  We investigate the coherent energy and thermal transport in a
temperature-biased long Josephson tunnel junction, when a Josephson vortex,
i.e., a soliton, steadily drifts driven by an electric bias current. We
demonstrate that thermal transport through the junction can be controlled by
the bias current, since it determines the steady-state velocity of the drifting
soliton. We study the effects on thermal transport of the damping affecting the
soliton dynamics. In fact, a soliton locally influences the power flowing
through the junction and can cause the variation of the temperature of the
device. When the soliton speed increases approaching its limiting value, i.e.,
the Swihart velocity, we demonstrate that the soliton-induces thermal effects
significantly modify. Finally, we discuss how the appropriate material
selection of the superconductors forming the junction is essential, since short
quasiparticle relaxation times are required to observe fast thermal effects.
