4Structure of integrable PDEs
II Integrable Systems
4.4 From Lax pairs to zero curvature
Lax pairs are very closely related to the zero curvature. Recall that we had this
isospectral flow theorem — if Lax’s equation
L
t
= [L, A],
is satisfied, then the eigenvalues of
L
are time-independent. Also, we found that
our eigensolutions satisfied
˜
ψ = ψ
t
+ Aψ = 0.
So we have two equations:
Lψ = λψ
ψ
t
+ Aψ = 0.
Now suppose we reverse this — we enforce that
λ
t
= 0. Then differentiating the
first equation and substituting in the second gives
L
t
= [L, A].
So we can see Lax’s equation as a compatibility condition for the two equations
above. We will see that given any equations of this form, we can transform it
into a zero curvature form.
Note that if we have
L =
∂
∂x
n
+
n−1
X
j=0
u
j
(x, t)
∂
∂x
j
A =
∂
∂x
n
+
n−1
X
j=0
v
j
(x, t)
∂
∂x
j
then
Lψ = λψ
means that derivatives of order
≥ n
can be expressed as linear combinations of
derivatives < n. Indeed, we just have
∂
n
x
ψ = λψ −
n−1
X
j=0
u
j
(x, t)∂
j
x
ψ.
Then differentiating this equation will give us an expression for the higher
derivatives in terms of the lower ones.
Now by introducing the vector
Ψ = (ψ, ∂
x
ψ, ··· , ∂
n−1
x
ψ),
The equation Lψ = λψ can be written as
∂
∂x
Ψ = U(λ)Ψ,
where
U(λ) =
0 1 0 ··· 0
0 0 1 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 ··· 1
λ − u
0
−u
1
−u
2
··· −u
n−1
Now differentiate “ψ
t
+ Aψ = 0” i times with respect to x to obtain
(∂
i−1
x
ψ)
t
+ ∂
i−1
x
m−1
X
j=0
v
j
(x, t)
∂
∂x
j
ψ
| {z }
P
n
j=1
V
ij
(x,t)∂
j−1
x
ψ
= 0
for some
V
ij
(
x, t
) depending on
v
j
, u
i
and their derivatives. We see that this
equation then just says
∂
∂t
Ψ = V Ψ.
So we have shown that
L
t
= [L, A] ⇔
(
Lψ = λψ
ψ
t
+ Aψ = 0
⇔
(
Ψ
x
= U(λ)Ψ
Ψ
t
= V (λ)Ψ
⇔
∂U
∂t
−
∂V
∂x
+ [U, V ] = 0.
So we know that if something can be written in the form of Lax’s equation, then
we can come up with an equivalent equation in zero curvature form.