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Kink dynamics in the \(\phi^4\) model: asymptotic stability for odd perturbations in the energy space. (English) Zbl 1387.35419

The authors consider the one-dimensional \(\phi^4\) model, \[ \partial^2_t\phi-\partial^2_x\phi=\phi-\phi^3, \quad\quad (t,x)\in\mathbb{R}\times\mathbb{R} \] and the examine the stability of odd perturbations to the kink stationary solution \[ \phi^*(x)=\tanh\left(\frac{x}{\sqrt{2}}\right). \] The main result shows the asymptotic stability of the kink with respect to odd perturbations in the energy space. The result joins the proof of the orbital stability of the kink with respect to small perturbations [D. B. Henry et al., Commun. Math. Phys. 85, 351–361 (1982; Zbl 0546.35062)] to describe the long time behavior of solutions.

MSC:

35L71 Second-order semilinear hyperbolic equations
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35B35 Stability in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations

Citations:

Zbl 0546.35062
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References:

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