Exact solution to the pairing equations for bosons in a one-dimensional model

Pairing equations for bosons interacting with a delta-function potential in one dimension are reduced to two coupled equations involving complete elliptic integrals. These equations are solved numerically and the results are compared to the exact Lieb and Liniger solution for the problem of one-dimensional bosons with delta-function interactions. One finds that the pairing solution gives too large a potential energy and too small a kinetic energy. We also compare the exact and paired excitation spectra. We discuss the relevance of these results to realistic problems.