On some special solutions to periodic Benjamin-Ono equation with discrete Laplacian
| Autor: | Jun'ichi Shiraishi, Yohei Tutiya |
|---|---|
| Rok vydání: | 2012 |
| Předmět: |
Numerical Analysis
Pure mathematics Mathematics::Combinatorics Nonlinear Sciences - Exactly Solvable and Integrable Systems General Computer Science Integrable system Applied Mathematics Degenerate energy levels Mathematical analysis FOS: Physical sciences Benjamin–Ono equation Theoretical Computer Science Macdonald polynomials Modeling and Simulation Limit (mathematics) Exactly Solvable and Integrable Systems (nlin.SI) Laplacian matrix Koornwinder polynomials Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Mathematics and Computers in Simulation. 82:1341-1347 |
| ISSN: | 0378-4754 |
| DOI: | 10.1016/j.matcom.2010.05.006 |
| Popis: | 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. 10 pages, no figure |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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