Nonlinear Phase Diffusion Equation

Derivation of the nonlinear phase diffusion equation. The longwave nature of the two basic instabilities of roll patterns described above shows that longwave distortions of rolls are of major interest. Let us consider longwave solutions of the NWS equation (73)... 

At the next order, a system of coupled equations for R2 and Oq is obtained. After eliminating R2, one can obtain the following nonlinear phase diffusion equation [49] (below we drop the subscript 0) ... 

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

The nonlinear phase diffusion equation (3.3.5) now takes the explicit form dy/... 

Generalization of the Nonlinear Phase Diffusion Equation dQtit)... 

We are now ready to take ep(X) as representing a diffusion term, and to generalize the nonlinear phase diffusion equation (3.3.5). To say that 6 is equivalent to saying that the operator V carries the smallness factor j/e, that is, whenever a spatial derivative appears, it generates a small quantity of order /s. In fact, V always appears as the combination j/e V in the theory below. 

As a preliminary to investigating two-dimensional wave patterns, we first make a brief inspection of some particular solutions of the nonlinear phase diffusion equation in one dimension. 

Note that (6.6.3) reduces to the nonlinear phase diffusion equation if we neglect the space dependence of R, The previous numerical simulation suggests that R... 

Although some physical implications of the nonlinear phase diffusion equation with positive a have been discussed in Sect. 6.2, we have not yet discussed the same equation in relation to the wavefront dynamics this should be done before going into the phase turbulence equation. Let the wavefront form a straight line which is slightly non-parallel to the y direction (Fig. 7.9). Then the nonlinear phase diffusion equation becomes... 

When a is positive, (7.3.15) is consistent with the ordinary picture that locally convex fronts tend to be flattened. If a given front is concave, the flattening effect will ultimately be balanced with the sharpening effect (coming from the very fact that the front has a finite propagation velocity), so that formation of a shocklike structure is expected (Fig. 7.10). We already know, in fact, that the nonlinear phase diffusion equation admits a family of shock solutions (though in a different physical context see Sect. 6.2). In the present notation, the shock solutions (6.2.6) are expressed as... 

Although the derivation of the phase equations has been sketched in the weakly nonlinear regime, i.e. starting from the amplitude equations, they are far more general and may be derived far from the onset of the structures. The starting point is then the finite amplitude planform the wavevectors of which are allowed to vary slowly over the extent of the reactor. In this context, that follows the work of Witham on nonlinear wavetrains, the phase diffusion equations appear as solvability conditions [28,61]. 

Let us remember that Eqns. (12.22) and (12.23) have to be coupled to the diffusion equations in the a and 0 phases in order to complete the total set of kinetic equations for the phase transformation (Le., the advancement of the interface). This set is very complicated and nonlinear and may lead to non-monotonic behavior of vb and the chemical potentials of the components in space and time, as has been observed experimentally (Figs. 10-13 and 12-9). Coherency stresses and other complications such as plastic flow have been neglected in this discussion. 

It would appear that the study of the diffusion equation subject to a phase change at one boundary is in a relatively satisfactory state, provided simple boundary conditions of the first, second, or third kind are specified. From a mathematical point of view, the interesting features of the problem arise from the nonlinearity, exhibited for all but a few particular boundary motions. A wide variety of approximate and numerical methods have been employed, and it has frequently been difficult for workers in one specialized field of activity to become conversant with similar approaches made by investigators in other areas. It is hoped that the present work will, to some extent, alleviate this problem. 

NONLINEAR EQUATION SOLVER PROGRAM Liquid-phase diffusivity of benzene at 55°C (cm /s) ... 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

The effect of laser phase fluctuations on PIER4 has been considered by Agarwal, and detailed line shapes were calculated. In a recent publication we have derived (within the phase diffusion model) equations of motion in the limit of short correlation times. The effect of the stochastic phase fluctuations was shown to be similar to T2 dephasing processes, and a procedure was given for the inclusion of this similarity in many nonlinear processes. In particular, two predictions were made ... 

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