  We present a detailed analysis of the relaxation dynamics in an extended
bosonic Josephson junction. We show that stochastic classical field simulations
using Gross-Pitaevskii equations 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, induced by the longitudinal confinement potential and persisting even at
zero temperature. Finally, we analyze different experimentally relevant
trapping geometries to mitigate these effects. Our results provide the basis
for future experimental implementations aiming to study nonlinear and quantum
effects of the relaxation in extended bosonic Josephson junctions.
