  The exact factorization of an electron-nuclear wavefunction [A. Abedi, N. T.
Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us to
define the rigorous nuclear time-dependent Schr\"{o}dinger equation (TDSE) with
a time-dependent potential-energy surface (TDPES) that fully accounts for the
coupling to the electronic motion and drives the nuclear wavepacket dynamics.
Here, we study whether the propagation of multiple classical trajectories can
reproduce the quantum nuclear motion in strong-field processes when their
motions are governed by the quantum Hamilton-Jacobi equation derived by
applying Bohmian mechanics to this exact nuclear TDSE. We demonstrate that
multiple classical trajectories propagated by the force from the gradient of
the exact TDPES plus the Bohmian quantum potential can reproduce the
strong-field dissociation dynamics of a one-dimensional model of the H$_2^+$
molecule. Our results show that the force from the Bohmian quantum potential
plays a non-negligible role in yielding quantum nuclear dynamics in the
strong-field process studied here, where ionization and/or splitting of nuclear
probability density occurs.
