  We establish dispersion engineering rules that allow us to control the
thermalization process and the thermal state of an initial beam propagating in
a multimode nonlinear photonic circuit. To this end, we have implemented a
kinetic equation (KE) approach in systems whose Bloch dispersion relation
exhibits bands and gaps. When the ratio between the gap-width to the band-width
is larger than a critical value, the KE has stationary solutions which differ
from the standard Rayleigh-Jeans (RJ) distribution. The theory also predicts
the relaxation times above which such non-conventional thermal states occur. We
have tested the validity of our results for the prototype SSH model whose
connectivity between the composite elements allows to control the band-gap
structure. These spectral engineering rules can be extended to more complex
photonic networks that lack periodicity but their spectra consist of groups of
modes that are separated by spectral gaps.
