We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences to the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue prob...

The paper addresses the stability analysis of linear circuits and systems under interval parameter uncertainties. The problem is equivalent of estimating the eigenvalues of matrices, which elements are nonlinear functions of interval parameters. A method for obtaining the exact range of the eigenvalues is suggested. It can be applied if certain monotonicity conditions are fulfilled. A method fo...

Kinematical conservation laws (KCL) is a system of conservation laws governing the evolution of a curve in a plane or a surface in space, even if the curve or the surface has singularities on it. In our recent publication [1] we have developed a mathematical theory to study the successive positions and geometry of a 3-D weakly nonlinear wavefront by adding an energy transport equation to KCL. T...

The paper addresses the stability analysis of linear continuous systems under interval uncertainties. This problem initially described by the characteristic polynomial is transformed into a corresponding eigenvalue problem. The final analysis is equivalent to estimating the eigenvalues of matrices which elements are nonlinear functions of interval parameters. A method for obtaining the exact ra...

In this presentation we review iterative projection methods for sparse nonlinear eigenvalue problems which have proven to be very efficient. Here the eigenvalue problem is projected to a subspace V of small dimension which yields approximate eigenpairs. If an error tolerance is not met then the search space V is expanded in an iterative way with the aim that some of the eigenvalues of the reduc...

We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schr6dinger equation. Devillard and SouiUard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such timeharmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerica...

Spectra of nonlinear waves in infinite-dimensional Hamiltonian systems are investigated. We establish a connection via the Krein signature between the number of negative directions of the second variation of the energy and the number of potentially unstable eigenvalues of the linearization about a nonlinear wave. We apply our results to determine the effect of symmetry-breaking on the spectral ...