Student Seminar: Classical and Quantum Integrable Systems

Indeed, if we differentiate this equation over t we get
∂ t Ψ(2π, t) = ∂ t TΨ(0, t) + T∂ t Ψ(0, t) ,
which, according to the fundamental linear system, gives
L t (2π, t)TΨ(0, t) = ∂ t TΨ(0, t) + TL t (0, t)Ψ(0, t) .
This leads to the same equation for the time evolution of the monodromy matrix as
found before:
∂ t T = [L t , T] .
4.4 Local integrals of motion
The Lax representation of the two-dimensional PDE allows one to exhibit an infinite
number of conservation laws. The procedure to derive the conservation laws from
the Lax representation is a direct analogue of the Zakharov-Shabat construction for
the finite-dimensional case. It is called the abelianization procedure.
Once again we start from the zero-curvature condition
∂ t U − ∂ x V − [V, U] = 0 .
We assume that the matrices U(x, t, λ) and V (x, t, λ) depend on the spectral parameter
λ in a rational way and they have poles at constant, i.e. x, t-independent, values
of λ k . Thus, we can write
U = U 0 + ∑ k
V = V 0 + ∑ k
U k , U k =
V k , V k =
∑−1
r=−n k
U k,r (x, t)(λ − λ r ) r ,
∑−1
r=−m k
V k,r (x, t)(λ − λ r ) r .
The same counting as in the finite-dimensional case shows that the zero-curvature
equations are always compatible: there is one more variable than the number of equations,
but there is a gauge transformation which leads the zero-curvature condition
invariant.
To understand solutions of the zero-curvature condition we will perform a local
analysis around a pole λ = λ k . Our aim is to show that around each singularity one
can perform a gauge transformation which brings the matrices U(λ) and V (λ) to a
diagonal form. Finally, to make the consideration as simple as possible we assume
that the pole is located at zero.
In the neighbourhood of λ = 0 the functions U and V can be expanded into
Laurent series
∞∑
∞∑
U(x, t, λ) = U r (x, t)λ r , V (x, t, λ) = V r (x, t)λ r .
r=−n
r=−m
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