Printed in Great Britain 235 Instability and breaking of a solitary wave

The result of a linear stability calculation of solitary waves which propagate steadily along the free surface of a liquid layer of constant depth is examined numerically by employing a time-stepping scheme based on a boundary-integral method. The initial growth rate that is found for sufficiently small perturbations agrees well with the growth rate expected from the linear stability calculation. In calculating the later ‘nonlinear ’ stage of the instability, it is found that two distinct types of long-time evolution are possible. These depend only on the sign of the unstable normal-mode perturbatioli that is superimposed initially on the steady wave. The growth of the perturbation ultimately leads to breaking for one sign. Unexpectedly, for the opposite sign, there is a monotonic decrease in the total height of the wave. In this latter case there is a smooth evolution to a stable solitary wave of lesser amplitude but very nearly the same energy. 1