An h-expansion of the Toda hierarchy

作者：Takasaki Kanehisa; Takebe Takashi^*

来源：Analysis and Mathematical Physics, 2012, 2(2): 171-214.

DOI：10.1007/s13324-012-0026-5

A construction of general solutions of the h-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L, M) and ((L) over bar, (M) over bar) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and (W) over bar. The dressing operators are set in an exponential form as W = e(X/h) and (W) over bar = e(phi/h) e((X) over bar /h), and the auxiliary operators X, (X) over bar and the function phi are assumed to have h-expansions X = X-0 + hX(1) + ..., (X) over bar = (X) over bar (0) + h (X) over bar (1) + ... and phi = phi(0) + h phi(1) + ... The coefficients of these expansions turn out to satisfy a set of recursion relations. X, (X) over bar and phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form psi = e(S/h) and (psi) over bar = e((S) over bar /h), which leads to an h-expansion of the logarithm of the tau function.

• 出版日期2012-6