scholarly journals Pattern Formation of a Keller-Segel Model with the Source Termup(1-u)

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shengmao Fu ◽  
Fenli Cao
Keyword(s):  
Nonlinear Dynamics ◽  
Pattern Formation ◽  
Finite Number ◽  
Time Scale ◽  
Source Term ◽  
Initial Perturbation ◽  
Constant Equilibrium ◽  
General Perturbation

Nonlinear dynamics near an unstable constant equilibrium in a Keller-Segel model with the source termup(1-u)is considered. It is proved that nonlinear dynamics of a general perturbation is determined by the finite number of linear growing modes over a time scale ofln(1/δ), whereδis a strength of the initial perturbation.

scholarly journals A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Shengmao Fu ◽  
Ji Liu
Keyword(s):  
Boundary Value Problem ◽  
Source Term ◽  
Nonlinear Evolution ◽  
Initial Perturbation ◽  
Neumann Boundary ◽  
Chemical Signal ◽  
Linear Dynamics ◽  
Neumann Boundary Value Problem ◽  
Time Period ◽  
Signal Substance

This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in ad-dimensional box(d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitudeδ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

scholarly journals Building better oscillators using nonlinear dynamics and pattern formation

Pramana ◽  
2015 ◽  
Vol 84 (3) ◽  
pp. 455-471
Author(s):  
M C CROSS ◽  
EYAL KENIG ◽  
JOHN-MARK A ALLEN
Keyword(s):  
Nonlinear Dynamics ◽  
Pattern Formation

Implementing New Nonlinear Term in Third Generation Wave Models

2014 ◽  
Author(s):  
Odin Gramstad ◽  
Alexander Babanin
Keyword(s):  
Kinetic Equation ◽  
Spread Spectrum ◽  
Time Scale ◽  
Source Term ◽  
Wave Spectrum ◽  
Wave Model ◽  
Third Generation ◽  
Spectral Evolution ◽  
Linear Interaction ◽  
Wavewatch Iii

The non-linear interaction term is one of the three key source functions in every third-generation spectral wave model. An update of physics of this term is discussed. The standard statistical/phase-averaged description of the nonlinear transfer of energy in the wave spectrum (wave-turbulence) is based on Hasselmann’s kinetic equation [1]. In the derivation of the kinetic equation (KE) it is assumed that the evolution takes place on the slow O(ε−4) time scale, where ε is the wave steepness. This excludes the effects of near-resonant quartet interactions that may lead to spectral evolution on the ‘fast’ O(ε−2) time scale. Generalizations of the KE (GKE) that enable description of spectral evolution on the O(ε−2) time scale [2–4] are discussed. The GKE, first solved numerically in [4], is implemented as a source term in the third generation wave model WAVEWATCH-III. The new source term (GKE) is tested and compared to the other nonlinear-interaction source terms in WAVEWATCH-III; the full KE (WRT method) and the approximate DIA method. It is shown that the GKE gives similar results to the KE in the case of a relatively broad banded and directional spread spectrum, while it shows somewhat larger difference in the case of a more narrow banded spectrum with narrower directional distribution. We suggest that the GKE may be a suitable replacement to the KE in situations where ‘fast’ spectral evolution takes place.

Lattice Boltzmann for reactive flows

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Heterogeneous Catalysis ◽  
Pattern Formation ◽  
Chemical Reactions ◽  
Lattice Boltzmann ◽  
Source Term ◽  
Reaction Diffusion ◽  
Reactive Flows ◽  
Fluid Equations

The dynamics of reactive flows lies at the heart of several important applications, such as combustion, heterogeneous catalysis, pollutant conversion, pattern formation in biology and many others. In general, LB is well suited to describe reaction-diffusion applications with flowing species. This chapter provides the basic guidelines to include reactive phenomena within the LBE formalism. Reactive flows obey the usual fluid equations, augmented with a reactive source term, accounting for species transformations due to chemical reactions. Such term comes typically in the form of a polynomial product of the mass densities of the reacting species.

The Planar to Cellular Transition and the Long Time Scale Dynamics of Pattern Formation During Thin-Film Directional Solidification

1990 ◽  
Vol 205 ◽  
Author(s):  
John T.C. Lee ◽  
Robert A. Brown
Keyword(s):  
Thin Film ◽  
Spectral Analysis ◽  
Pattern Formation ◽  
Directional Solidification ◽  
Time Scale ◽  
Linear Stability Theory ◽  
Morphological Instability ◽  
Thin Sample ◽  
Long Time ◽  
Crystal Interface

AbstractExperimental observations of the melt/crystal interface in the directional solidification of a thin sample of a binary alloy are reported for conditions only slightly beyond those for the onset of morphological instability. Spectral analysis for long times shows dynamics leading to a band of most probable wavelengths with sharply selected peaks at wavelengths significantly below the most dangerous wavelength predicted by linear stability theory. These large-amplitude cells develop first in packets that spread to fill the interface.

scholarly journals NONLINEAR DYNAMICS OF LONG-WAVE PERTURBATIONS OF THE INVISCID KOLMOGOROV FLOW

2019 ◽  
Vol 47 (1) ◽  
pp. 64-65
Author(s):  
M.V. Kalashnik ◽  
M.V. Kurgansky
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Oscillations ◽  
Initial Perturbation ◽  
Analytical Formula ◽  
Basic Research ◽  
Elliptic Functions ◽  
Structural Features ◽  
Kolmogorov Flow ◽  
Long Wave ◽  
The Galerkin Method

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).

scholarly journals The Moisture Mode in the Quasi-Equilibrium Tropical Circulation Model. Part II: Nonlinear Behavior on an Equatorial β Plane

2009 ◽  
Vol 66 (6) ◽  
pp. 1525-1542 ◽  
Author(s):  
Masahiro Sugiyama
Keyword(s):  
Nonlinear Dynamics ◽  
Time Scale ◽  
Large Scale ◽  
Circulation Model ◽  
Scaling Analysis ◽  
Quasi Equilibrium ◽  
Tropical Circulation ◽  
Large Scale Disturbance ◽  
Nonlinear Advection ◽  
Moisture Mode

Abstract Numerical calculations of a simplified quasi-equilibrium tropical circulation model (QTCM) on the equatorial β plane have been performed to explore the nonlinear regime of the moisture mode. Sensitivity tests have examined the effects of nonlinear advection and nonlinear wind-induced surface heat exchange (WISHE). Starting from a spatially homogeneous radiative–convective equilibrium with some background gustiness, the model develops quasi-stationary moisture modes, as expected from linear analysis. Upon nonlinear saturation due to thermodynamic limiting processes, a different regime emerges. A classical Gill model augmented with a prognostic humidity variable is found to be able to capture the nonlinear dynamics of the moisture mode; the time evolution solely resides in the humidity variable, and it is possible to understand its dynamics by examining the humidity budget. A scaling analysis shows that the approximation with a Gill model is valid for disturbances whose time scale is much longer than the damping time scale of equatorial waves. When nonlinear WISHE is included, a large-scale disturbance of wavenumber 1 grows and moves westward because evaporation is enhanced to the west of increased convection. Turning on nonlinear advection leads to disturbances of wavenumber 10 that translate eastward via advection of dry air by Rossby gyres. Combining nonlinear WISHE and nonlinear advection leads to gross moist instability and prohibits long-term numerical integration, which suggests that QTCM must be refined to fully describe the nonlinear dynamics of the moisture mode.

Sign in / Sign up

Export Citation Format

Share Document