相关论文: Graphical Krein Signature Theory and Evans-Krein F…
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it is especially important to locate not only the unstable eigenvalues (i.e., those with positive real part), but also those which are purely imaginary but have…
We explain the concept of Krein signature in Hamiltonian and $\mathcal{PT}$-symmetric systems on the case study of the one-dimensional Gross-Pitaevskii equation with a real harmonic potential and an imaginary linear potential. These…
We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill's equation.…
In its original formulation the Krein matrix was used to locate the spectrum of first-order star-even polynomial operators where both operator coefficients are nonsingular. Such operators naturally arise when considering first-order-in-time…
Frequently encountered in nature, internal solitary waves in stratified fluids are well-observed and well-studied from the experimental, the theoretical, and the numerical perspective. From the mathematical point of view, these waves are…
We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretation and generalization of the index theory using the…
We derive a counting formula for the eigenvalues of Schr\"odinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue…
We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach.…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
In this work, we systematically generalize the Evans function methodology to address vector systems of discrete equations. We physically motivate and mathematically use as our case example a vector form of the discrete nonlinear Schrodinger…
Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes…
In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability…
We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers…
The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts.…
In this article we develop a graphical calculus for stable invariants of Riemannian manifolds akin to the graphical calculus for Rozansky-Witten invariants for hyperk\"ahler manifolds; based on interpreting trivalent graphs with colored…
We study the spectrum of the linearization around standing wave profiles for two quantum hydrodynamics systems with linear and nonlinear viscosity. The essential spectrum for such profiles is stable; we investigate the point spectrum using…
We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schr\"odinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it…
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the…
The nonlinear Schr\"{o}dinger equation with a linear periodic potential and a nonlinearity coefficient $\Gamma$ with a discontinuity supports stationary localized solitary waves with frequencies inside spectral gaps, so called surface gap…
The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros…