Student Seminar: Classical and Quantum Integrable Systems

7.7 Seminar 7
Exercise 1
Consider the classical Heisenberg model. Show that the formula for the Poisson
brackets between the components of the Lax matrix
[
]
{U(x, λ), U(y, µ)} = r(λ, µ), U(x, λ) ⊗ I + I ⊗ U(y, µ) δ(x − y) ,
with the classical r-matrix
r(λ, µ) = 1 σ i ⊗ σ i
2 λ − µ .
implies that the Poisson bracket between the components of the monodromy matrix
is of the form
[ ∫ 2π ]
T(λ) = P exp dx U(x, λ)
0
{T(λ) ⊗ T(µ)} =
[
]
r(λ, µ), T(λ) ⊗ T(µ) .
Exercise 2
Show that the Jacobi identity for the Poisson bracket
[
]
{T(λ) ⊗ T(µ)} = r(λ, µ), T(λ) ⊗ T(µ) .
implies the classical Yang-Baxter equation for the r-matrix sckew-symmetric r 12 (λ, µ) =
−r 21 (µ, λ):
[r 12 (λ, µ), r 13 (λ, ν)] + [r 12 (λ, µ), r 13 (µ, ν)] + [r 13 (λ, ν), r 23 (µ, ν)] = 0
Check (e.g. by using Mathematica) that the r-matrix
solves the classical Yang-Baxter equation.
r(λ, µ) = 1 σ i ⊗ σ i
2 λ − µ .
Exercise 3
Consider the zero-curvature representation for the KdV equation:
( )
(
)
0 1
u
U =
, V =
x
4λ − 2u
λ + u 0
4λ 2 + 2λu + u xx − 2u 2 − u x
Using abelianization procedure around the pole λ = ∞ find the first four integrals
of motion.
.
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