Generating functions that are solutions to integrable hierarchies

Integrable hierarchies of partial differential equations arose as a tool describing behavior of waves of special form. However, their solutions appeared to include formal ones that are generating functions for natural enumerating problems. According to Sato's construction, such solutions can be expressed in terms of Young diagrams and Schur polynomials.
A spectacular example of such a solution is the Witten–Kontsevich potential, which generates certain geometric characteristics of moduli spaces of complex curves. For solutions of this kind, the equations of the hierarchy can be treated as recurrence relations allowing for efficient computations of the coefficients of the formal power series expansions.
It will be explained how to construct solutions to the Kadomtsev–Petviashvili hierarchy by means of Schur polynomials, and examples will be given, including those found during the last years, of important enumeration problems leading to these solutions.