  Optical bound state in the continuum (BIC) is characterized by infinitely
high quality factor resulting in drastic enhancement of light-matter
interaction phenomena. We study the optical response of a one-dimensional
photonic crystal slab with Kerr focusing nonlinearity in the vicinity of BIC
analytically and numerically. We predict a strong nonlinear response including
multistable behaviour, self-tuning of BIC to the frequency of incident wave,
and breaking of symmetry protected BIC. We show that all of these phenomena can
be observed in silicon photonic structure at the pump power of several
$\mu$W/cm$^2$. We also analyze the modulation instability of the obtained
solutions and the effect of the finite size of the structure on the stability.
Our findings have strong implications for nonlinear photonics and integrated
optical circuits.
