  Weyl semimetals (WSMs) have unusual optical response originated from unique
topological properties of their bulk and surface electron states. Their
third-order optical nonlinearity is expected to be very strong, especially at
long wavelengths, due to linear dispersion and high Fermi velocity of
three-dimensional Weyl fermions. Here we derive the third-order nonlinear
optical conductivity of WSMs in the long-wavelength limit and calculate the
intensity of the nonlinear four-wave mixing signal as it is transmitted through
the WSM film or propagates away from the surface of the material in the
reflection geometry. All results are analytic and show the scaling of the
signal intensity with variation of all relevant parameters. The nonlinear
generation efficiency turns out to be surprisingly high for a lossy material,
of the order of several mW per W$^3$ of the incident pump power. Optimal
conditions for maximizing the nonlinear signal are realized in the vicinity of
bulk plasma resonance. This indicates that ultrathin WSM films of the order of
skin depth in thickness could find applications in compact optoelectronic
devices.
