  We solve the mixing-demixing transition in repulsive one-dimensional
bose-bose mixtures. This is done numerically by means of the continuous matrix
product states variational ansatz. We show that the effective low-energy
bosonization theory is able to detect the transition whenever the Luttinger
parameters are exactly computed. We further characterize the transition by
calculating the ground-state energy density, the field-field fluctuations and
the density correlations.
