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A review on asymptotic stability of solitary waves in nonlinear dispersive problems in dimension one

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arxiv 2410.04508 v1 pith:RGNVTX3M submitted 2024-10-06 math.AP

Pierre Germain This is my paper · ORCID
classification math.AP
keywords asymptoticnonlinearsolitarystabilitydispersiveequationsfocusmethods
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We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrodinger equation; second, the notion of full asymptotic stability (which states that perturbations of a solitary wave decompose globally into a solitary wave and a decaying solution); and third, spectral methods. Besides this focus, we summarize the state of the art in a broader context, including nonlinear Klein-Gordon equations, the notion of local asymptotic stability, and virial methods.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model

    math.AP 2026-08 conditional novelty 8.0 of 10

    Small equivariant perturbations of the degree-one vortex decay with radiation rate, while the internal mode damps like epsilon squared over one plus Gamma epsilon squared t, proving asymptotic stability.

  2. Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordstr\"om black hole

    math.AP 2025-01 reject novelty 7.0 of 10

    For the Bizon-Kahl Yang-Mills kink in an extremal Reissner-Nordström background, globally bounded perturbations are shown to converge locally in space, and a finite-codimensional stable manifold is built.

  3. Stability of the catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensions

    math.AP 2024-11 conditional novelty 7.0 of 10

    In four spatial dimensions, small codimension-1 perturbations of catenoid initial data yield global HVMC solutions that converge modulo translation and boost to a boosted/translated catenoid with explicit decay rates.

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