  The linear energy dispersion of graphene electrons leads to a strongly
nonlinear electromagnetic response of this material. We develop a general
quantum theory of the third-order nonlinear local dynamic conductivity of
graphene $\sigma_{\alpha\beta\gamma\delta}(\omega_1,\omega_2,\omega_3)$, which
describes its nonlinear response to a uniform electromagnetic radiation. The
derived analytical formulas describe a large number of different nonlinear
phenomena such as the third harmonic generation, the four wave mixing, the
saturable absorption, the second harmonic generation stimulated by a dc
electric current, etc., which may be used in different terahertz and
optoelectronic devices.
