We study a relation between harmonic spheres in loop spaces of compact Lie groups and Yang-Mills fields on the Euclidean four-space $\mathbb R^4$.

Quantization of the electromagnetic field in non-stationary media (linear with respect to $E$, with negative differential conductivity) is investigated. The dynamical invariants and statistical properties of the field are found in such media. It is shown that in the eigenstates of linear dynamical invariant, the Schrödinger uncertainty relation is minimized. The time evolution of the tree independent second-order statistical moments (quantum fluctuations: covariance cov(q,p), var(q) and var(p)) are found out.

We present a summary of: 1) the non-uniqueness problem of the Hamiltonian and energy operators associated, in any given coordinate system, with the generally-covariant Dirac equation, 2) two different ways to restrict the gauge freedom so as to solve that problem, 3) the application of these two ways to the case of a uniformly rotating reference frame in Minkowski spacetime. We find that a spin-rotation coupling term is there only with one of these two ways.

For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable system using separating variables. The variables are chosen in such way that half of them parameterizes the Lagrangian manifold, which coincides with the Liouville torus of the integrable system. The obtained representation is indecomposable and non-exponentiated.

We show that if $\psi$ is an $f$-biharmonic map from a compact Riemannian manifold into a Riemannian manifold with non-positive curvature satisfying a condition, then $\psi$ is an $f$-harmonic map. We prove that if the $f$-tension field $\tau_f(\psi)$ of a map $\psi$ of Riemannian manifolds is a Jacobi field and $\phi$ is a totally geodesic map of Riemannian manifolds, then $\tau_f( \phi\circ \psi)$ is a Jacobi field. We finally investigate the stress $f$-bienergy tensor, and relate the divergence of the stress $f$-bienergy of a map $\psi$ of Riemannian manifolds with the Jacobi field of the $\tau_f (\psi)$ of the map.

This is a review of multiparameter families of contact transformations and their relationship with the generalized Hamiltonian system. We derive the integrability conditions for the generalized Hamiltonian system and show that when they are satisfied the solutions to this system determine a family of multiparameter contact transformations of the initial conditions. We prove a necessary and sufficient condition for a multiparameter family of contact transformations to be a group and a characterization of the function which describes the group multiplication rule.

Our work has been inspired among others by the work of Arnold, Kozlov and Neihstadt. Our goal is to carry out a thorough analysis of the geometric problems we are faced with in the dynamics of affinely rigid bodies. We examine two models: classical dynamics description by d'Alembert and vakonomic one. We conclude that their results are quite different. It is not yet clear which model is practically better.

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.

Following the tradition of the nano and picosecond optics, the basic theoretical studies continue to investigate the processes of propagation of femtosecond and attosecond laser pulses through the corresponding envelope equation for narrow-band laser pulses, working in paraxial approximation. We should point out here that this approximation is not valid for large band pulses. In air due to the small dispersion the wave equation as well as the $3D+1$ amplitude equation describe more accurate the pulse dynamics. New exact localized solutions of the linear wave and amplitude equations are presented. The solutions discover non-paraxial semi-spherical diffraction of single-cycle and half-cycle laser pulses and a new class of spherically symmetric solutions of the wave equation. The propagation of large band optical pulses in nonlinear vacuum is investigated also in the frame of a system of nonlinear wave vector equations. We obtained exact vector solution with its own angular momentum in the form of a shock wave.

An interesting relation between the mKdV equation and the cylindrical equilibrium shapes of fluid membranes is observed. In our setup mKdV arises from the study of the evolution of planar curves in the normal direction.

Here we consider the Helfrich's membrane shape model from a group-theoretical viewpoint. By making use of the conformal metric on the associated surface the model is represented by a system of four second order nonlinear partial differential equations. In order to construct the determining system for the symmetries of the metric we rely on the previously developed package LieSymm-PDE within ${\tt Mathematica}^{\!®}$. In this way we have obtained the determining system consisting of 206 equations. Using the above mentioned programs we have solved the equations in a semi-automatic way. As a result we end up with an infinite dimensional symmetry Lie algebra of the Helfrich`s model in conformal metric representation which we present here in explicit form.

The problem of describing or determining the image of the exponential map $ \exp :\mathfrak{g}\rightarrow G$ of a Lie group $G$ is important and it has many applications. If the group $G$ is compact, then it is well-known that the exponential map is surjective, hence the exponential image is $G$. In this case the problem is reduced to the computation of the exponential and the formulas strongly depend on the group $G$. In this paper we discuss the generalization of Rodrigues formulas for computing the exponential map of the special orthogonal group ${\rm SO}(n) $, which is compact, and of the special Euclidean group ${\rm SE}(n)$, which is not compact but its exponential map is surjective, in the case $ n\geq 4$.

In this paper, we have reintroduced a new approach to conformal geometry developed and presented in two previous papers, in which we show that all $n$-dimensional pseudo-Riemannian metrics are conformal to a flat $n$-dimensional manifold as well as an $n$-dimensional manifold of constant curvature when Riemannian normal coordinates are well-behaved in the origin and in their neighborhood. This was based on an approach developed by French mathematician Elie Cartan. As a consequence of geometry, we have reintroduced the classical and quantum angular momenta of a particle and present new interpretations. We also show that all $n$-dimensional pseudo-Riemannian metrics can be embedded in a hyper-cone of a flat $(n+2)$-dimensional manifold.

The paper is motivated by gauge theories of gravitation and condensed matter, tetrad models of gravitation and generalized Born-Infeld type nonlinearity. The main idea is that any generally-covariant and $\mathrm{GL}(n, \mathbb R)$-invariant theory of the n-leg field (tetrad field when $n=4$) must have the Born-Infeld structure. This means that Lagrangian is given by the square root of the determinant of some second-order twice covariant tensor built in a quadratic way of the field derivatives. It is shown that there exist interesting solutions of the group-theoretical structure. Some models of the interaction between gravitation and matter are suggested. It turns out that in a sense the space-time dimension $n=4$, the normal-hyperbolic signature and velocity of light are integration constants of our differential equations.

We study derivative nonlinear Schrödinger equations related to symmetric spaces of the type A.III. We discuss the spectral properties of the corresponding Lax operator and develop the direct scattering problem connected to it. By applying an appropriately chosen dressing factor we derive soliton solutions to the nonlinear equation. We find the integrals of motion by using the method of diagonalization of Lax pair.

By using spectral methods, we present seven classes of stable and unstable structures that occur in a dissipative media. By varying parameters and initial conditions we find ranges of existence of stable structures (spinning elliptic, pulsating, stationary, organized exploding), and unstable structures (filament, disorganized exploding, creeping). By varying initial conditions, vorticity, and parameters of the equation, we find a reacher behavior of solutions in the form of creeping-vortex (propellers), spinning rings and spinning “bean-shape” solitons. Each class differentiates from the other by distinctive features of their shape and energy evolution, as well as domain of existence.

In this work, a continuum model is used to determine in analytic form a class of unduloid-like equilibrium shapes of single-wall carbon nanotubes subjected to uniform hydrostatic pressure. The parametric equations of the profile curves of the foregoing shapes are presented in explicit form by means of elliptic functions and integrals.

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.