On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation
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- by Jonas Lührmann and Wilhelm Schlag;
- Comm. Amer. Math. Soc. 4 (2024), 230-356
- DOI: https://doi.org/10.1090/cams/32
- Published electronically: February 28, 2024
- HTML | PDF
Abstract:
We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms (see Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820]).
The main result of this work establishes decay estimates up to exponential time scales for small “codimension one type” perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from Lührmann and Schlag [Duke Math. J. 172 (2023), pp. 2715–2820], and an implementation of a version of an adapted functional framework introduced by Germain and Pusateri [Forum Math. Pi 10 (2022), p. 172].
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Bibliographic Information
- Jonas Lührmann
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: luhrmann@math.tamu.edu
- Wilhelm Schlag
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- MR Author ID: 313635
- Email: wilhelm.schlag@yale.edu
- Received by editor(s): February 26, 2023
- Received by editor(s) in revised form: January 8, 2024, and January 9, 2024
- Published electronically: February 28, 2024
- Additional Notes: The first author was partially supported by NSF grant DMS-1954707. The second author was partially supported by NSF grant DMS-1902691.
- © Copyright 2024 by the authors under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License (CC BY NC ND 4.0)
- Journal: Comm. Amer. Math. Soc. 4 (2024), 230-356
- MSC (2020): Primary 35P25; Secondary 35Q51, 42B37
- DOI: https://doi.org/10.1090/cams/32
- MathSciNet review: 4710855