An ħ-dependent formulation of the Kadomtsev-Petviashvili hierarchy

We briefly review a recursive construction of ħ-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients Xn of an ħ-expansion of the operator X = X0 + ħX1 + ħ2X2 + ... for which the dressing operator W is expressed in the exponential form W = eX/ħ. The wave function Ψ associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form Ψ = eS/kh, and the coefficients Sn of the ħ-expansion S = S0 + ħS1 + ħ2S2 + ... are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an ħ-expansion of the form log τ = ħ−2F0 + ħ−1F1 + F2 + ....