  This paper investigates bright quantum-matter-wave solitons beyond the
Gross-Pitaevskii equation (GPE). As proposals for interferometry and creating
nonlocal quantum superpositions have been formed, it has become necessary to
investigate effects not present in mean-field models. We investigate the effect
of harmonic confinement on the internal degrees of freedom, as the ratio of
zero-point harmonic oscillator length to classical soliton length, for
different numbers of atoms. We derive a first-order energy correction for the
addition of a harmonic potential to the many-body wave function and use this to
create a variational technique based on energy minimization of this wave
function for an arbitrary number of atoms, and include numerics based on
diagonalization of the Hamiltonian in a basis of harmonic oscillator Fock
states. Finally we compare agreement between a Hartree product ground state and
the Bethe ansatz solution with a Gaussian envelope localizing the center of
mass and show a region of good agreement.
