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arXiv · 1112.0601

An hbar-expansion of the Toda hierarchy: a recursive construction of solutions

Kanehisa Takasaki·Takashi Takebe
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Abstract

A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{ϕ/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function ϕare assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and ϕ= ϕ_0 + \hbarϕ_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\bar X and ϕare recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form Ψ= e^{S/\hbar} and \barΨ= e^{\bar S/\hbar}, which leads to an \hbar-expansion of the logarithm of the tau function.

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BibTeXRIS

Kanehisa Takasaki, Takashi Takebe. 2011-12-02. An hbar-expansion of the Toda hierarchy: a recursive construction of solutions. https://doi.org/10.1007/s13324-012-0026-5

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