We investigate the way in which the pattern of fringes in a coherent pair of two-dimensional Bose condensed clouds of ultra-cold atoms traveling in opposite directions subject to a harmonic trapping potential can seed the irreversible formation of internal excitations in the clouds, notably solitons and vortices. We identify under, over and critically damped regimes in the dipole oscillations of the condensates according to the balance of internal and centre-of-mass energies of the clouds. We carry out simulations of the collision of two clouds with respect to different initial phase differences in these regimes to investigate the creation of internal excitations. We distinguish the behaviour of this system from previous studies of quasi one-dimensional BEC's. In particular we note that the nature of the internal excitations is only essentially sensitive to an initial phase difference between the clouds in the overdamped regime.

We investigate the thermodynamic limit of the su(n)-invariant spin chain models with unparallel boundary fields. It is found that the contribution of the inhomogeneous term in the associated T − Q relation to the ground state energy does vanish in the thermodynamic limit. This fact allows us to calculate the boundary energy of the system. Taking the su(2) (or the XXX) spin chain and the su(3) spin chain as concrete examples, we have studied the corresponding boundary energies of the models. The method used in this paper can be generalized to study the thermodynamic properties and boundary energy of other high rank models with non-diagonal boundary fields.

The XXX Gaudin model with generic integrable boundaries specified by the most general non-diagonal K-matrices is studied by the off-diagonal Bethe ansatz method. The eigenvalues of the associated Gaudin operators and the corresponding Bethe ansatz equations are obtained.