Kink dynamics in the 𝜙⁴ model: Asymptotic stability for odd perturbations in the energy space

Kowalczyk, Michał¹ ; Martel, Yvan² ; Muñoz, Claudio³

¹ Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
² CMLS, École polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
³ CNRS and Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

Journal of the American Mathematical Society, Tome 30 (2017) no. 3, pp. 769-798

Résumé

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.

Détail

DOI : 10.1090/jams/870

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