  We study elastic shear waves of small but finite amplitude, composed of an
anti-plane shear motion and a general in-plane motion. We use a multiple scales
expansion to derive an asymptotic system of coupled nonlinear equations
describing their propagation in all isotropic incompressible non-linear elastic
solids, generalizing the scalar Zabolotskaya equation of compressible nonlinear
elasticity. We show that for a general isotropic incompressible solid, the
coupling between anti-plane and in-plane motions cannot be undone and thus
conclude that linear polarization is impossible for general nonlinear
two-dimensional shear waves. We then use the equations to study the evolution
of a nonlinear Gaussian beam in a soft solid: we show that a pure (linearly
polarised) shear beam source generates only odd harmonics, but that introducing
a slight in-plane noise in the source signal leads to a second harmonic, of the
same magnitude as the fifth harmonic, a phenomenon recently observed
experimentally. Finally, we present examples of some special shear motions with
linear polarisation.
