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Article

An h-dependent formulation of the Kadomtsev-Petviashvili hierarchy

Theoretical and Mathematical Physics. 2012. Vol. 171. No. 2. P. 683-690.
Takasaki K., Takebe T.

We briefly review a recursive construction of hbar-dependent solutions of the Kadomtsev-Petviashvili hierarchy. We give recurrence relations for the coefficients X_n of an ħ-expansion of the operator X = X_0 + hbar X_1 + hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = exp(X/hbar). The wave function Psi associated with W turns out to have the WKB (Wentzel-Kramers-Brillouin) form Psi=exp(S/hbar), and the coefficients S_n of the ħ-expansion S = S_0 + hbar S_1 + hbar^2 S_2 + ... 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 tau = hbar^{-2} (F_0 + hbar F_1 + hbar^2 F_2 + ...).