Browsing by Author "Sanei Kashani, Kourosh"
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Item Open Access Validity and attractivity of amplitude equations(2016) Sanei Kashani, Kourosh; Schneider, Guido (Prof. Dr.)Whitham's equations and the Ginzburg-Landau equation belong to a set of famous amplitude equations containing the KdV equation, the NLS equation, Burgers equation, and so-called phase diffusion equations. They play an important role in the description of spatially extended dissipative or conservative physical systems. Except of Whitham's system for all other amplitude equations there exists a satisfying mathematical theory showing that the original system behaves approximately as predicted by the associated amplitude equation. In the first part of this work we therefore derive Whitham's equations for a coupled system of equations, namely a Klein-Gordon-Boussinesq model. Subsequently we prove the validity of Whitham's equations for this system. The combination of our scaled ansatz adapted to Whitham's equations with the resonance structure of our system poses a new challenge. In order to prove the approximation results for Whitham's equations we will require some infinite series of normal transformations, for which we need to prove the convergence. In the second part we prove the attractivity of the Ginzburg-Landau manifold for a toy problem inspired by Marangoni convection. In comparison to the previous classical situation in our case the curve of eigenvalues possesses additionally a marginally stable mode at the origin. Therefore, we will need to modify the requirements for the attractivity result and the method of proof.