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This article is cited in 2 scientific papers (total in 2 papers)
An $\hbar$-dependent formulation of the Kadomtsev–Petviashvili hierarchy
K. Takasakia, T. Takebeb a Graduate School of Human and
Environmental Studies, Kyoto University,
Yoshida, Sakyo, Kyoto, Japan
b Faculty of Mathematics, Higher School of Economics,
Moscow, Russia
Abstract:
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 $\ln\tau=\hbar^{-2}F_0+ \hbar^{-1}F_1+F_2+\dots$.
Keywords:
$\hbar$-expansion, Riemann–Hilbert problem, quantization, recurrence relation.
Received: 30.04.2011
Citation:
K. Takasaki, T. Takebe, “An $\hbar$-dependent formulation of the Kadomtsev–Petviashvili hierarchy”, TMF, 171:2 (2012), 303–311; Theoret. and Math. Phys., 171:2 (2012), 683–690
Linking options:
https://www.mathnet.ru/eng/tmf6905https://doi.org/10.4213/tmf6905 https://www.mathnet.ru/eng/tmf/v171/i2/p303
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Abstract page: | 386 | Full-text PDF : | 154 | References: | 54 | First page: | 13 |
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