Formal solutions to the KP hierarchy

Abstract

We find all formal solutions to theh-dependent KP hierarchy. They are characterized by certain Cauchy-like data. The solutions are found in the form of formal series for the tau-function of the hierarchy and for its logarithm (the F -function). An explicit combinatorial description of the coefficients of the series is provided.

Genus expansion of matrix models and ћ expansion of KP hierarchy

“…We do this explicitly in great detail, so as to dispel any doubts. Finally, we show that recent papers considering -formulation of KP hierarchy [2,3] do coincide with original Takasaki-Takebe deformation.…”

“…The -formulation can be explicitly performed for all structures in the KP theory: Lax operator, -symmetries and an element of the GL(∞) group. Following Takasaki-Takebe, Natanzon and Zabrodin recently introduced formulation of -KP [2] in terms of common equations on deformed -functions and F-functions ( = 2 log , note extra factor 2 for correct limit → 0) with one more parameter , which is the shift of first variable 1 → 1 + . They managed to obtain explicit solution for the F-function in terms of Cauchy-like data and explicit combinatorial constants.…”

“…Natanzon and Zabrodin were following Takasaki-Takebe to introduce their rescaling of the KP hierarchy [2]. This expansion allows -functions to depend on arbitrary powers of formal parameter .…”

“…For this reason one will need deformed variables and combinatorial coefficients . Detailed description of these objects can be found in [2]. The solution in terms of logarithm of the -function is determined by arbitrary set of functions f as well.…”

On the expansion coefficients of Tau-function of the BKP hierarchy

“…Recently Giambelli and Jacobi-Trudi type formulae for the expansion coefficients attract much attention in relation to the study of 2 dimensional solvable lattice models [1,12,6] and higher genus theta functions [4,8]. It is interesting to study relations of our results to such subjects.…”