Five Lectures on Soliton Equations

which commute with the flow defined by (0.1) (we set τ1 = z and τ2 = τ). The important fact is that the right hand sides Pn of equations (0.2) are polynomials in v and its derivatives, i.e., differential polynomials in v. This fact allows us to treat the KdV hierarchy (0.2) in the following way. Consider the ring R = C[v]n≥0 of polynomials in the variables v (n) (we shall write v for v(0)). Let ∂z be the derivation of R defined by the formula ∂z · v (n) = v(n+1) (so that v(n) = ∂ z v). Note that any derivation on R is uniquely determined by its values on v(n)’s via the Leibnitz rule. Any equation of the form ∂τv = P [v, vz , . . . ] gives rise to an evolutionary derivation D of R (i.e., such that commutes with ∂z) defined by the formula D · v = P ∈ R. The condition of commutativity with ∂z implies that D · v (n) = ∂ z · P , so that D can be written as