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 $${\bar {W} = e^{\phi/\hbar}e^{\bar {X}/\hbar}}$$ , and the auxiliary operators $${X, \bar {X}}$$ and the function $${\phi}$$ are assumed to have $${\hbar}$$ -expansions $${X = X_0 + \hbar X_1 + . . . , \bar {X}= \bar {X}_0 + \hbar\bar {X}_1 + . . .}$$ and $${\phi = \phi_0 + \hbar\phi_1 + . . .}$$ . The coefficients of these expansions turn out to satisfy a set of recursion relations. $${X, \bar {X}}$$ and $${\phi}$$ are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form $${\Psi = e^{S/\hbar}}$$ and $${\bar {\Psi}= e^{\bar {S}/\hbar}}$$ , which leads to an $${\hbar}$$ -expansion of the logarithm of the tau function.
Analysis and Mathematical Physics – Springer Journals
Published: Feb 18, 2012
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