4Structure of integrable PDEs
II Integrable Systems
4.3 Zero curvature representation
There is a more geometric way to talk about integrable systems, which is via
zero-curvature representations.
Suppose we have a function
u
(
x, t
), which we currently think of as being fixed.
From this, we construct
N × N
matrices
U
=
U
(
λ
) and
V
=
V
(
λ
) that depend
on
λ
,
u
and its derivatives. The
λ
will be thought of as a “spectral parameter ”,
like the λ in the eigenvalue problem Lϕ = λϕ.
Now consider the system of PDE’s
∂
∂x
v = U(λ)v,
∂
∂t
v = V (λ)v, (†)
where v = v(x, t; λ) is an N-dimensional vector.
Now notice that here we have twice as many equations as there are unknowns.
So we need some compatibility conditions. We use the fact that
v
xt
=
v
tx
. So
we need
0 =
∂
∂t
U(λ)v −
∂
∂x
V (λ)v
=
∂U
∂t
v + U
∂v
∂t
−
∂V
∂x
v − V
∂v
∂x
=
∂U
∂t
v + UV x −
∂V
∂x
v − V Uv
=
∂U
∂t
−
∂V
∂x
+ [U, V ]
v.
So we know that if a (non-trivial) solution to the PDE’s exists for any initial
v
0
,
then we must have
∂U
∂t
−
∂V
∂x
+ [U, V ] = 0.
These are known as the zero curvature equations.
There is a beautiful theorem by Frobenius that if this equation holds, then
solutions always exist. So we have found a correspondence between the existence
of solutions to the PDE, and some equation in U and V .
Why are these called the zero curvature equations? In differential geometry,
a connection
A
on a tangent bundle has a curvature given by the Riemann
curvature tensor
R = ∂Γ − ∂Γ + ΓΓ − ΓΓ,
where Γ is the Christoffel symbols associated to the connection. This equation is
less silly than it seems, because each of the objects there has a bunch of indices,
and the indices on consecutive terms are not equal. So they do not just outright
cancel. In terms of the connection A, the curvature vanishes iff
∂A
j
∂x
i
−
∂A
i
∂x
j
+ [A
i
, A
j
] = 0,
which has the same form as the zero-curvature equation.
Example. Consider
U(λ) =
i
2
2λ u
x
u
x
−2λ
, V (λ) =
1
4iλ
cos u −i sin u
i sin u −cos u
.
Then the zero curvature equation is equivalent to the sine–Gordon equation
u
xt
= sin u.
In other words, the sine–Gordon equation holds iff the PDEs (
†
) have a solution.
In geometry, curvature is an intrinsic property of our geometric object, say a
surface. If we want to to compute the curvature, we usually pick some coordinate
systems, take the above expression, interpret it in that coordinate system, and
evaluate it. However, we could pick a different coordinate system, and we get
different expressions for each of, say,
∂A
j
∂x
i
. However, if the curvature vanishes in
one coordinate system, then it should also vanish in any coordinate system. So
by picking a new coordinate system, we have found new things that satisfies the
curvature equation.
Back to the real world, in general, we can give a gauge transformation that
takes some solution (
U, V
) to a new (
˜
U,
˜
V
) that preserves the zero curvature
equation. So we can use gauge transformations to obtain a lot of new solutions!
This will be explored in the last example sheet.
What are these zero-curvature representations good for? We don’t have time
to go deep into the matter, but these can be used to do some inverse-scattering
type things. In the above formulation of the sine–Gordon equation. If
u
x
→
0
as |x| → ∞, we write
v =
ψ
1
ψ
2
.
Then we have
∂
∂x
ψ
1
ψ
2
=
i
2
2λ u
x
u
x
−2λ
ψ
1
ψ
2
= iλ
ψ
1
−ψ
2
.
So we know
ψ
1
ψ
2
= A
1
0
e
iλx
+ B
0
1
e
−iλx
as
|x| → ∞
. So with any
v
satisfying the first equation in (
†
), we can associate
to it some “scattering data”
A, B
. Then the second equation in (
†
) tells us how
v
, and thus
A, B
evolves in time, and using this we can develop some inverse
scattering-type way of solving the equation.