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On quadratic eigenvalue problems arising in stability of discrete vortices. (English) Zbl 1176.15010

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)].

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

Citations:

Zbl 1088.39005
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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)
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