The family of Macdonald operators is regarded as a commutative subring sitting inside the Ding-Iohara algebra acting on the Fock space. Taking the classical limit, the Ding-Iohara algebra becomes commutative, allowing us to induce the Poisson algebra structure. This gives us the Hamiltonian description of the Macdonald system (with infinitely many particles) in the classical limit. This system is a discrete analogue of the Benjamin-Ono (BO) hydrodynamic equation. We can introduce the tau-functions and Hirota bilinear equation for the BO equation. The motion of the zeros of the tau-functions obey the Hamiltonian equations induced form the Macdonald operators (with finitely many particles).

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Ingemar Bengtsson and Edwin Langmann