4Structure of integrable PDEs
II Integrable Systems
4.2 Bihamiltonian systems
So far, this is not too interesting, as we just generalized the finite-dimensional
cases in sort-of the obvious way. However, it is possible that the same PDE
might be able to be put into Hamiltonian form for different
J
’s. These are
known as bihamiltonian systems.
Definition
(Bihamiltonian system)
.
A PDE is bihamiltonian if it can be written
in Hamiltonian form for different J.
It turns out that when this happens, then the system has infinitely many first
integrals in involution! We will prove this later on. This is rather miraculous!
Example. We can write the KdV equation in Hamiltonian form by
u
t
= J
1
δH
1
, J
1
=
∂
∂x
, H
1
[u] =
Z
1
2
u
2
x
+ u
3
dx.
We can check that this says
u
t
=
∂
∂x
∂
∂u
− D
x
∂
∂u
x
1
2
u
2
x
+ u
3
= 6uu
x
− u
xxx
,
and this is the KdV equation.
We can also write it as
u
t
= J
0
δH
0
, J
0
= −
∂
3
∂x
3
+ 4u∂
x
+ 2u
x
, H
0
[u] =
Z
1
2
u
2
dx.
So KdV is bi-Hamiltonian. We then know that
J
1
δH
1
= J
0
δH
0
.
We define a sequence of Hamiltonians {H
n
}
n≥0
via
J
1
δH
n+1
= J
0
δH
n
.
We will assume that we can always solve for
H
n+1
given
H
n
. This can be proven,
but we shall not. We then have the miraculous result.
Theorem.
Suppose a system is bi-Hamiltonian via (
J
0
, H
0
) and (
J
1
, H
1
). It is
a fact that we can find a sequence {H
n
}
n≥0
such that
J
1
δH
n+1
= J
0
δH
n
.
Under these definitions,
{H
n
}
are all first integrals of the system and are in
involution, i.e.
{H
n
, H
m
} = 0
for all n, m ≥ 0, where the Poisson bracket is taken with respect to J
1
.
Proof. We notice the following interesting fact: for m ≥ 1, we have
{H
n
, H
m
} = hδH
n
, J
1
δH
m
i
= hδH
n
, J
0
δH
m−1
i
= −hJ
0
δH
n
, δH
m−1
i
= −hJ
1
δH
n+1
, δH
m−1
i
= hδH
n+1
, J
1
δH
m−1
i
= {H
n+1
, H
m−1
}.
Iterating this many times, we find that for any n, m, we have
{H
n
, H
m
} = {H
m
, H
n
}.
Then by antisymmetry, they must both vanish. So done.