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Zbl 1231.35016
Arioli, Gianni; Koch, Hans
Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation.
(English)
[J] Arch. Ration. Mech. Anal. 197, No. 3, 1033-1051 (2010). ISSN 0003-9527; ISSN 1432-0673/e

The authors develop computer-assisted techniques for the analysis of stationary solutions, of stability, and of bifurcation diagrams of parabolic equations of the form $\partial_t u + (i\partial_x)^m u+H_{\alpha} (u, \partial_x u, \dots, \partial_x^{m-1} u)=0$ with even positive $m$, $H_{\alpha}$ real analytic, and $u(x,t)$ periodic in $x$. As a case of study, these methods are applied to the Kuramoto-Sivashinski equation. The authors rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. The dimension of the unstable manifold for the flow is determined at some stationary solution in each branch.
MSC 2000:
*35B32 Bifurcation (PDE)
35K55 Nonlinear parabolic equations
37M20 Computational methods for bifurcation problems
70K50 Transition to stochasticity (general mechanics)

Keywords: Kuramoto-Sivashinski equation; bifurcation diagram

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