Student Seminar: Classical and Quantum Integrable Systems

We can add to M a polynomial of L to shift the pole in λ from infinity to the zero
point. In fact one has to take
P (L) = λ(αL 2 + βL + γ) ,
where
α = − 1
I 1 I 2 I 3
,
β = I2 1 + I 2 2 + I 3 3
2I 1 I 2 I 3
,
With this choice we get
γ = (I 1 + I 2 + I 3 )(I 2 + I 3 − I 1 )(I 1 + I 2 − I 2 )(I 1 + I 2 − I 3 )
16I 1 I 2 I 3
.
M(λ) → λI + Ω − P (L) = Ω − α(I 2 J + JI 2 ) − βJ − α } {{ } λ J 2 .
=0
Thus we have a new Lax pair
L(λ) = I 2 + 1 λ J , M(λ) = −α λ J 2 .
Check
˙L = 1 λ ˙ J = [M, L] = − α λ [I2 , J 2 ]
Thus, we should get
These are precisely the Euler equations
J ˙ = − 1 [I 2 , J 2 ] .
I 1 I 2 I 3
Here
dJ 1
dt = a 1J 2 J 3 ,
dJ 2
dt = a 2J 3 J 1 ,
dJ 3
dt = a 3J 1 J 2 .
a 1 = I 2 − I 3
I 2 I 3
, a 2 = I 3 − I 1
I 1 I 3
, a 3 = I 1 − I 2
I 1 I 2
.
The eigenvalues of J are (0, i√
⃗J 2
, −i√
⃗J 2
) and they are non-dynamical since ⃗ J 2
belongs to the center of the Poisson structure.
4. Two-dimensional integrable PDEs
Here we introduce some interesting examples of infinite-dimensional Hamiltonian
systems which appear to be integrable.
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