KP族的hbar展开：解的递归构造
hbar-expansion of KP hierarchy: Recursive construction of solutions
摘要：
依赖于\hbar的KP层级是KP层级的公式，其依赖于普朗克常量\hbar，并简化为无色散KP层级为\hbar -> 0。本文利用Lax和Orlov-Schulman算子对（L，M）的Riemann-Hilbert问题，给出了其解的递归构造。本文将Riemann-Hilbert问题转化为算子X = X_0 + \hbar X_1 + \hbar^2 X_2 +.的\hbar展开式的系数X_n的一组递推关系。其中修整算子W以指数形式W = \exp（X/ \hbar）表示。给定最低阶项X_0，可以求解递归关系以获得高阶项。与W相关联的波函数δ证明具有WKB形式δ = \exp（S/ \hbar），并且\hbar展开式S = S_0 + \hbar S_1 + \hbar^2 S_2 +.的系数S_n，也是由一组递归关系决定的。这种WKB形式用于表明相关的tau函数具有形式为\log τ= \hbar^{-2} F_0 + \hbar^{-1} F_1 + F_2 +.的\hbar -展开。
英文摘要：
The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). Given the lowest order term X_0, one can solve the recursion relations to obtain the higher order terms. The wave function Ψassociated with W turns out to have the WKB form Ψ= \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S = S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an \hbar-expansion of the form \logτ= \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 + ...
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