In this paper, we consider an initial-value problem for the Korteweg-de Vries equation with time dependent coefficients. The normalized variable coefficient Korteweg-de Vries equation considered is given by
u
t
+
Φ
(
t
)
u
u
x
+
Ψ
(
t
)
u
x
x
x
=
0
,
−
∞
>
x
>
∞
,
t
>
0
,
\begin{equation*} u_{t}+ \Phi (t) u u_{x}+ \Psi (t) u_{xxx}=0, \quad -\infty >x>\infty , \quad t>0, \end{equation*}
where
x
x
and
t
t
represent dimensionless distance and time respectively, whilst
Φ
(
t
)
\Phi (t)
,
Ψ
(
t
)
\Psi (t)
are given functions of
t
(
>
0
)
t (>0)
. In particular, we consider the case when the initial data has a discontinuous expansive step, where
u
(
x
,
0
)
=
u
+
u(x,0)=u_{+}
for
x
≥
0
x \ge 0
and
u
(
x
,
0
)
=
u
−
u(x,0)=u_{-}
for
x
>
0
x>0
. We focus attention on the case when
Φ
(
t
)
=
t
δ
\Phi (t)=t^{\delta }
(with
δ
>
−
2
3
\delta >-\frac {2}{3}
) and
Ψ
(
t
)
=
1
\Psi (t)=1
. The constant states
u
+
u_{+}
,
u
−
u_{-}
(
>
u
+
>u_{+}
) and
δ
\delta
are problem parameters. The method of matched asymptotic coordinate expansions is used to obtain the large-
t
t
asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in
x
≥
u
−
(
δ
+
1
)
t
(
δ
+
1
)
x \ge \frac {u_{-} }{(\delta +1)}t^{(\delta +1)}
as
t
→
∞
t \to \infty
, while the solution is oscillatory in
x
>
u
−
(
δ
+
1
)
t
(
δ
+
1
)
x>\frac {u_{-}}{(\delta +1)}t^{(\delta +1)}
as
t
→
∞
t \to \infty
. We conclude with a brief discussion of the structure of the large-
t
t
solution of the initial-value problem when the initial data is step-like being continuous with algebraic decay as
|
x
|
→
∞
|x| \to \infty
, with
u
(
x
,
t
)
→
u
+
u(x,t) \to u_{+}
as
x
→
∞
x \to \infty
and
u
(
x
,
t
)
→
u
−
(
>
u
+
)
u(x,t) \to u_{-} (>u_{+})
as
x
→
−
∞
x \to -\infty
.