Abstract
We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.
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Communicated by G. Friesecke
This work was supported in part by MIUR project “Metodi variazionali ed Equazioni Differenziali Non Lineari”.
This work was supported in part by the National Science Foundation under Grants No. DMS-0088935 and DMS-0322962.
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Arioli, G., Koch, H. Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation. Arch Rational Mech Anal 197, 1033–1051 (2010). https://doi.org/10.1007/s00205-010-0309-7
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DOI: https://doi.org/10.1007/s00205-010-0309-7