Brin, Leon Q.; Zumbrun, Kevin Analytically varying eigenvectors and the stability of viscous shock waves. (English) Zbl 1044.35057 Mat. Contemp. 22, 19-32 (2002). 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. Cited in 1 ReviewCited in 42 Documents 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 Keywords:Evans function; matrix; eigenvalue; eigenvector; stability of traveling waves; shock stability PDFBibTeX XMLCite \textit{L. Q. Brin} and \textit{K. Zumbrun}, Mat. Contemp. 22, 19--32 (2002; Zbl 1044.35057)