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Localization and approximation of attractors for the Ginzburg-Landau equation. (English) Zbl 0751.34036

This paper studies the initial-value problem \(u'-(\nu+i\alpha)\Delta u+(\kappa+i\beta)| u|^ 2u-\gamma u=0\), \(u(0)=u_ 0\), for \(u:\Omega\times\mathbb{R}_ +\to\mathbb{C}\), \(\Omega\) a bounded domain in \(\mathbb{R}^ n\), \(n=1,2\). This is the Ginzburg-Landau equation. The boundary conditions considered are either \(u\) zero on the boundary of \(\Omega\), \(\partial\Omega\), \(\partial u/\partial n\) zero on \(\partial\Omega\), or \(u\) periodic on \(\Omega=(0,L)^ n\). The paper constructs approximate inertial manifolds for this equation, thereby capturing the large time dynamics of the equation. Explicit details of the construction of the approximate inertial manifolds are given. The last section of the paper compares numerically the solutions found from the approximate inertial manifold method and a direct solution found by a Galerkin technique.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34D45 Attractors of solutions to ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
34C45 Invariant manifolds for ordinary differential equations
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