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2013
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# N. Ruijsenaars S.

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

## Publications:

 Ruijsenaars S. N. M. A New Class of Reflectionless Second-order $\mathrm{A}\Delta\mathrm{Os}$ and Its Relation to Nonlocal Solitons 2002, vol. 7, no. 4, pp.  351-391 Abstract 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 DOI:10.1070/RD2002v007n04ABEH000217