Page 247 -

230 5. Some worked examples arising from physical problems

and then (5.86) and (5.85) in equation (5.81) produces, at leading order, the equation
for

This equation can be solved in general (by introducing and expressing
the solution in terms of once is expressed in these same variables); this is
left as an exercise. We present a simple example: for which the
solution can be written (on introducing

which satisfies on as required by the condition at the
origin.

This problem of finding the path of a light ray has used the idea of multiple scales in
a less routine way; we now examine an equation for which a more familiar approach
(§4.2) is applicable.

E5.13 Raman scattering: a damped Morse oscillator
Under certain circumstances, a small fraction of the incidence light propagating through
a medium may be scattered so that the wavelength of this light differs from that of
the incident light—usually it is of greater wavelength. This is called Raman scattering.
An example of this (Lie & Yuan, 1986), which incorporates the Morse (exponential)
model for the potential energy of atoms as a function of their separation, is

This equation also includes a (weak) linear damping term, which we will characterise
by we wish to find a solution, subject to the initial conditions

Because equation (5.87) is nonlinear (although the underlying solution—valid for
be expressed in terms of elementary functions), we must expect a development
along the lines of that described in E4.3.
We introduce fast and slow scales according to

and then seek a solution which has a constant period (in T). Equation (5.87) therefore