An $\hbar$-dependent formulation of the Kadomtsev–Petviashvili hierarchy

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 $\hbar$-expansion of the operator $X=X_0+\hbar X_1+\hbar^2X_2+\cdots$ for which the dressing operator $W$ is expressed in the exponential form $W=e^{X/\hbar}$. The wave function $\Psi$ associated with $W$ turns out to have the WKB {(}Wentzel–Kramers–Brillouin{\rm)} form $\Psi=e^{S/\hbar}$, and the coefficients $S_n$ of the $\hbar$-expansion $S=S_0+\hbar S_1+\hbar^2S_2+\cdots$ are also determined by a set of recurrence relations. We use this WKB form to show that the associated tau function has an $\hbar$-expansion of the form $\ln\tau=\hbar^{-2}F_0+ \hbar^{-1}F_1+F_2+\dots$.