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Teoreticheskaya i Matematicheskaya Fizika, 2012, Volume 171, Number 2, Pages 303–311
DOI: https://doi.org/10.4213/tmf6905
(Mi tmf6905)
 

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
Full-text PDF (392 kB) Citations (2)
References:
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
English version:
Theoretical and Mathematical Physics, 2012, Volume 171, Issue 2, Pages 683–690
DOI: https://doi.org/10.1007/s11232-012-0065-y
Bibliographic databases:
Language: Russian
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
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/tmf6905
  • https://doi.org/10.4213/tmf6905
  • https://www.mathnet.ru/eng/tmf/v171/i2/p303
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теоретическая и математическая физика Theoretical and Mathematical Physics
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