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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.

MSC:

35B32 Bifurcations in context of PDEs
35K55 Nonlinear parabolic equations
37M20 Computational methods for bifurcation problems in dynamical systems
70K50 Bifurcations and instability for nonlinear problems in mechanics
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