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 i₁ and i₂ 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→0, i.e. the limit where macdonald polynomials tend to the hall–littlewood polynomials.