# An $${\hbar}$$ -expansion of the Toda hierarchy

An $${\hbar}$$ -expansion of the Toda hierarchy 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# An $${\hbar}$$ -expansion of the Toda hierarchy

, Volume 2 (2) – Feb 18, 2012
44 pages

/lp/springer-journals/an-hbar-expansion-of-the-toda-hierarchy-p1GWCWAaLD
Publisher
Springer Journals
Subject
Mathematics; Mathematical Methods in Physics; Analysis
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-012-0026-5
Publisher site
See Article on Publisher Site

### 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 $${\bar {W} = e^{\phi/\hbar}e^{\bar {X}/\hbar}}$$ , 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 18, 2012