asymptotics technique introduced in Lecture I, to two problems of water waves: (i) gravity-capillary solitary waves in shallow water for Bond number < 1/3; (ii) bifurcation of gravity-capillary solitary waves at the minimum phase speed on water of finite or infinite depth.
In (i), the classical weakly nonlinear long-wave expansion fails to capture the short-scale oscillations which arise at the tails due to capillary waves that propagate at the same speed as the main hump. Using the wavenumber technique, these tails are computed asymptotically, in good agreement with direct numerical computations. Thus, in the presence of small surface tension (Bond < 1/3), solitary waves on shallow water are non-local.
In (ii), the bifurcating solitary waves are in the form of envelope solitons of the nonlinear Schrödinger (NLS) equation with carrier (crests) propagating at the same speed as the envelope. Here, exponential asymptotics is required in order to specify the phase of the crests relative to the envelope. It is shown that only two branches of symmetric solitary waves bifurcate at zero amplitude. By contrast, there is a rich variety of symmetric and asymmetric solitary wave solutions that bifurcate at small but finite amplitude. The asymptotic predictions are confirmed by direct numerical computations.