Spectral stability via the Maslov index and computer assisted proofs
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Abstract
In the scalar Swift-Hohenberg equation with quadratic-cubic nonlinearity, it is known that symmetric pulse solutions exist for certain parameter regions. In this dis- sertation, we develop a method to determine the spectral stability of these solutions. We first associate a Maslov index to each solution and then argue that this index co- incides with the number of unstable eigenvalues for the linearized evolution equation. This requires extending the method of computing the Maslov index introduced by Robbin and Salamon to so-called degenerate crossings. We extend their formulation of the Maslov index to degenerate crossings of general order. Furthermore, we develop a numerical method to compute the Maslov index associated to symmetric pulse so- lutions. We consider several solutions to the Swift-Hohenberg equation and use our method to characterize their stability. Additionally, we work towards the framework of a computer assisted proof that can be used to rigorously prove the spectral stability or instability of a pulse solution to the Swift-Hohenberg equation. We provide a novel computer assisted proof of the existence of such solutions and a novel technique for computing exponential dichotomy subspaces in the presence of a so-called external resonances that builds on the parameterization method developed by Cabré, Fontich and de la Llave.
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2024