  We consider nonlinear effects in scattering of light by a periodic structure
supporting optical bound states in the continuum. In the spectral vicinity of
the bound states the scattered electromagnetic field is resonantly enhanced
triggering optical bistability. Using coupled mode approach we derive a
nonlinear equation for the amplitude of the resonant mode associated with the
bound state. We show that such an equation for the isolated resonance can be
easily solved yielding bistable solutions which are in quantitative agreement
with the full-wave solutions of Maxwell's equations. The coupled mode approach
allowed us to cast the the problem into the form of a driven nonlinear
oscillator and analyze the onset of bistability under variation of the incident
wave. The results presented drastically simplify the analysis nonlinear
Maxwell's equations and, thus, can be instrumental in engineering optical
response via bound states in the continuum.
