A new proof of a Liouville theorem for the one dimensional Gross-Pitaevskii equation
Micha{\l} Kowalczyk, Yvan Martel
TL;DR
This paper offers an alternative proof of Liouville theorems related to the stability of solitons in the one-dimensional Gross-Pitaevskii equation, simplifying spectral analysis through a new factorization identity.
Contribution
It introduces a novel proof method for Liouville theorems, bypassing complex spectral analysis by using a factorization identity for the linearized operator.
Findings
Simplified proof of Liouville theorems for the Gross-Pitaevskii equation
New factorization identity for the linearized operator
Enhanced understanding of soliton stability analysis
Abstract
The asymptotic stability of the black and dark solitons of the one-dimensional Gross-Pitaevskii equation was proved by B\'ethuel, Gravejat and Smets (Ann. Sci. \'Ec. Norm. Sup\'er. 48 (2015)) and Gravejat and Smets (Proc. Lond. Math. Soc. 111 (2015)), using a rigidity property in the vicinity of solitons. We provide an alternate proof of the Liouville theorems in the above articles using a factorization identity for the linearized operator which trivializes the spectral analysis.
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