A new proof of a Liouville theorem for the one dimensional Gross–Pitaevskii equation

The asymptotic stability of the black and dark solitons of the one-dimensional Gross–Pitaevskii equation was proved by Béthuel, Gravejat and Smets ( Ann. Sci. Éc. Norm. Supér. , 2015), and Gravejat and Smets ( Proc. Lond. Math. Soc. , 2015), using a rigidity property in the vicinity of solitons. We provide an alternate proof of the Liouville theorems in those two articles, using a factorization identity for the linearized operator which trivializes the spectral analysis.