Kuramoto-Sivashinsky equation

In order to investigate such front instabilities quantitatively one may derive an equation for the profile ( i(y, t) of the front directly from the reaction-diffusion equation. This Kuramoto-Sivashinsky equation 1691... 

I. Procaccia, M. H. Jensen, V. S. L vov, K. Sneppen, R. Zeitak. Surface roughening and the long-wavelength properties of the Kuramoto-Sivashinsky equation. Phys Rev A 45 3220, 1992. 

The leading order equation governing the evolution of the phase is the Kuramoto-Sivashinsky equation. ... 

It exhibits solutions in the form of spatially-irregular cells" splitting and merging in a chaotic manner in time. An example of spatio-temporal behavior of a chaotic solution of the Kuramoto-Sivashinsky equation (158) as well as a snapshot of this solution at a particular moment of time are shown in Fig.20. 

Conventionally used Darcy s law and Richards equation (2 order partial differential equations) are exanq>les of a system with a positive feedback, but witiiout a negative feedback conqmnent. Negative and positive feedback mechanisms are taken into account by using tiie difference-differential equation for soil-moisture balance (75) and the Kuramoto-Sivashinsky equation 74) (see below). 

The phase space analysis of both inlet and outlet capillary pressures produce a zero Lyapunov exponent (I), itr lying that this dynamic system can be described by differential equations (50). A conq>arison of the pseudo-phase-space three-dunensional attractors for the inlet and outlet capillary pressures shows (Figures 14b and 14c) that these attractors are analogous to those described using the solution of the Kuramoto-Sivashinsky equation (Figures 14d and 14e) discussed below. 

Kuramoto-Sivashinsky Equation for Chaotic Flow through Fractures... 

Figure 14. (a) Temporal variations of inlet and outlet capillary pressures from Experiment C of Persoff and Pruess (25) using Stripa natural rock under controlled gas-liquid volumetric flow ratio of two (the data are noise-reduced using Fourier transformation (b) and (c) 3D attractors of inlet (time delay, ir=12) and outlet (r=7) capillary pressures and (d) and (e) 3D attractors of the solution of the Kuramoto-Sivashinsky equation (7) for the upper boundary of the flow domain ( t=6) and the lower boundary of the flow domain (r=7). Time delays were determined using the average mutual information function (50). 

When g = 0, eq.(149) provides an extension of the Kuramoto-Sivashinsky equation to the case presented here. There is a new term, Z (C ) due to the surface tension gradient-driven Marangoni stress. 

In the previous sections, we briefly introduced a number of different specific models for crystal growth. In this section we will make some further simplifications to treat some generic behavior of growth problems in the simplest possible form. This usually leads to some nonlinear partial differential equations, known under names like Burgers, Kardar-Parisi-Zhang (KPZ), Kuramoto-Sivashinsky, Edwards-Anderson, complex Ginzburg-Landau equation and others. 

A 4" -order partial differential equation, called die Kuramoto-Sivashinsky (K-S) equation, is given in a canonical form by (74)... 

This evolution equation (6.6), of the Benney type , contains the small parameter e, a fact which is due to that H(i , x) has not been expanded at it should in a fully consistent asymptotic approach through an expansion with respect to e. Of course we may expand H(t, x) in different ways and we shall investigate, here only the same kind of phenomenon that the one which led the the Kuramoto - Sivashinsky (so-called, KS ) equation. In order to achieve this, we put in (6.6) ... 

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