Asymptotic stability of the phi-four kink without odd symmetry
Consider the $\phi^4$ model $$\phi_{tt} - \phi_{xx} = \phi - \phi^3, \qquad (t, x) \in \mathbb{R} \times \mathbb{R},$$ with the kink $H(x) = \tanh(x/\sqrt{2})$. By Lorentz invariance, for $|c| < 1$ the function $H_c(x - ct)$ with $H_c(y) = H(y/\sqrt{1 - c^2})$ is also a solution. Henry, Perez and Wreszinski proved that the kink is orbitally stable under small perturbations in the energy space $H^1 \times L^2$. Kowalczyk, Martel and Muñoz proved that it is asymptotically stable under small odd perturbations in the energy space, in the sense that the perturbation tends to zero in $H^1(I) \times L^2(I)$ for every bounded interval $I$. Besides the eigenvalue $0$ from translation, the operator linearized about $H$ has the internal mode eigenvalue $3/2$ below its continuous spectrum $[2, \infty)$. Kowalczyk, Martel, Muñoz and Van Den Bosch gave a sufficient condition for asymptotic stability of kinks of general scalar field models without parity assumptions, which excludes models with an internal mode such as this one.
Problem. Does there exist $\varepsilon > 0$ such that every initial datum with $$\|\phi(0) - H\|_{H^1} + \|\phi_t(0)\|_{L^2} < \varepsilon$$ has the following asymptotics? There exist $c_\infty \in (-1, 1)$ and $a \in C^1([0, \infty))$ with $a'(t) \to c_\infty$ such that, for every $R > 0$, as $t \to \infty$, $$\|\phi(t, a(t) + \cdot) - H_{c_\infty}\|_{H^1(-R, R)} + \|\phi_t(t, a(t) + \cdot) + c_\infty H_{c_\infty}'\|_{L^2(-R, R)} \to 0.$$ No parity condition is imposed on either component of the perturbation, and $\phi_t$ is the time derivative before shifting coordinates.
Source: Kowalczyk, Martel and Muñoz (2017), Remark 1.2, who conjecture asymptotic stability of the kink for general perturbations in the energy space. Stated by AIM — Open Applied Problems with convergence to a moving kink.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, asymptotic stability, kinks, scalar field equations
References
@article{kowalczyk2017,
author = {Kowalczyk, Micha{\l} and Martel, Yvan and Mu{\~n}oz, Claudio},
title = {Kink dynamics in the $\phi^4$ model: asymptotic stability for odd perturbations in the energy space},
journal = {J. Amer. Math. Soc.},
volume = {30},
number = {3},
year = {2017},
pages = {769--798},
doi = {10.1090/jams/870},
eprint = {1506.07420},
archivePrefix = {arXiv},
}
@article{henry1982,
author = {Henry, Daniel B. and Perez, J. Fernando and Wreszinski, Walter F.},
title = {Stability theory for solitary-wave solutions of scalar field equations},
journal = {Comm. Math. Phys.},
volume = {85},
number = {3},
year = {1982},
pages = {351--361},
doi = {10.1007/BF01208719},
}
@article{kowalczyk2021,
author = {Kowalczyk, Micha{\l} and Martel, Yvan and Mu{\~n}oz, Claudio and Van Den Bosch, Hanne},
title = {A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models},
journal = {Ann. PDE},
volume = {7},
number = {1},
year = {2021},
pages = {10},
doi = {10.1007/s40818-021-00098-y},
eprint = {2008.01276},
archivePrefix = {arXiv},
}
@article{kowalczyk2024,
author = {Kowalczyk, Micha{\l} and Martel, Yvan},
title = {Kink dynamics under odd perturbations for (1+1)-scalar field models with one internal mode},
journal = {Math. Res. Lett.},
volume = {31},
number = {3},
year = {2024},
pages = {795--832},
doi = {10.4310/mrl.241113042759},
eprint = {2203.04143},
archivePrefix = {arXiv},
}
@misc{aim-phi-four-general-kink-stability,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Asymptotic stability of the phi-four kink without odd symmetry}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/232-phi-four-general-kink-stability.md},
note = {Entry 232 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00632, "Asymptotic stability of the phi-four kink without odd symmetry", clopen.solutions.