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.
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