4–2
OCIAM Mathematical Institute
University of Oxford
then, along these curves,
d
p
d
τ
=

∂F
∂x

p
∂u
,
d
q
d
τ
=

∂y

q
.
(4.8)
We therefore have a system of four ODEs for
x
,
y
,
p
and
q
satisﬁed along the rays. Recall,
though, that in general
F
depends on
u
also, so to close the system we also need an ODE for
u
along the rays, namely
d
u
d
τ
=
d
x
d
τ
+
d
y
d
τ
=
p
∂p
+
q
∂q
.
(4.9)
In summary, we have the following system of ODEs for
x
,
y
,
p
,
q
and
u
, known as
Charpit’s
equations
:
d
x
d
τ
=
,
(4.10a)
d
y
d
τ
=
,
(4.10b)
d
p
d
τ
=


p
,
(4.10c)
d
q
d
τ
=


q
,
(4.10d)
d
u
d
τ
=
p
+
q
.
(4.10e)
It is easily veriﬁed that these reduce to the usual characteristic equations
d
x
d
τ
=
a,
d
y
d
τ
=
b,
d
u
d
τ
=
c,
(4.11)
for quasilinear equations where
F
takes the form (4.4).
4.3
Boundary data
As for quasilinear scalar equations, Cauchy data speciﬁes
u
along some curve Γ in the
(
x,y
)plane:
x
=
x
0
(
s
)
,
y
=
y
0
(
s
)
,
u
=
u
0
(
s
)
,
(4.12)
for
s
in some (possibly inﬁnite) interval. We also require initial conditions for
p
and
q
, say
p
=
p
0
(
s
),
q
=
q
0
(
s
), which are obtained by diﬀerentiating
u
0
with respect to
s
and using the
PDE (4.1):
d
u
0
d
s
=
p
0
d
x
0
d
s
+
q
0
d
y
0
d
s
,
F
(
p
0
,q
0
,u
0
,x
0
,y
0
) = 0
.
(4.13)
By the implicit function theorem, the two equations (4.13) may be solved uniquely (in prin
ciple, if not explicitly) for
p
0
and
q
0
provided the condition
d
x
0
d
s
0

d
y
0
d
s
0
6
= 0
(4.14)