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Alternate proof of the Liouville theorem for black and dark solitons in the gross–pitaevskii equation, highlighting spectral analysis implications.
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.
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Kowalczyk et al. (2025) studied this question.
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