hbar-expansion of KP hierarchy: Recursive construction of solutions

TL;DR

This paper develops a recursive method to construct solutions of the ar-dependent KP hierarchy using a Riemann-Hilbert problem, revealing the ar-expansion structure of the tau function and wave functions.

Contribution

It introduces a recursive construction of solutions for the ar KP hierarchy based on a Riemann-Hilbert problem, connecting the ar-expansion to the tau function and wave functions.

Findings

Recursive relations for ar-expansion coefficients are derived.

The wave function exhibits a WKB form with coefficients determined recursively.

The tau function's ar-expansion is explicitly characterized.

Abstract

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 \Psi associated with W turns out to have the WKB form \Psi = \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 \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 + ...