  We present a detailed analysis of the relaxation dynamics in an extended
bosonic Josephson junction. We show that stochastic classical field simulations
in three spatial dimensions reproduce the main experimental findings of M.
Pigneur et al., Phys. Rev. Lett. 120, 173601 (2018). We give an analytic
solution describing the short time evolution through multimode dephasing. For
longer times, the observed relaxation to a phase locked state is caused by
nonlinear dynamics beyond the sine-Gordon model, persisting even at zero
temperature. Finally, we analyze different experimentally relevant trapping
geometries and their potential for analogue simulation of the sine-Gordon model
out of equilibrium. Our results provide the basis for future experimental
implementations aiming to study quantum effects of extended bosonic Josephson
junctions.
