Instability and breaking of a solitary wave

Abstract

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 perturbation 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.