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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 $\partial_t u + (i\partial_x)^m u+H_{\alpha} (u, \partial_x u, \dots, \partial_x^{m-1} u)=0$ 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.

35B32Bifurcation (PDE)
35K55Nonlinear parabolic equations
37M20Computational methods for bifurcation problems
70K50Transition to stochasticity (general mechanics)
Full Text: DOI
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