T. R. Akylas

On asymmetric gravity–capillary solitary waves

Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrodinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.

Envelope solitons with stationary crests

Recent analytical and numerical work has shown that gravity–capillary surface waves as well as other dispersive wave systems support symmetric solitary waves with decaying oscillatory tails, which bifurcate from linear periodic waves at an extremum value of the phase speed. It is pointed out here that, for small amplitudes, these solitary waves can be interpreted as particular envelope‐soliton solutions of the nonlinear Schrodinger equation, such that the wave crests are stationary in the reference frame of the wave envelope. Accordingly, these waves (and their three‐dimensional extensions) are expected to be unstable to oblique perturbations.

On gravity–capillary lumps

Two-dimensional (plane) solitary waves on the surface of water are known to bifurcate from linear sinusoidal wavetrains at specific wavenumbers

Solitary internal waves in a rotating channel: A numerical study

k\,{=}\,k_{0}

Stability of steep gravity–capillary solitary waves in deep water

where the phase speed

On the formation of bound states by interacting nonlocal solitary waves

c(k)

WEAKLY NONLOCAL GRAVITY-CAPILLARY SOLITARY WAVES

attains an extremum

On the excitation of nonlinear water waves by a moving pressure distribution oscillating at resonant frequency

(\left. \hbox{d}c/\hbox{d}k \right |_{0}\,{=}\,0)

On the stability of lumps and wave collapse in water waves

and equals the group speed. In particular, such an extremum occurs in the long-wave limit

From non-local gap solitary waves to bound states in periodic media

k_{0}\,{=}\,0