Nonlinear waves and one-dimensional solitons of the Zakharov–Kuznetsov equation are unstable in two dimensions. Although the wavevector K of a perturbation leading to an instability covers a whole region in (Kx, Ky) parameter space, two classes are of particular interest. One corresponds to the perpendicular, Benjamin–Feir instability (Kx = 0). The second is the wave-length-doubling instability. These two are the only purely growing modes. We concentrate on them. Both analytical and numerical methods for calculating growth rates are employed and results compared. Once a nonlinear wave or soliton breaks up owing to one of these instabilities, an array of cylindrical and/or spherical solitons can emerge. We investigate the interaction of these entities numerically.