<p>Several new results on the bifurcation and instability of nonlinear periodic travelling waves, at the interface between two fluids in relative motion, in a parametric neighbourhood of a Kelvin–Helmholtz unstable equilibrium are presented. The organizing centre for the analysis is a canonical Hamiltonian formulation of the Kelvin–Helmholtz problem presented in Part 1. When the density ratio of the upper and lower fluid layers exceeds a critical value, and surface tension is present, a pervasive superharmonic instability is found, and as <i>u</i> [rightward arrow] <i>u</i><sub>0</sub>, where <i>u</i> is the velocity difference between the two layers and u0 is the Kelvin–Helmholtz threshold, the amplitude at which the superharmonic instability occurs scales like (<i>u</i><sub>0</sub> - <i>u</i>)<sup>1/2</sup> with <i>u</i> > <i>u</i><sub>0</sub>. Other results presented herein include <i>(a)</i> new results on the structure of the superharmonic instability, <i>(b)</i> the discovery of isolated branches and intersecting branches of travelling waves near a critical density ratio, <i>(c)</i> the appearance of Benjamin–Feir instability along branches of waves near the Kelvin–Helmholtz instability threshold and <i>(d)</i> the interaction between the Kelvin–Helmholtz, superharmonic and Benjamin–Feir instability at low amplitude.</p
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