Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations

Details Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations Proof of the marginal stability bound for the Swift-Hohenberg equation and related equations

2000-05-12

arxiv.org/abs/nlin/0005024v1

Nonlinear Sciences

Pattern Formation and Solitons

16 pages, 0 figures

Scientific paper

We prove that if the initial condition of the Swift-Hohenberg equation $\partial_t u(x,t)=\bigl(\epsilon^2-(1+\partial_ x^2)^2\bigr) u(x,t) -u^3(x,t)$ is bounded in modulus by $Ce^{-\beta x}$ as $x\to+\infty $, the solution cannot propagate to the right with a speed greater than $\sup_{0<\gamma\le\beta}\gamma^{-1}(\epsilon ^2+4\gamma^2+8\gamma^4).$ 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.

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