Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations
| dc.creator | Collet, Pierre | |
| dc.creator | Eckmann, Jean-Pierre | |
| dc.date | 2000-05-12 | |
| dc.date.accessioned | 2026-07-07T05:32:50Z | |
| dc.date.available | 2026-07-07T05:32:50Z | |
| dc.description | We prove that if the initial condition of the Swift-Hohenberg equation $\partial_t u(x,t)=\bigl(ε^2-(1+\partial_ x^2)^2\bigr) u(x,t) -u^3(x,t)$ is bounded in modulus by $Ce^{-βx}$ as $x\to+\infty $, the solution cannot propagate to the right with a speed greater than $\sup_{0<γ\leβ}γ^{-1}(ε^2+4γ^2+8γ^4).$ This settles a long-standing conjecture about the possible asymptotic propagation speed of the Swift-Hohenberg equation. The proof does not use the maximum principle and is simple enough to generalize easily to other equations. We illustrate this with an example of a modified Ginzburg-Landau equation, where the minimal speed is not determined by the linearization alone. | |
| dc.description | 16 pages, 0 figures | |
| dc.identifier | https://arxiv.org/abs/nlin/0005024 | |
| dc.identifier | http://arxiv.org/abs/nlin/0005024 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79801 | |
| dc.subject | Pattern Formation and Solitons | |
| dc.subject | Mathematical Physics | |
| dc.subject | Fluid Dynamics | |
| dc.title | Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations | |
| dc.type | text |