Quasi-periodic solutions to hierarchies of nonlinear integrable equations and bilinear relations

This is a short review of the construction of quasi-periodic (algebraic-geometrical) solutions to hierarchies of nonlinear integrable equations. As is well known, the solutions are expressed through Riemann's theta-functions associated with algebraic curves. It is explained how solutions from this class can be treated within the framework of the approach to the integrable hierarchies developed by the Kyoto school. Three representative examples are considered in detail: the Kadomtsev-Petviashvili hierarchy, the 2D Toda lattice hierarchy and the B-version of the Kadomtsev-Petviashvili hierarchy.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This is a short review of the construction of quasi-periodic solutions, which are expressed through Riemann's theta-functions associated with algebraic curves, and how they can be treated within the framework of the integrable hierarchies developed by the Kyoto school.