The dynamics of a one-dimensional parabolic problem versus the dynamics of its discretization

In this paper we prove that the spatial discretization of a one dimensional system of parabolic equations. with suitably small step size, contains exactly the same asymptotic dynamics as the continuous problem. (C) 2000 Academic Press.

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for the all proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.

Quantum and semiclassical Husimi distributions for a one-dimensional resonant system

We compare exact and semiclassical Husimi distributions for the single eigenstates of a one-dimensional resonant Hamiltonian. We find that both distributions concentrate near the unstable fixed points even when these points are made complex by suitably varying a parameter. © 1992 The American Physical Society.

Stable spike clusters for the one-dimensional Gierer-Meinhardt system

We consider the Gierer-Meinhardt system with precursor inhomogeneity in a one-dimensional interval. A spike cluster is the combination of several spikes which all approach the same point in the singular limit of small activator diffusivity. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point of the smooth inhomogeneity, where N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O(1)) and small (of order o(1)). The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity), (ii) it contains three different spatial scales two of which are small. (iii) all the expressions can be made explicit and often have a particularly simple form.

Continuous structural transitions in quasi-one-dimensional classical Wigner crystals

We study the structural phase transitions in confined systems of strongly interacting particles. We consider infinite quasi-one-dimensional systems with different pairwise repulsive interactions in the presence of an external confinement following a power law. Within the framework of Landau's theory, we find the necessary conditions to observe continuous transitions and demonstrate that the only allowed continuous transition is between the single-and the double-chain configurations and that it only takes place when the confinement is parabolic. We determine analytically the behavior of the system at the transition point and calculate the critical exponents. Furthermore, we perform Monte Carlo simulations and find a perfect agreement between theory and numerics.

Phase diagram of the one-dimensional Holstein model of spinless fermions

The one-dimensional Holstein model of spinless fermions interacting with dispersionless phonons is studied using a new variant of the density matrix renormalization group. By examining various low-energy excitations of finite chains, the metal-insulator phase boundary is determined precisely and agrees with the predictions of strong coupling theory in the antiadiabatic regime and is consistent with renormalization group arguments in the adiabatic regime. The Luttinger liquid parameters, determined by finite-size scaling, are consistent with a Kosterlitz-Thouless transition.

One-dimensional point interaction with three complex parameters

For a pair of non-Hermitian Hamiltonian H and its Hermitian adjoint H(dagger), there are situations in which their eigenfunctions form a biorthogonal system. We illustrate such a situation by means of a one-particle system with a one-dimensional point interaction in the form of the Fermi pseudo-potential. The interaction consists of three terms with three strength parameters g(i) (i = 1, 2 and 3), which are all complex. This complex point interaction is neither Hermitian nor PT-invariant in general. The S-matrix for the transmission reflection problem constructed with H (or with H(dagger)) in the usual manner is not unitary, but it conforms to the pseudo-unitarity that we define. The pseudounitarity is closely related to the biorthogonality of the eigenfunctions. The eigenvalue spectrum of H with the complex interaction is generally complex but there are cases where the spectrum is real. In such a case H and H(dagger) form a pseudo-Hermitian pair.

Synchronization and Basins of Synchronized in Two-Dimensional Piecewise Maps Via Coupling Three Pieces of One-Dimensional Maps

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

Controlled resolution reconstruction of one-dimensional permittivity profiles

Electromagnetic scattering inverse problems, microwave imaging, reconstruction of dielectric media, remote sensing, tomography

A one-dimensional Keller-Segel equation with a drift issued from the boundary

We investigate in this note the dynamics of a one-dimensional Keller-Segel type model on the half-line. On the contrary to the classical configuration, the chemical production term is located on the boundary. We prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass, they blow-up in finite time above the critical mass, and they converge to some equilibrium at the critical mass. Entropy techniques are presented which aim at providing quantitative convergence results for the subcritical case. This note is completed with a brief introduction to a more realistic model (still one-dimensional).

Spectral functions of half-filled one-dimensional Hubbard rings with varying boundary conditions

We study the effect of varying the boundary condition on: the spectral function of a finite one-dimensional Hubbard chain, which we compute using direct (Lanczos) diagonalization of the Hamiltonian. By direct comparison with the two-body response functions and with the exact solution of the Bethe ansatz equations, we can identify both spinon and holon features in the spectra. At half-filling the spectra have the well-known structure of a low-energy holon band and its shadow-which spans the whole Brillouin zone-and a spinon band present for momenta less than the Fermi momentum. Features related to the twisted boundary condition are cusps in the spinon band. We show that the spectral building principle, adapted to account for both the finite system size and the twisted boundary condition, describes the spectra well in terms of single spinon and holon excitations. We argue that these finite-size effects are a signature of spin-charge separation and that their study should help establish the existence and nature of spin-charge separation in finite-size systems.

Bistable systems with stochastic noise: virtues and limits of effective one-dimensional Langevin equations

The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcings and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.

Black-box decomposition approach for computational hemodynamics: One-dimensional models

Increasing efforts exist in integrating different levels of detail in models of the cardiovascular system. For instance, one-dimensional representations are employed to model the systemic circulation. In this context, effective and black-box-type decomposition strategies for one-dimensional networks are needed, so as to: (i) employ domain decomposition strategies for large systemic models (1D-1D coupling) and (ii) provide the conceptual basis for dimensionally-heterogeneous representations (1D-3D coupling, among various possibilities). The strategy proposed in this article works for both of these two scenarios, though the several applications shown to illustrate its performance focus on the 1D-1D coupling case. A one-dimensional network is decomposed in such a way that each coupling point connects two (and not more) of the sub-networks. At each of the M connection points two unknowns are defined: the flow rate and pressure. These 2M unknowns are determined by 2M equations, since each sub-network provides one (non-linear) equation per coupling point. It is shown how to build the 2M x 2M non-linear system with arbitrary and independent choice of boundary conditions for each of the sub-networks. The idea is then to solve this non-linear system until convergence, which guarantees strong coupling of the complete network. In other words, if the non-linear solver converges at each time step, the solution coincides with what would be obtained by monolithically modeling the whole network. The decomposition thus imposes no stability restriction on the choice of the time step size. Effective iterative strategies for the non-linear system that preserve the black-box character of the decomposition are then explored. Several variants of matrix-free Broyden`s and Newton-GMRES algorithms are assessed as numerical solvers by comparing their performance on sub-critical wave propagation problems which range from academic test cases to realistic cardiovascular applications. A specific variant of Broyden`s algorithm is identified and recommended on the basis of its computer cost and reliability. (C) 2010 Elsevier B.V. All rights reserved.

One-dimensional trapped atoms: critical coupling parameter and critical number of particles

In this paper we consider the case of a Bose gas in low dimension in order to illustrate the applicability of a method that allows us to construct analytical relations, valid for a broad range of coupling parameters, for a function which asymptotic expansions are known. The method is well suitable to investigate the problem of stability of a collection of Bose particles trapped in one- dimensional configuration for the case where the scattering length presents a negative value. The eigenvalues for this interacting quantum one-dimensional many particle system become negative when the interactions overcome the trapping energy and, in this case, the system becomes unstable. Here we calculate the critical coupling parameter and apply for the case of Lithium atoms obtaining the critical number of particles for the limit of stability.