Generalization of the Nonlinear Phase Diffusion Equation

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. 

Thus our reaction-diffusion equations (with e inserted before D V A) may be expressed as 

One may alternatively make an expansion in powers of ]/e rather than e. However, all odd powers of ]/ prove to vanish identically. In any case (4.2.12, 13) are reduced to 

Remember that V carriers the smallness factor ]/e. This means that the coefficient A v [0] of in the -expansion of some quantity A [0] must consist of terms in which the operator V appears 2 v times in all possible combinations. For example, 

As in these expressions, the coefficients of different types of terms for a given Ay[0] will be specified by a superscript (ct) By sorting out various 

---

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

---