N. Ruijsenaars S.

Centre for Mathematics and Computer Science, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands
Centre for Mathematics and Computer Science


Ruijsenaars S. N. M.
We study an extensive class of second-order analytic difference operators admitting reflectionless eigenfunctions. The eigenvalue equation for our $\mathrm{A}\Delta\mathrm{Os}$ may be viewed as an analytic analog of a discrete spectral problem studied by Shabat. Moreover, the nonlocal soliton evolution equation we associate to the $\mathrm{A}\Delta\mathrm{Os}$ is an analytic version of a discrete equation Boiti and coworkers recently associated to Shabat's problem. We show that our nonlocal solitons $G(x,t)$ are positive for $(x,t) \in \mathbb{R}^2$ and obtain evidence that the corresponding $\mathrm{A}\Delta\mathrm{Os}$ can be reinterpreted as self-adjoint operators on $L^2(\mathbb{R},dx)$. In a suitable scaling limit the KdV solitons and reflectionless Schrodinger operators arise.
Citation: Ruijsenaars S. N. M.,  A New Class of Reflectionless Second-order $\mathrm{A}\Delta\mathrm{Os}$ and Its Relation to Nonlocal Solitons, Regular and Chaotic Dynamics, 2002, vol. 7, no. 4, pp. 351-391

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