In this paper we survey variational characterizations of eigenvalues of nonlinear eigenproblems, i.e. generalizations of Rayleigh’s principle, the minmax characterization of Poincaré, and the maxmin characterization of Courant, Fischer and Weyl to eigenvalue problems containing the eigenparameter nonlinearly. In this note we consider the nonlinear eigenvalue problem

The linear FEAST algorithm is a method for solving linear eigenvalue problems. It uses complex contour integration to calculate the eigenvectors whose eigenvalues that are located inside some user-defined region in the complex plane. This makes it possible to parallelize the process of solving eigenvalue problems by simply dividing the complex plane into a collection of disjoint regions and cal...

Abst rac t -Dynamic rent-seeking games with nonlinear cost functions are analyzed. The local asymptotic stability of the solution is first examined. We show that in the absence of a dominant agent, all eigenvalues of the Jacobian are real. Conditions are given for the local asymptotic stability as well as for the local instability of the equilibrium. In the presence of a dominant agent, complex...

We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin’s Invariant Subspace theorem...

2-ns waveforms with 7 eigenvalues and their QPSK-modulated spectral amplitudes were optimized by taking constraints of link, transmitter, and receiver into account. In experiment these signals were transmitted with a BER of 3.2E-3 over 1440-km of NZ-DSF fiber spans. Introduction In recent years the modulation of the nonlinear spectrum or detection in this nonlinear domain has been re-discovered...

We present elements of a stability theory for small, stochastic, nonlinear chemical reaction networks. Steady state probability distributions are computed with zero-information (ZI) closure, a closure algorithm that solves chemical master equations of small arbitrary nonlinear reactions. Stochastic models can be linearized around the steady state with ZI-closure, and the eigenvalues of the Jaco...

This work considers eigenvalue problems that are nonlinear in the eigenvalue parameter. Given such a nonlinear eigenvalue problem T , we are concerned with finding the minimal backward error such that T has a set of prescribed eigenvalues with prescribed algebraic multiplicities. While the usual resolvent norm addresses this question for a single eigenvalue of multiplicity one, the general sett...

We consider the Hamiltonian first-order coupled-mode system that occurs in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupled-mode system, which is equivalent to a four-by-four Dirac system with sign-indefinite metric. In the special class of symmetric nonlinear potentials, we construct a block-diagonal repr...

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1Laplacian. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The graphic feature of eigenvalues are also studied. In particular...

We consider the Hamiltonian coupled-mode system that occur in nonlinear optics, photonics, and atomic physics. Spectral stability of gap solitons is determined by eigenvalues of the linearized coupled-mode system, which is equivalent to a four-by-four Dirac system with sign-indefinite metric. In the special class of symmetric nonlinear potentials, we construct a blockdiagonal representation of ...