  We consider the generalized hydrodynamics including the recently introduced
diffusion term for an initially inhomogeneous state in the Lieb-Liniger model.
We construct a general solution to the linearized hydrodynamics equation in
terms of the eigenstates of the evolution operator and study two prototypical
classes of initial states: delocalized and localized spatially. We exhibit some
general features of the resulting dynamics, among them, we highlight the
difference between the ballistic and diffusive evolution. The first one governs
a spatial scrambling, the second, a scrambling of the quasi-particles content.
We also go one step beyond the linear regime and discuss the evolution of the
zero momentum mode that does not evolve in the linear regime.
