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

2025/07/05 by Kowalczyk, Michał, Martel, Yvan
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.03966

Abstract

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. 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.

Citations

• Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes
• Kink dynamics under odd perturbations for (1+1)-scalar field models with one internal mode
• Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal modes
• Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

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• On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation
• Stability and instability of the quasilinear Gross--Pitaevskii dark solitons