On quadratic eigenvalue problems arising in stability of discrete vortices

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, Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Physica D 212 (2005) 20–53]. 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, V.G.Yakhno, Generation of collective–activity structures in a homogeneous neuron-like medium, Int. J. Bifurcation and Chaos 6 (1996) 81–87].