Evans function and Fredholm determinants

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The Evans function and transmission coefficient, which are both finite determinants, are natural tools for both analytic and numerical determination of eigenvalues of such linear operators. However, inverting the eigenvalue problem by the free state operator generates a natural linear integral eigenvalue problem whose solvability is determined through the corresponding infinite Fredholm determinant. The relationship between all three determinants has received a lot of recent attention. We focus on the case when the underlying Fredholm operator is a trace class perturbation of the identity. Our new results include: (i) clarification of the sense in which the Evans function and transmission coefficient are equivalent; and (ii) proof of the equivalence of the transmission coefficient and Fredholm determinant, in particular in the case of distinct far fields.Comment: 26 page

Interference features in scanning gate conductance maps of quantum point contacts with disorder

We consider quantum point contacts (QPCs) defined within disordered two-dimensional electron gases as studied by scanning gate microscopy. We evaluate the conductance maps in the Landauer approach and wave function picture of electron transport for samples with both low and high electron mobility at finite temperatures. We discuss the spatial distribution of the impurities in the context of the branched electron flow. We reproduce the surprising temperature stability of the experimental interference fringes far from the QPC. Next, we discuss -- previously undescribed -- funnel-shaped features that accompany splitting of the branches visible in previous experiments. Finally, we study elliptical interference fringes formed by an interplay of scattering by the point-like impurities and by the scanning probe. We discuss the details of the elliptical features as functions of the tip voltage and the temperature, showing that the first interference fringe is very robust against the thermal widening of the Fermi level. We present a simple analytical model that allows for extraction of the impurity positions and the electron gas depletion radius induced by the negatively charged tip of the atomic force microscope, and apply this model on experimental scanning gate images showing such elliptical fringes

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

Exact Solution of the Zakharov-Shabat Scattering Problem for Doubly-Truncated Multi-Soliton Potentials

Recent studies have revealed that multi-soliton solutions of the nonlinear Schr\"odinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multi-solitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multi-solitonic signal and provide the exact solution of the associated Zakharov-Shabat scattering problem for the windowed/doubly-truncated multi-soliton potential. This method further allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multi-solitonic signals. The method devised in this work also applies to general type of signals and may prove to be a useful tool in the theoretical analysis of such systems.Comment: The manuscript is revised for submission to PRE. Also, some typos have been correcte