Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2009-11-26
Nonlinear Sciences
Exactly Solvable and Integrable Systems
10 pages, no figure
Scientific paper
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.
Shiraishi Jun'ichi
Tutiya Yohei
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