308 4 Slow modulation of dispersive waves
4.1.3 Matching between the NLS and KdV equations
The two fundamental equations for weakly nonlinear waves that we have
introduced are the KdV and NLS equations. The former equation
describes long waves, which can be obtained by letting 8 -> 0 and
e -> 0 with 8
2
—
O(e); see Section
2.9.1.
Alternatively, and more gener-
ally, we use a suitable rescaling of the variables which allows us to obtain
the KdV equation for arbitrary
8.
However, this transformation results in
the replacement of 8
2
by e in the governing equations (see equations
(3.10)—(3.15)) with s -• 0; thus the transformation, coupled with e -» 0,
is equivalent to 8 -> 0: long waves. On the other hand, the NLS equation
uses scaled variables which are defined with respect to e only, with
8 (= O(l)) retained as a parameter throughout. Thus, at least for a
class of waves, we have two representations:
rj(x, t\ e, 8) with e -> 0, 8 -> 0 - KdV;
rj(x, t; e, 8) with e -> 0, 8 fixed - NLS.
We might, therefore, suppose that the two descriptions satisfy some
appropriate matching condition in 8. That is, the KdV representation
with 8 -> oo might match with the NLS representation with 8 -> 0. So
we take the short-wave limit of the KdV equation (but, as we shall see,
written in an appropriate form) and the long-wave limit of the NLS
equation.
Let us first construct the limiting form of the NLS equation as 8 -> 0;
this requires that we determine the dominant behaviours of the coeffi-
cients of the equation
-2ikc
p
A
0r
+
<xA
m
+
0A
o
\A
o
\
2
= 0,
where a and ft are given in equations (4.33) and (4.34). (The details of this
calculation, for the DS equations and then for the NLS equation, are
rehearsed in Q4.6 but we shall record the salient features here.) From
Q4.5 we have that
Cp ~ 1 - \&&\ c
g
^ 1 ~ \&& as 8-+0 (4.43)
(cf. equation (2.137) et seq.; the behaviours of c
p
and c
g
, as functions of
8k, are also shown in Figure 4.1), and so
-2\kc
p
~ -2ik