Setting the components of the Lax connection to be L x = ( 0 1 ϕ ϕ − − λ λ 0 ) , L t = ( − − ϕ ϕ x 2 ϕ ϕ + 4 λ λ 2 ϕ ϕ 2 − − ϕ ϕ x x + 2 ϕ ϕ λ λ − − 4 λ λ 2 ϕ ϕ x ) , the KdV equation is equivalent to the zero-curvature equation for the Lax connection, ∂ ∂ t L x − − ∂ ∂ x L t + [ L x , L t ] = 0.