ELEVATION SOLITARY WAVES WITH SURFACE-TENSION

Elevation solitary waves propagating at the surface of a fluid of finite depth are computed numerically by a boundary integral equation method. Both gravity and surface tension are included in the dynamic boundary condition. The solitary waves are approximated by long periodic waves whose wavelengths are about 100 times the depth of the fluid. Previous numerical and analytical results are confirmed and extended. For a given value of the Bond number 0 < tau < 1/3, there is a two-parameter family of waves with a train of ripples in the far field. It is shown that, for sufficiently small solitary waves, there are solutions for which the amplitude of the ripples vanish. These particular solutions form, for each value of 0 < tau < 1/3, a one-parameter family of solutions. Moreover they are accurately described by the classical solution of the Korteweg-de Vries equation.