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.

We review here a method, recently introduced by the authors, that can be used to construct completely integrable Classical and Quantum Hamiltonian Systems from representations of coalgebras with Casimir element(s). As a prototype example, we choose the spin $1/2$ Calogero–Gaudin system and its $q$-deformation. Possible drawbacks and generalizations of the method are outlined.

The present paper reviews some new results in the study of the bispectral problem. We describe all bispectral operators in the Weyl algebra and the Darboux transformations of them. The dynamics of their poles is shown to be connected with the KP-hierarchy. The exposition is intended for readers with some experience in the subject. On the other hand being self-contained it can be used for a first acquaintance with the subject of bispectral operators.

In this paper we derive the KdV–Burgers and higher order nonlinear Schrodinger equations as an Euler–Poincaré flow on the joint space of Hill's and first order differential operators on circle. We also study a quasi-hamiltonian pair of involution equations one member of which is the KdV–Burger equation.

The problem of three vortex lines in a homogeneous layer of an ideal incompressible fluid is generalized to the case of a two-layer liquid with constant density values in each layer. For zero-complex-momentum systems the theory of the roundabout two-layer tripole is built. When the momentum is different from zero, based on the phase portraits in trilinear coordinates, a classification of possible relative motions of a system composed of three discrete (or point) vortices is provided. One vortex is situated in the upper layer, and the other two in the lower layer; their total intensity is zero. More specifically, a model of a two-layer tripole is constructed, and existence conditions for stationary solutions are found. These solutions represent a uniform translational motion of the following vortex structures: 1) a stable collinear configuration triton, a discrete analog of the vortex structure modon+rider, 2) an unstable triangular configuration. Features of the absolute motion of the system of three discrete vortices were studied numerically.
We compared the dynamics of a system of three point vortices with the dynamics of three finite-core vortices (vortex patches). In studying the evolution of the vortex patch system, a two-layer version of the Contour Dynamics Method (CDM) was used. The applicability of discrete-vortex theory to the description of the finite-size vortex behavior is dicussed. Examples of formation of vortical configurations are given. Such configurations appear either after merging of vortices of the same layer or as a result of instability of the two-layer vortex structure.