  Dark matter models involving a very light bosonic particle, generally known
as Fuzzy Dark Matter (FDM), have been recently attracting great interest in the
cosmology community, as their wave-like phenomenology would simultaneously
explain the longstanding mis-detection of a dark matter particle and help
easing the small-scale issues related to the standard Cold Dark Matter (CDM)
scenario. With the present work, we initiate a series of papers aiming at
investigating the evolution of FDM structures in a cosmological framework
performed with our N-body code AX-GADGET, detailing for the first time in the
literature how the actual scaling relations between solitonic cores and host
haloes properties are significantly affected by the dynamical state, morphology
and merger history of the individual systems. In particular, in this first
paper we confirm the ability of AX-GADGET to correctly reproduce the typical
FDM solitonic core and we employ it to study the non-linear evolution of eight
FDM haloes in their cosmological context through the zoom-in simulation
approach. We find that the scaling relations identified in previous works for
isolated systems are generally modified for haloes evolving in a realistic
cosmological environment, and appear to be valid only as a limit for the most
relaxed and spherically symmetric systems.
