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The Evans Function for nth-Order Operators on the Real Line

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Spectral and Dynamical Stability of Nonlinear Waves

Part of the book series: Applied Mathematical Sciences ((AMS,volume 185))

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Abstract

The primary goal of this chapter is the construction of the Evans function for eigenvalue problems associated with nth-order, exponentially asymptotic linear operators acting on L2(R). The construction, through the Jost solutions, is distinguished from the construction for second-order operators by the fact that the matrix eigenvalues and associated eigenvectors for the nth-order problem may not be analytic in the natural domain of the Evans function. Moreover, while it is relatively easy to determine the essential spectrum for these problems, the matrix eigenvalues and the absolute spectrum do not generally have an explicit representation. We sidestep these issues via an analytic extension of the stable and unstable spaces of the asymptotic matrix which leads to the construction of Jost matrices.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Calvin College, Grand Rapids, Michigan, USA

    Todd Kapitula

  2. Department of Mathematics, Michigan State University, East Lansing, Michigan, USA

    Keith Promislow

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  1. Todd Kapitula
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Kapitula, T., Promislow, K. (2013). The Evans Function for nth-Order Operators on the Real Line. In: Spectral and Dynamical Stability of Nonlinear Waves. Applied Mathematical Sciences, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6995-7_10

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