Quantum depletion of a soliton condensate

We present rigorous results on the diagonalization of Bogoliubov Hamiltonian for a soliton condensate. Using the complete and orthonormalized set of eigenfunction for the Bogoliubov-de Gennes equations, we calculate exactly the quantum depletion of the condensate and show that two degenerate zero-modes, which originate from a U(1) gauge- and a translational-symmetry breaking of the system, induce the quantum diffusion and transverse instability of the soliton condensate.