Periodic Benjamin-Ono equation with discrete Laplacian and 2D-Toda Hierarchy

TL;DR

This paper explores the connection between the periodic Benjamin-Ono equation with discrete Laplacian and the 2D Toda hierarchy, introducing tau-functions, integrals of motion, and verifying bilinear equations.

Contribution

It establishes a link between the periodic Benjamin-Ono equation and the 2D Toda hierarchy through tau-functions and integrals of motion, and confirms bilinear equations from hierarchy reductions.

Findings

Low-lying bilinear equations match 2D Toda hierarchy reductions

Constructed tau-functions for the periodic Benjamin-Ono equation

Derived integrals of motion for the system

Abstract

We study the relation between the periodic Benjamin-Ono equation with discrete Laplacian and the two dimensional Toda hierarchy. We introduce the tau-functions tau_pm(z) for the periodic Benjamin-Ono equation, construct two families of integrals of motion {M_1,M_2,cdots}, {overline{M}_1,overline{M}_2,cdots}, and calculate some examples of the bilinear equations using the Hamiltonian structure. We confirmed that some of the low lying bilinear equations agree with the ones obtained from a certain reduction of the 2D Toda hierarchy.