  Dual-unitary quantum circuits have recently attracted attention as an
analytically tractable model of many-body quantum dynamics. Consisting of a
1+1D lattice of 2-qudit gates arranged in a 'brickwork' pattern, these models
are defined by the constraint that each gate must remain unitary under swapping
the roles of space and time. This dual-unitarity restricts the dynamics of
local operators in these circuits: the support of any such operator must grow
at the effective speed of light of the system, along one or both of the edges
of a causal light cone set by the geometry of the circuit. Using this property,
it is shown here that for 1+1D dual-unitary circuits the set of width-$w$
conserved densities (constructed from operators supported over $w$ consecutive
sites) is in one-to-one correspondence with the set of width-$w$ solitons -
operators which, up to a multiplicative phase, are simply spatially translated
at the effective speed of light by the dual-unitary dynamics. A number of ways
to construct these many-body solitons (explicitly in the case where the local
Hilbert space dimension $d=2$) are then demonstrated: firstly, via a simple
construction involving products of smaller, constituent solitons; and secondly,
via a construction which cannot be understood as simply in terms of products of
smaller solitons, but which does have a neat interpretation in terms of
products of fermions under a Jordan-Wigner transformation. This provides
partial progress towards a characterisation of the microscopic structure of
complex many-body solitons (in dual-unitary circuits on qubits), whilst also
establishing a link between fermionic models and dual-unitary circuits,
advancing our understanding of what kinds of physics can be explored in this
framework.
