Spontaneous instability of nonlinear surface waves lying on the upper solution branch of Figure 12(a), where and (interface mismatch parameters are and ). (a) Evolution in of the peak amplitude of the beam. (b), (c), and (d) show the full numerical solution of (5) when the nonlinearity exponent is , 2, and 3, respectively. Note that the profiles of the input waves in (b) and (d) correspond to the upper-branch solutions shown in Figure 11(a).