  We consider an impurity in a one-dimensional weakly-interacting Bose gas and
analytically calculate the density profile of the Bose gas. Within the
mean-field approximation, by increasing the distance from the impurity, the
Bose gas density saturates exponentially fast to its mean thermodynamic-limit
value at distances beyond the healing length. The effect of quantum
fluctuations drastically changes this behavior, leading to a power law decay of
the density deviation from the mean density. At distances longer than the
healing length and shorter than a new length scale proportional to the impurity
coupling strength, the power-law exponent is $2$, while at longest distances
the corresponding exponent becomes $3$. The latter crossover does not exist in
two special cases. The first one is realized for infinitely strongly coupled
impurity; then the density deviation always decays with the exponent $2$. The
second special case occurs when the new length scale is smaller than the
healing length, i.e., at weak impurity coupling; then the density deviation
always decays with the exponent $3$. The obtained results are exact in the
impurity coupling strength and account for the leading order in the interaction
between the particles of the Bose gas.
