The spectrum of the quantum discrete nonlinear Schrödinger equation, or boson Hubbard Hamiltonian, on a periodic 1D lattice shows some interesting detailed band structure, which may be interpreted as the quantum signature of a two-breather interaction in the classical case. This fine structure is studied using degenerate perturbation theory. We also present a modification to this model, which increases the mobility of bound states.
Dorignac, J., Eilbeck, J. C., Salerno, M., & Scott, A. C. (2004). Quantum signatures of breather-breather interactions. Physical Review Letters, 93(2), . https://doi.org/10.1103/PhysRevLett.93.025504