Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 
 

 

Kink dynamics in the $ \phi^4$ model: Asymptotic stability for odd perturbations in the energy space


Authors: Michał Kowalczyk, Yvan Martel and Claudio Muñoz
Journal: J. Amer. Math. Soc. 30 (2017), 769-798
MSC (2010): Primary 35L71; Secondary 35Q51, 37K40
DOI: https://doi.org/10.1090/jams/870
Published electronically: September 27, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a classical equation known as the $ \phi ^4$ model in one space dimension. The kink, defined by $ H(x)=\tanh (x/{\sqrt {2}})$, is an explicit stationary solution of this model. From a result of Henry, Perez and Wreszinski it is known that the kink is orbitally stable with respect to small perturbations of the initial data in the energy space. In this paper we show asymptotic stability of the kink for odd perturbations in the energy space. The proof is based on Virial-type estimates partly inspired from previous works of Martel and Merle on asymptotic stability of solitons for the generalized Korteweg-de Vries equations. However, this approach has to be adapted to additional difficulties, pointed out by Soffer and Weinstein in the case of general Klein-Gordon equations with potential: the interactions of the so-called internal oscillation mode with the radiation, and the different rates of decay of these two components of the solution in large time.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35L71, 35Q51, 37K40

Retrieve articles in all journals with MSC (2010): 35L71, 35Q51, 37K40


Additional Information

Michał Kowalczyk
Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Email: kowalczy@dim.uchile.cl

Yvan Martel
Affiliation: CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
Email: yvan.martel@polytechnique.edu

Claudio Muñoz
Affiliation: CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Email: claudio.munoz@math.u-psud.fr, cmunoz@dim.uchile.cl

DOI: https://doi.org/10.1090/jams/870
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: July 18, 2016
Published electronically: September 27, 2016
Additional Notes: The first author was partially supported by Chilean research grants Fondecyt 1130126, Fondo Basal CMM-Chile and ERC 291214 BLOWDISOL. The author would like to thank Centre de mathématiques Laurent Schwartz at the Ecole Polytechnique and the Université Cergy-Pontoise where part of this work was done.
The second author was partially supported by ERC 291214 BLOWDISOL
The third author would like to thank the Laboratoire de Mathématiques d’Orsay where part of this work was completed. His work was partly funded by ERC 291214 BLOWDISOL, and Chilean research grants FONDECYT 1150202, Fondo Basal CMM-Chile, and Millennium Nucleus Center for Analysis of PDE NC130017
Article copyright: © Copyright 2016 American Mathematical Society