  Dissipative Kerr solitons (DKSs) in optical microresonators have been
intensely studied from the perspective of both fundamental nonlinear physics
and portable and low power technological applications in communications,
sensing, and metrology. In parallel, synthetic dimensions offer the promise of
studying physical phenomena with a dimensionality beyond that imposed by
geometry, and have been implemented in optics. The interplay of DKS physics
with synthetic dimensions promises to unveil numerous new physical and
technological insights, yet many fundamental challenges remain. In particular,
DKSs intrinsically rely on dispersion to exist while the creation of synthetic
frequency lattices typically needs a dispersion-less system. We present a
change of paradigm with the creation of a synthetic frequency lattice in the
eigenfrequency space of a dispersive multi-color soliton through all-optical
nonlinear coupling -- compatible with octave spanning microcombs -- harnessing
the interplay between the cavity dispersion and the dispersion-less nature of
the DKS. We examine theoretically and experimentally the nonlinear coupling
mechanism in a 1~THz repetition rate resonator and demonstrate four-wave mixing
Bragg scattering between the different wavepackets forming the multi-color
soliton, with the microcomb ranging over 150~THz, yielding a complex
all-optical and integrated synthetic frequency lattice.
