Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion (“accelerations equal forces”) are integrable, i.e., they allow the explicit exhibition of $N$ constants of motion in terms of the dependent variables and their time-derivatives. Some of these models are moreover solvable by purely algebraic operations, by (explicitly performable) quadratures and, finally, by functional inversions. The techniques to manufacture these models are not new; some of these models are themselves new; others are reinterpretations of known models.

Here we use an extension of Rodrigues' vector parameter construction for pseudo-rotations in order to obtain explicit formulae for the generalized Euler decomposition with arbitrary axes for the structure groups in the classical models of hyperbolic geometry. Although the construction is projected from the universal cover $\,\mathsf{SU}(1,1)\simeq\mathsf{SL}(2,\mathbb{R})$, most attention is paid to the $2+1$ Minkowski space model, following the close analogy with the Euclidean case, and various decompositions of the restricted Lorentz group $\mathsf{SO}^+(2,1)$ are investigated in detail. At the end we propose some possible applications in special relativity and scattering theory.

We derive a model suitable for computer simulations of a weak ion beam with solitons of the Korteweg-de Vries (KdV) equation. This type of interactions arise in experiments on soliton generation in double plasma devices and include soliton growth, damping, or decay. Our simplified model aims at capturing only the essential physics of these interactions. The model is formulated in the context of plasma physics in the electrostatic approximation. The bulk plasma is described by cold fluid ions and warm, massless electrons. The ion beam is included as a separate plasma species and is coupled to the bulk plasma through Poisson's equation. The derivation uses the Lagrangian of the system of plasma and beam and an expansion in small amplitude perturbations around an equilibrium. The Korteweg-de Vries equation arises from this expansion naturally. The model is thus applicable to general weakly non-linear ion-acoustic plasma waves, of which solitons are a particular case. A novel feature of our method is that it includes both the evolution of the wave and the perturbation while in previous analyses the perturbation is kept fixed. The computational advantages of such description other approaches, such as fluid description of both plasma and beam or all kinetic description, are that in the former case particle trapping cannot be fully simulated while in the latter case the computational time is much longer and the numerical noise is higher than in our hybrid approach.

The phase-space formulation of the nonrelativistic quantum mechanics is constructed on the basis of a deformation of the classical mechanics by the $\ast$-product. We have taken up the MIC-Kepler problem in which Iwai and Uwano have interpreted its wave-function as the cross section of complex line bundle associated with a principal fibre bundle in the conventional operator formalism. We show that its Green's function, which is derived from the $\ast$-exponential corresponds to unitary operator through the Weyl application, is equal to the infinite series that consists of its wave-functions. Finally, we obtain its Wigner function.

In this article, a generalized Kleinian sigma function for an affine $(3,4,5)$ space curve of genus two was constructed as the simplest example of the sigma function for an affine space curve, and in terms of the sigma function, the Jacobi inversion formulae for the curve are obtained. An interesting relation between a space curve with a semigroup generated by $(6,13,14,15,16)$ and Norton number associated with Monster group is also mentioned in the Appendix by the second author.

In certain scaling limits the higher-dimensional Euclideanized Kerr-NUT-$(A){\rm d}S$ metrics are related to the Einstein-Sasaki ones. The complete set of Killing forms of the Einstein-Sasaki spaces are presented. It is pointed out the existence of two additional Killing forms on these spaces associated with the complex volume form of the Calabi-Yau cone manifold. As a concrete example we present the complete set of Killing-Yano tensors on the five-dimensional Einstein-Sasaki $Y(p,q)$ spaces.

We consider the Caudrey-Beals-Coifman linear problem and the theory of the Recursion Operators (Generating Operators) related to it in the presence of $\mathbb{Z}_p$ reduction of Mikhailov type.