Exponential dichotomies for nonlocal differential operators with infinite range interactions

Schouten-Straatman, Willem M.; Hupkes, Hermen Jan

doi:10.1016/j.jde.2021.08.019

Cited by 2 publications

(1 citation statement)

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“…On the other hand, it appears to be difficult to adapt this formalism to yield spreading speeds also in the oscillatory case ω dr = 0, and to multidimensional problems. Similarly, the pointwise formulation adapted here relies strongly on a "local" formulation in x, excluding to some extend spatially nonlocal coupling that does not permit a formulation as a first-order spatial ODE through linearization of the matrix pencil in ∂ x ; see however [13,14,40] for techniques that recover "pointwise" descriptions in this nonlocal setting. Similarly, effective computational tools to analyze multi-dimensional problems in this pointwise context do not appear to be available; see for instance [7] for a discussion of pointwise instabilities in constant-coefficient, multi-dimensional problems.…”

Section: Discussionmentioning

confidence: 99%

Nonlinear eigenvalue methods for linear pointwise stability of nonlinear waves

Scheel¹

2022

Preprint

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We propose an iterative method to find pointwise growth exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends analytically on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value to a given reference point in the complex plane. The method finds branch points, eigenvalues, and resonance poles without a priori knowledge.2 Pointwise nonlinear eigenvalue problems from linearization at heteroclinic profiles First-order ODEs from eigenvalue problemsWe consider eigenvalues problems that arise in the linearization at traveling waves, of the formwith matrix coefficients A(x; λ) ∈ C N ×N , continuous in x and analytic in λ. We focus on the simplest case of asymptotically constant coefficients lim x→±∞ A(x; λ) = A ± (λ).

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