3Inverse scattering transform
II Integrable Systems
3.3 Lax pairs
The final ingredient to using the inverse scattering transform is how to relate
the evolution of the potential to the evolution of the scattering data. This is
given by a lax pair.
Recall that when we studied Hamiltonian systems at the beginning of the
course, under a Hamiltonian flow, functions evolve by
df
dt
= {f, H}.
In quantum mechanics, when we “quantize” this, in the Heisenberg picture, the
operators evolve by
i~
dL
dt
= [L, H].
In some sense, these equations tell us
H
“generates” time evolution. What we
need here is something similar — an operator that generates the time evolution
of our operator.
Definition (Lax pair). Consider a time-dependent self-adjoint linear operator
L = a
m
(x, t)
∂
m
∂x
m
+ ···+ a
1
(x, t)
∂
∂x
+ a
0
(x, t),
where the
{a
i
}
(possibly matrix-valued) functions of (
x, t
). If there is a second
operator A such that
L
t
= LA − AL = [L, A],
where
L
t
= ˙a
m
∂
m
∂x
m
+ ···+ ˙a
0
,
denotes the derivative of L with respect to t, then we call (L, A) a Lax pair.
The main theorem about Lax pairs is the following isospectral flow theorem:
Theorem
(Isospectral flow theorem)
.
Let (
L, A
) be a Lax pair. Then the
discrete eigenvalues of
L
are time-independent. Also, if
Lψ
=
λψ
, where
λ
is a
discrete eigenvalue, then
L
˜
ψ = λ
˜
ψ,
where
˜
ψ = ψ
t
+ Aψ.
The word “isospectral” means that we have an evolving system, but the
eigenvalues are time-independent.
Proof.
We will assume that the eigenvalues at least vary smoothly with
t
, so
that for each eigenvalue
λ
0
at
t
= 0 with eigenfunction
ψ
0
(
x
), we can find some
λ(t) and ψ(x, t) with λ(0) = λ
0
, ψ(x, 0) = ψ
0
(x) such that
L(t)ψ(x, t) = λ(t)ψ(x, t).
We will show that in fact
λ
(
t
) is constant in time. Differentiating with respect
to t and rearranging, we get
λ
t
ψ = L
t
ψ + Lψ
t
− λψ
t
= LAψ −ALψ + Lψ
t
− λψ
t
= LAψ −λAψ + Lψ
t
− λψ
t
= (L − λ)(ψ
t
+ Aψ)
We now take the inner product ψ, and use that kψk = 1. We then have
λ
t
= hψ, λ
t
ψi
= hψ, (L −λ)(ψ
t
+ A
ψ
)i
= h(L − λ)ψ, ψ
t
+ A
ψ
i
= 0,
using the fact that L, hence L − λ is self-adjoint.
So we know that
λ
t
= 0, i.e. that
λ
is time-independent. Then our above
equation gives
L
˜
ψ = λ
˜
ψ,
where
˜
ψ = ψ
t
+ Aψ.
In the case where
L
is the Schr¨odinger operator, the isospectral theorem tells
us how we can relate the evolution of some of the scattering data (namely the
χ
n
), to some differential equation in
L
(namely the Laxness of
L
). For a cleverly
chosen
A
, we will be able to relate the Laxness of
L
to some differential equation
in
u
, and this establishes our first correspondence between evolution of
u
and
the evolution of scattering data.
Example. Consider
L = −∂
2
x
+ u(x, t)
A = 4∂
3
x
− 3(u∂
x
+ ∂
x
u).
Then (L, A) is a Lax pair iff u = u(x, t) satisfies KdV. In other words, we have
L
t
− [L, A] = 0 ⇐⇒ u
t
+ u
xxx
− 6uu
x
= 0.