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Analytically varying eigenvectors and the stability of viscous shock waves. (English) Zbl 1044.35057

Summary: Given a matrix \(A(\cdot)\) and a simple eigenvalue \(a(\cdot)\), both depending analytically on a complex parameter \(\lambda\) within a simply connected domain \(\Lambda\), we present a simple algorithm, based on a classical result of Kato, for finding an associated eigenvector \(V(\cdot)\) that likewise varies analytically with respect to \(\lambda\).
This is useful for numerical approximation of the Evans function/numerical determination of stability of traveling waves, as we demonstrate by an application to stability of viscous shock waves. Indeed, it extends to general traveling waves/systems of equations an efficient and robust ‘shooting’ method developed by Brin for the determination of shock stability in special cases where the relevant eigenvector \(V(\cdot)\) is explicitly available.

MSC:

35Q35 PDEs in connection with fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
15A18 Eigenvalues, singular values, and eigenvectors
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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