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

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 I1I1 and I2I2 corresponding to these solutions. It is found that there exists a strong resemblance between them and the spectra for the Macdonald qq-difference operators. To better understand the connection between these classical and quantum integrable systems, we consider the special degenerate case corresponding to q=0q=0 in more detail. Namely, we give general solutions to this degenerate periodic BO, obtain explicit formulas representing all the integrals of motions InIn (n=1,2,…n=1,2,…), and successfully identify it with the eigenvalues of Macdonald operators in the limit q→0q→0, i.e. the limit where Macdonald polynomials tend to the Hall–Littlewood polynomials.