Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Abstract

Summary. Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small-as, for instance, can be the case in doublelayer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the 'Landau reduction' to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called 'localised structures' in the underlying system: They connect simple periodic patterns at x → ±∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.

A Geometric Approach to Singularly Perturbed Nonlocal Reaction-Diffusion Equations

“…Such equations arise in microwave heating applications [18], activator-inhibitor chemical systems [20], thermistor [19], and ballast resistor problems [3], among other places. Nonlocal equations also are of interest because a higher-dimensional system can often be recast into a lower-dimensional nonlocal system [5,7,9,14]. Existence of stationary solutions for scalar nonlocal equations has been considered by [3,7,9,18,19].…”

“…Nonlocal equations also are of interest because a higher-dimensional system can often be recast into a lower-dimensional nonlocal system [5,7,9,14]. Existence of stationary solutions for scalar nonlocal equations has been considered by [3,7,9,18,19]. Stability of solutions for scalar nonlocal equations has been studied by [1,3,5,10,11,12,14].…”

“…Indeed, there are examples in the literature in which what appear to be asymptotic approximations of solutions are derived, but in fact there are no true solutions nearby at all. In the spirit of the approach that we shall take, Doelman and Rottschäfer [7] have studied a nonlocal reduction of the Ginzburg-Landau equations from a geometric point of view. The main purpose of their work is to compare a nonlocal model to a singularly perturbed one to determine whether the former is a good approximant of the latter.…”

Accessible solitons in complex Ginzburg-Landau media

“…Such equations were derived in diverse physical settings, including, in particular, combustion [18], the Faraday's parametric instability [19], electrochemistry [20], and waves of reaction-diffusion waves [21]. Various forms of nonlocal CGL equations were also proposed as mathematical models for the description of nonlinear-dissipative media with long-range interactions [22,23].…”

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

“…Fourth-order differential equations occur in various areas of mathematics such as viscoelastic and inelastic flows, beam theory, Lifshitz point in phase transition physics (e.g., nematic liquid crystal, crystals, and ferroelectric crystals) [1], the rolls in a Rayleigh-Benard convection cell (two parallel plates of different temperature with a liquid in between) [2], spontaneous pattern formation in second-order materials (e.g., polymeric fibres) [3], the waves on a suspension bridge [4,5], geological folding of rock layers [6], buckling of a strut on a nonlinear elastic foundation [7], traveling water waves in a shallow channel [8], pulse propagation in optical fibers [9], system of two reaction diffusion equation [10], and so forth.…”