Archive for Rational Mechanics and Analysis

, Volume 197, Issue 3, pp 1033–1051

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation


DOI: 10.1007/s00205-010-0309-7

Cite this article as:
Arioli, G. & Koch, H. Arch Rational Mech Anal (2010) 197: 1033. doi:10.1007/s00205-010-0309-7


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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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica and MOXPolitecnico di MilanoMilanItaly
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA