Lax pairs of time-dependent Gross-Pitaevskii equation

We calculate the Lax pairs of homogeneous and inhomogeneous one-dimensional time-dependent Gross-Pitaevskii equations with time-dependent scattering length. The inhomogeneity corresponds to linear and quadratic potentials. Our approach introduces a systematic method of searching for the Lax pair corresponding to a given differential equation. We derive known Lax pairs for the Gross-Pitaevskii equation with homogeneous and quadratic potentials and time-dependent scattering length. We also derive new Lax pairs corresponding to a Gross-Pitaevskii equation with a linear potential. Using the resulting Lax pairs, the Darboux transformation can be performed and exact solutions of the Gross-Pitaevskii equation can be obtained for experimentally relevant cases such as solitonic solutions.