We study the initial value problem for the Korteweg-de Vries equation
\[
(
i
)
u
t
−
6
u
u
x
+
ε
2
u
x
x
x
=
0
({\text {i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0
\]
in the limit of small dispersion, i.e.,
ε
→
0
\varepsilon \to 0
. When the unperturbed equation
\[
(
ii
)
u
t
−
6
u
u
x
=
0
({\text {ii}})\quad {u_t} - 6u{u_x} = 0
\]
develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as
ε
→
0
\varepsilon \to 0
. b. We show that in the neighborhood of a point
(
x
,
t
)
(x,\,t)
the solution
u
(
x
,
t
,
ε
)
u(x,\,t,\,\varepsilon )
can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order
O
(
1
/
ε
)
O(1/\varepsilon )
. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of
x
x
and
t
t
. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.