  We show that the Bose-glass phase of a one-dimensional disordered Bose fluid
exhibits a chaotic behavior, i.e., an extreme sensitivity to external
parameters. Using bosonization, the replica formalism and the nonperturbative
functional renormalization group, we find that the ground state is unstable to
any modification of the disorder configuration ("disorder" chaos) or variation
of the Luttinger parameter ("quantum" chaos, analog to the "temperature" chaos
in classical disordered systems). This result is obtained by considering two
copies of the system, with slightly different disorder configurations or
Luttinger parameters, and showing that inter-copy statistical correlations are
suppressed at length scales larger than an overlap length
$\xi_{\mathrm{ov}}\sim |\epsilon|^{-1/\alpha}$ ($|\epsilon|\ll 1$ is a measure
of the difference between the disorder distributions or Luttinger parameters of
the two copies). The chaos exponent $\alpha$ can be obtained by computing
$\xi_{\mathrm{ov}}$ or by studying the instability of the Bose-glass fixed
point for the two-copy system when $\epsilon\neq 0$. The renormalized,
functional, inter-copy disorder correlator departs from its fixed-point value
-- characterized by cuspy singularities -- via a chaos boundary layer, in the
same way as it approaches the Bose-glass fixed point when $\epsilon=0$ through
a quantum boundary layer. Performing a linear analysis of perturbations about
the Bose-glass fixed point, we find $\alpha=1$.
