Student Seminar: Classical and Quantum Integrable Systems

4.3 Zero-curvature representation
The inverse scattering method (the method of finding certain class of solutions of
a non-linear integrable PDE) is based on the following remarkable observation. A
two-dimensional PDE appears as the consistency condition of the overdetermined
system of equations
∂Ψ
= U(x, t, λ)Ψ ,
∂x
∂Ψ
= V (x, t, λ)Ψ .
∂t
for a proper choice of the matrices U(x, t, λ) and V (x, t, λ). The consistency condition
arises upon differentiation the first equation w.r.t. t and the second w.r.t. x:
∂ 2 Ψ
(
)
∂t∂x = ∂ tU(x, t, λ)Ψ + U(x, t, λ)∂ t Ψ = ∂ t U(x, t, λ) + U(x, t, λ)V (x, t, λ) Ψ ,
∂ 2 Ψ
(
)
∂x∂t = ∂ xV (x, t, λ)Ψ + V (x, t, λ)∂ x Ψ = ∂ x V (x, t, λ) + V (x, t, λ)U(x, t, λ) Ψ ,
which implies the fulfilment of the following relation
∂ t U − ∂ x V + [U, V ] = 0 .
If we introduce a gauge field L α with components L x = U, L t = V , then the last
relation is the condition of vanishing of the curvature of L α :
F αβ (L ) ≡ ∂ α L β − ∂ β L α − [L α , L β ] = 0 .
Example: KdV equation. Introduce the following 2 × 2 matrices
( )
(
)
0 1
u
U =
, V =
x 4λ − 2u
λ + u 0
4λ 2 + 2λu + u xx − 2u 2 .
−u x
Show by direct computation that
(
)
0 0
∂ t U − ∂ x V + [U, V ] =
.
u t + 6uu x − u xxx 0
Example: Sine-Gordon equation. Introduce the following 2 × 2 matrices
U = β 4i φ tσ 3 + k 0
i
V = β 4i φ xσ 3 + k 1
i
βφ
sin
2 σ 1 + k 1 βφ
cos
i 2 σ 2
βφ
sin
2 σ 1 + k 0 βφ
cos
i 2 σ 2 ,
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