Chugunova, Marina; Pelinovsky, Dmitry On quadratic eigenvalue problems arising in stability of discrete vortices. (English) Zbl 1176.15010 Linear Algebra Appl. 431, No. 5-7, 962-973 (2009). Authors’ summary: We develop a count of unstable eigenvalues in a finite-dimensional quadratic eigenvalue problem arising in the context of stability of discrete vortices in a multi-dimensional discrete nonlinear Schrödinger equation [D. E. Pelinovsky, P.G. Kevrekidis, D. J. Frantzeskakis, Physica D 212, No. 1–2, 20–53 (2005; Zbl 1088.39005)]. The count is based on the Pontryagin invariant subspace theorem and the parameter continuation arguments. Another application of the method is given in the context of front-pulse solutions of neuron networks with piecewise constant nonlinear functions [D. E. Pelinovsky and V. G. Yakhno, Generation of collective-activity structures in a homogeneous neuron-like medium, Int. J. Bifurcation Chaos 6, 81–87 (1996)]. Reviewer: Jaspal Singh Aujla (Jalandhar) Cited in 5 Documents MSC: 15A18 Eigenvalues, singular values, and eigenvectors 15A22 Matrix pencils 37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems 47A75 Eigenvalue problems for linear operators 39A12 Discrete version of topics in analysis Keywords:quadratic eigenvalue problem; generalized eigenvalue problem; symmetric operators in sign-indefinite spaces; unstable eigenvalues and bifurcation; nonlinear Schrödinger equation; Pontryagin invariant subspace theorem; neuron networks Citations:Zbl 1088.39005 PDFBibTeX XMLCite \textit{M. Chugunova} and \textit{D. Pelinovsky}, Linear Algebra Appl. 431, No. 5--7, 962--973 (2009; Zbl 1176.15010) Full Text: DOI References: [1] M. Chugunova, D. Pelinovsky, Count of eigenvalues in the generalized eigenvalue problem, 2006. Available from: <arXiv:math/0602386>.; M. Chugunova, D. Pelinovsky, Count of eigenvalues in the generalized eigenvalue problem, 2006. Available from: <arXiv:math/0602386>. [2] Gurski, K. F.; Kollar, R.; Pego, R. L., Slow damping of internal waves in a stably stratified fluid, Proc. Roy. Soc. Lond., 460, 977-994 (2004) · Zbl 1070.76020 [3] Kapitula, T.; Kevrekidis, P. G.; Sandstede, B., Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems, Physica D, 195, 263-282 (2004) · Zbl 1056.37080 [4] Kapitula, T.; Kevrekidis, P. G., Linear stability of perturbed Hamiltonian systems: theory and a case example, J. Phys. A: Math. Gen., 37, 7509-7526 (2004) · Zbl 1061.70012 [5] Lukas, M.; Pelinovsky, D.; Kevrekidis, P. G., Lyapunov-Schmidt reduction algorithm for three-dimensional discrete vortices, Physica D, 237, 339-350 (2008) · Zbl 1147.35097 [6] A.S. Markus, Introduction to the spectral theory of polynomial operator pencils, Trans. Math. Monographs, vol. 71, AMS, Providence, 1988.; A.S. Markus, Introduction to the spectral theory of polynomial operator pencils, Trans. Math. Monographs, vol. 71, AMS, Providence, 1988. · Zbl 0678.47005 [7] Pelinovsky, D. E.; Kevrekidis, P. G.; Frantzeskakis, D. J., Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Physica D, 212, 20-53 (2005) · Zbl 1088.39005 [8] Pelinovsky, D. E.; Yakhno, V. G., Generation of collective-activity structures in a homogeneous neuron-like medium, Int. J. Bifurcation and Chaos, 6, 81-87 (1996) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.