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

Yohei Tutiya and Shiraishi, Jun’ichi

Mathematics and Computers in Simulation (MATCOM), 2012, vol. 82, issue 7, 1341-1347

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 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 In (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.

Keywords: Benjamin-Ono equation; Nonlocal integrable system; Macdonald polynomial; Integrals of motion (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:82:y:2012:i:7:p:1341-1347

DOI: 10.1016/j.matcom.2010.05.006

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