On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian

TL;DR

This paper explores special solutions to a periodic Benjamin-Ono equation with a discrete Laplacian, revealing connections to Macdonald and Hall-Littlewood polynomials and providing explicit formulas for integrals of motion.

Contribution

It introduces explicit solutions and integrals of motion for a discrete Laplacian version of the periodic Benjamin-Ono equation and links these to Macdonald and Hall-Littlewood polynomial spectra.

Findings

Special solutions to the periodic BO equation are found.

Explicit formulas for integrals of motion are derived.

Connection established between classical solutions and quantum integrable systems.

Abstract

We investigate a periodic version of the Benjamin-Ono (BO) equation associated with a discrete Laplacian. We find some special solutions to this equation, and calculate the values of the first two integrals of motion $I_1$ and $I_2$ corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald $q$-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to $q=0$ in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions $I_n$ ($n=1,2,...$), and successfully identify it with the eigenvalues of Macdonald operators in the limit $q\to 0$, i.e. the limit where Macdonald polynomials tend to the Hall-Littlewood polynomials.