Hirota equation and Bethe ansatz

The paper is a review of recent works devoted to analysis of classical integrable structures in quantum integrable models solved by one or another version of the Bethe ansatz. Similarities between elements of the quantum and classical theories of integrable systems are discussed. Some key notions of the quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions known in the domain of classical soliton equations. Functional relations for quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation and all the basic results for spectral properties of quantum systems can be obtained by solving this classical equation. Vice versa, solutions of the latter with certain boundary conditions arise as typical Bethe ansatz formulas. The famous Baxter T -Q relation and its generalizations appear as a kind of auxiliary linear problems for the Hirota equation. Contents 1 Introduction 2 2 From Yang's R-m..