  Bound states in the continuum (BICs) are non-radiating solutions of the wave
equation with a spectrum embedded in the continuum of propagating waves of the
surrounding space. The complete decoupling of BICs from the radiation continuum
makes their excitation impossible from the far-field. Here, we develop a
general theory of parametric excitation of BICs in nonlinear systems with
Kerr-type nonlinearity via spontaneous symmetry breaking, which results in a
coupling of a BIC and a bright mode of the system. Using the temporal
coupled-mode theory and perturbation analysis, we found the threshold intensity
for excitation of a BIC and study the possible stable and unstable solutions
depending on the pump intensity and frequency detuning between the pump and
BIC. We revealed that at some parameters of the pump beam, there are no stable
solutions and the BIC can be used for frequency comb generation. Our findings
can be very promising for use in nonlinear photonic devices and all-optical
networks.
