Modulation equations near the Eckhaus boundary: the KdV equation

We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*} \partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi |\Psi|^2, \end{align*} near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters $\alpha$, $\beta$ a number of modulation equations can be derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the KdV approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping we work in spaces of analytic functions.Comment: 44 pages, 8 figure

A genomic approach to examine the complex evolution of laurasiatherian mammals

Recent phylogenomic studies have failed to conclusively resolve certain branches of the placental mammalian tree, despite the evolutionary analysis of genomic data from 32 species. Previous analyses of single genes and retroposon insertion data yielded support for different phylogenetic scenarios for the most basal divergences. The results indicated that some mammalian divergences were best interpreted not as a single bifurcating tree, but as an evolutionary network. In these studies the relationships among some orders of the super-clade Laurasiatheria were poorly supported, albeit not studied in detail. Therefore, 4775 protein-coding genes (6,196,263 nucleotides) were collected and aligned in order to analyze the evolution of this clade. Additionally, over 200,000 introns were screened in silico, resulting in 32 phylogenetically informative long interspersed nuclear elements (LINE) insertion events. The present study shows that the genome evolution of Laurasiatheria may best be understood as an evolutionary network. Thus, contrary to the common expectation to resolve major evolutionary events as a bifurcating tree, genome analyses unveil complex speciation processes even in deep mammalian divergences. We exemplify this on a subset of 1159 suitable genes that have individual histories, most likely due to incomplete lineage sorting or introgression, processes that can make the genealogy of mammalian genomes complex. These unexpected results have major implications for the understanding of evolution in general, because the evolution of even some higher level taxa such as mammalian orders may sometimes not be interpreted as a simple bifurcating pattern

Invariant manifolds for random dynamical systems with slow and fast variables

We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modelling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition

Weak side of strong topological insulators

Strong topological insulators may have nonzero weak indices. The nonzero weak indices allow for the existence of topologically protected helical states along line defects of the lattice. If the lattice admits line defects that connect opposite surfaces of a slab of such a “weak-and-strong” topological insulator, these states effectively connect the surface states at opposite surfaces. Depending on the phases accumulated along the dislocation lines, this connection results in a suppression of in-plane transport and the opening of a spectral gap or in an enhanced density of states and an increased conductivity

Spectral representation of Matsubara n-point functions: Exact kernel functions and applications

In the field of quantum many-body physics, the spectral (or Lehmann) representation simplifies the calculation of Matsubara n-point correlation functions if the eigensystem of a Hamiltonian is known. It is expressed via a universal kernel function and a system- and correlator-specific product of matrix elements. Here we provide the kernel functions in full generality, for arbitrary n, arbitrary combinations of bosonic or fermionic operators and an arbitrary number of anomalous terms. As an application, we consider bosonic 3- and 4-point correlation functions for the fermionic Hubbard atom and a free spin, respectively.Comment: 13 pages, 1 figure, comments welcome

Taming pseudo-fermion functional renormalization for quantum spins: Finite-temperatures and the Popov-Fedotov trick

The pseudo-fermion representation for $S=1/2$ quantum spins introduces unphysical states in the Hilbert space which can be projected out using the Popov-Fedotov trick. However, state-of-the-art implementation of the functional renormalization group method for pseudo-fermions have so far omitted the Popov-Fedotov projection. Instead, restrictions to zero temperature were made and absence of unphysical contributions to the ground-state was assumed. We question this belief by exact diagonalization of several small-system counterexamples where unphysical states do contribute to the ground state. We then introduce Popov-Fedotov projection to pseudo-fermion functional renormalization, enabling finite temperature computations with only minor technical modifications to the method. At large and intermediate temperatures, our results are perturbatively controlled and we confirm their accuracy in benchmark calculations. At lower temperatures, the accuracy degrades due to truncation errors in the hierarchy of flow equations. Interestingly, these problems cannot be alleviated by switching to the parquet approximation. We introduce the spin projection as a method-intrinsic quality check. We also show that finite temperature magnetic ordering transitions can be studied via finite-size scaling.Comment: 14 pages, 8 figures; minor clarifications, added reference

Moving modulating pulse and front solutions of permanent form in a FPU model with nearest and next-to-nearest neighbor interaction

We consider a nonlinear chain of coupled oscillators, which is a direct generalization of the classical FPU lattice and exhibits, besides the usual nearest neighbor interaction, also next-to-nearest neighbor interaction. For the case of nearest neighbor attraction and next-to-nearest neighbor repulsion we prove that such a lattice admits, in contrast to the classical FPU model, moving modulating front solutions of permanent form, which have small converging tails at infinity and can be approximated by solitary wave solutions of the Nonlinear Schrödinger equation. When the associated potentials are even, then the proof yields moving modulating pulse solutions of permanent form, whose profiles are spatially localized. Our analysis employs the spatial dynamics approach as developed by Iooss and Kirchgässner. The relevant solutions are constructed on a five-dimensional center manifold and their persistence is guaranteed by reversibility arguments