Synchronization of coupled noisy oscillators: Coarse-graining from continuous to discrete phases

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse-graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators

Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis

An initial-boundary value problem for a model of stimulated Raman scattering was considered in [Moskovchenko E.A., Kotlyarov V.P., J. Phys. A: Math. Theor. 43 (2010), 055205, 31 pages]. The authors showed that in the long-time range $t\to+\infty$ the $x>0$, $t>0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results by Moskovchenko and Kotlyarov

Periodic solutions of systems with asymptotically even nonlinearities

New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate ... next order ... terms are obtained for the 2 Pi-periodic problem for the scalar equation x'' +n2x=g(|x|)+f(t,x)+b(t) with bounded g(u) and f(t,x) -> 0 as |x| -> 0. The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory. AMS subject classifications: 47Hll, 47H30

Breakdown of the standard Perturbation Theory and Moving Boundary Approximation for "Pulled" Fronts

The derivation of a Moving Boundary Approximation or of the response of a coherent structure like a front, vortex or pulse to external forces and noise, is generally valid under two conditions: the existence of a separation of time scales of the dynamics on the inner and outer scale and the existence and convergence of solvability type integrals. We point out that these conditions are not satisfied for pulled fronts propagating into an unstable state: their relaxation on the inner scale is power law like and in conjunction with this, solvability integrals diverge. The physical origin of this is traced to the fact that the important dynamics of pulled fronts occurs in the leading edge of the front rather than in the nonlinear internal front region itself. As recent work on the relaxation and stochastic behavior of pulled fronts suggests, when such fronts are coupled to other fields or to noise, the dynamical behavior is often qualitatively different from the standard case in which fronts between two (meta)stable states or pushed fronts propagating into an unstable state are considered.Comment: pages Latex, submitted to a special issue of Phys. Rep. in dec. 9

A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation

We propose a novel second order in time numerical scheme for Cahn-Hilliard-Navier- Stokes phase field model with matched density. The scheme is based on second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is uncondition- ally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the weak coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme

Accurate macroscale modelling of spatial dynamics in multiple dimensions

Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction-diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the modelling of systems by a type of finite elements, and the `equation free' macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/PDE to as high an order as desired. Computer algebra handles the considerable algebraic details as seen in the specific application to the Ginzburg--Landau PDE. However, higher order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction-diffusion PDEs and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv admin note: substantial text overlap with arXiv:0904.085