  We explore the coherent thermal transport sustained by solitons through a
long Josephson junction, as a thermal gradient across the system is
established. We observe that a soliton causes the heat current through the
system to increase. Correspondingly, the junction warms up in correspondence of
the soliton, with temperature peaks up to, e.g., approximately 56 mK for a
realistic Nb-based proposed setup at a bath temperature Tbath = 4.2 K. The
thermal effects on the dynamics of the soliton are also discussed. Markedly,
this system inherits the topological robustness of the solitons. In view of
these results, the proposed device can effectively find an application as a
superconducting thermal router in which the thermal transport can be locally
mastered through solitonic excitations, which positions can be externally
controlled through a magnetic field and a bias current.
