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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 t u+(i x ) m u+H α (u, x u,, x m-1 u)=0 with even positive m, H α 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:
35B32Bifurcation (PDE)
35K55Nonlinear parabolic equations
37M20Computational methods for bifurcation problems
70K50Transition to stochasticity (general mechanics)
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