ON ASYMMETRIC GRAVITY-CAPILLARY SOLITARY WAVES

Symmetric gravity-capillary solitary waves with decaying oscillatory t ails are known to bifurcate from infinitesimal periodic waves at the m inimum value of the phase speed where the group velocity is equal to t he phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are de scribed 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 possibili ty is examined here on the basis of the fifth-order Korteweg-de Vries (KdV) equation, a model for gravity-capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of expon ential asymptotics beyond all orders of the NLS theory, it is shown th at asymmetric solitary waves of the form suggested by the NLS theory i n fact are not possible. On the other hand, an infinity of symmetric a nd asymmetric solitary-wave solution families comprising two or more N LS solitary wavepackets bifurcate at finite values of the amplitude pa rameter. The asymptotic results are consistent with numerical solution s of the fifth-order KdV equation. Moreover, the asymptotic theory sug gests that such multi-packet gravity-capillary solitary waves also exi st in the full water-wave problem near the minimum of the phase speed.