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Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation. (English) Zbl 1231.35016
The authors develop computer-assisted techniques for the analysis of stationary solutions, of stability, and of bifurcation diagrams of parabolic equations of the form ${\partial }_{t}u+{\left(i{\partial }_{x}\right)}^{m}u+{H}_{\alpha }\left(u,{\partial }_{x}u,\cdots ,{\partial }_{x}^{m-1}u\right)=0$ with even positive $m$, ${H}_{\alpha }$ real analytic, and $u\left(x,t\right)$ 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:
 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
##### References:
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