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Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: Analysis and computations. (English) Zbl 0704.58030
The paper is divided into two main parts, analytic and computational. The first part is devoted to the analysis of an approximate inertial manifold (AIM) passing through all the steady states of the Kuramoto-Sivashinsky equation (KSE). After expressing the KSE in functional form below, the authors derive some preliminary results regarding its solutions in section 2. In section 3 the authors introduce an AIM which passes through all the stationary solutions, and show in theorem 3.3 that all trajectories of the KSE approach a thin layer about this AIM at an exponential rate. In section 4 they give some motivation behind some other methods of approximation in the literature, their formal relation with various stages of the method presented in section 3, and tabulate the associated errors.
Section 5 contains the computational part of the paper. The authors present complete steady state bifurcation diagrams and tabulate all steady state bifurcations. Diagrams including certain periodic solutions are also presented.
They conclude the interpretation of the computational results with the discussion of a particular side effect the approximations appear to have on the size of the absorbing ball.
Reviewer: T.Rassias

MSC:
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37G99 Local and nonlocal bifurcation theory for dynamical systems
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
37A30 Ergodic theorems, spectral theory, Markov operators
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