We study certain classes of integrable nonlinear differential equations related to the type symmetric spaces. Our main examples concern equations related to A.III-type symmetric spaces. We use the Cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic Lax operators to this symmetric space. Next we outline the spectral theory of the reduced Lax operator $L$ and construct its fundamental analytic solutions. Analyzing the Wronskian relations we introduce the `squared solutions' of $L$ and derive the recursion operators by three different methods.

In this article we present an introduction in the geometrical theory of motion of curves and surfaces in $\mathbb{R}^3$, and its relations with the nonlinear integrable systems. The working frame is the Cartan's theory of moving frames together with Cartan connection. The formalism for the motion of curves is constructed in the Serret-Frenet frames as elements of the bundle of adapted frames. The motion of surfaces is investigated in the Gauss-Weingarten frame. We present the relations between types of motions and nonlinear equations and their soliton solutions.

Here we use the mathematical structure of group algebras and $H^{+}$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient ${\rm SO}(3,\mathbb{R})$. The proposed scheme is considered in two different contexts.

Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules and etc.

Secondly, the other promising area of applications is Schrödinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schrödinger framework the algebras of operators related to group algebras are a very useful tool.

Finally, we investigate also some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to the geometry of the coadjoint orbits of the Lie group SU(2).

The presentation is based on the general ideas of applying group-algebraic methods and extesive use of the Lie group structure. The papers ends with consideration of the special case of the group SU(2) and its quotient ${\rm SO}(3,\mathbb{R})$, which is the main subject in this paper, i.e., angular momentum problems. Formally, the scheme could be applied to the isospin systems. However, it is rather hard to imagine realistic quasiclassical isospin problems.

In this paper we review some interesting results obtained through the Newman-Janis algorithm, a solution generating technique employed in General Relativity. We also describe the use of this algorithm in different theories, namely f(R), Einstein-Maxwell-Dilaton gravity, Braneworld, Born-Infeld monopole and we focus on the validity of the results.

In this paper, we study the one-level Friedrichs model by using the quantum time super-operator that predicts the excited state decay inside the continuum. Its survival probability decays exponentially in time.

This work is concerned with the determination of stresses and strains in cell membranes subjected to micro-injections. For that purpose, a suitable variational statement of the problem is developed within a continuum mechanics approach to the analysis of cell membrane geometry and physics. In this setting, the cell membrane is regarded as an axially symmetric surface in the three-dimensional Euclidean space providing a stationary value of the bending energy functional under the constraint of fixed total area. The Euler-Lagrange equations and the natural boundary conditions associated with the foregoing variational problem are derived, analyzed and used to express the stresses and moments in the membrane. Several examples of such surfaces representing possible shapes of cell membranes subjected to micro injection are determined numerically.

In this paper we consider relativistic models that contain in their spectra of solutions extended topological defects. We find the geometrical constrains that describe deformed vortices and domain walls of constant width. Analytical form of these solutions in co-moving coordinates is identical with analytical form of the appropriate static solutions in the laboratory Cartesian coordinates. The geometrical constrains presented in this report describe fully the shape and the evolution of the vortices and domain walls of constant width.

We work out the general idea that a composite continuous physical system can be mathematically modelled locally as a completely integrable geometric distribution on a manifold, the time-recognizable subsystems to be modelled by corresponding subdistributions, and any local interaction between two subsystems of the physical system to be described in terms of the nonintegrability of the two subdistributions making use of the corresponding two curvature forms. As an illustration we present the corresponding description of photon-like objects, based on the notion that photon-like objects are real, massless time-stable physical objects with intrinsically compatible translational-rotational dynamical structure. The spatial propagation of the system follows some external/shuffling symmetry of the distribution.

This is a review of the variational principle proposed by Gustav Hergotz and resent results related to it. In that variational principle the functional is defined by a certain differential equation instead of an integral. The solutions of the equations for the extrema of the functional determine contact transformations. Some of those results are: two Noether-type theorems for finding conserved quantities and identities, a method for calculating symmetry groups of the functional and several applications.

Many exactly solvable models are based on Lie algebras. The pairing interaction is important in nuclear physics and its exact solution for identical particles in non-degenerate single-particle levels was first given by Richardson in 1963. His solution and its generalization to Richardson-Gaudin quasi-exactly solvable models have attracted the attention of many contemporary researchers and resulted in the exact solution of the isovector pn-pairing within the SO(5) RG-model and the equal strength spin-isospin pn-pairing within the SO(8) RG-model. Basic properties of the RG-models are summarized and possible applications to nuclear physics are emphasized.

Here we present a general scheme which leads effectively to the reconstruction of any plane curve whose curvature is specified by a function of the radial coordinate. As a concrete example we have derived two new parametrizations of the Cassinian ovals.

A family of star products parametrized by complex matrices is defined. Especially commutative associative star products are treated, and star exponentials with respect to these star products are considered. Jacobi's theta functions are given as infinite sums of star exponentials. As application, several concrete identities are obtained by properties of the star exponentials.

We formulate the equation of motion of a charged particle in a Riemannian manifold with a closed two form. Since a two-step nilpotent Lie group has natural left-invariant closed two forms, it is natural to consider the motion of a charged particle in a simply connected two-step nilpotent Lie groups with a left invariant metric. We study the behavior of the motion of a charged particle in the above spaces.

The Baily-Borel compactification $\widehat{{\mathbb B} / \Gamma}$ of an arithmetic ball quotient admits projective embeddings by $\Gamma$-modular forms of sufficiently large weight. We are interested in the target and the rank of the projective map $\Phi$, determined by $\Gamma$-modular forms of weight one. This paper concentrates on the finite $H$-Galois quotients ${\mathbb B} / \Gamma _H$ of a specific ${\mathbb B} / \Gamma _{-1}^{(6,8)}$, birational to an abelian surface $A_{-1}$. Any compactification of ${\mathbb B} / \Gamma _H$ has non-positive Kodaira dimension. The rational maps $\Phi ^H$ of $\widehat{{\mathbb B} / \Gamma _H}$ are studied by means of the $H$-invariant abelian functions on $A_{-1}$.

The stability properties of the Einstein Static solution of General Relativity is altered when corrective terms arising from modifications of the underlying gravitational theory appear in the cosmological equations. Employing dynamical system techniques and numerical integrations, we discuss the stability of static cosmological solutions in the framework of two recently proposed quantum gravity models, namely Loop Quantum Cosmology and Horava-Lifshitz gravity.

The constant mean curvature surfaces in three-dimensional space-forms are examples of isothermic constrained Willmore surfaces, characterized as the constrained Willmore surfaces in three-space admitting a conserved quantity. Both constrained Willmore spectral deformation and constrained Willmore Bäcklund transformation preserve the existence of a conserved quantity. The class of constant mean curvature surfaces in three-dimensional space-forms lies, in this way, at the intersection of several integrable geometries, with classical transformations of its own, as well as constrained Willmore transformations and transformations as a class of isothermic surfaces. Constrained Willmore transformation is expected to be unifying to this rich transformation theory.

We study the linear part of the monodromy of completely integrable Hamiltonian systems via Bohr-Sommerfeld Geometric Quantization. We relate monodromy to the ambiguity in the choice of the pre-quantum connection and to the action of the (connected component of the) gauge group.

Solutions of vacuum Einstein’s field equations, for the class of pseudo-Riemannian four-metrics admitting a non Abelian two dimensional Lie algebra of Killing fields, are explicitly described. When the distribution orthogonal to the orbits is completely integrable and the metric is not degenerate along the orbits, these solutions are parameterized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a three dimensional Lie algebra of Killing fields with two dimensional leaves. Metrics, corresponding to the case in which the commutator of the two Killing fields is isotropic, represent nonlinear gravitational waves.

We consider the recursion operator approach to the soliton equations related to a $\mathfrak{sl}(3,\mathbb{C})$ generalized Zakharov-Shabat auxiliary linear system in pole gauge and show that the recursion operator can be identified with the dual to a Nijenhuis tensor for a Poisson-Nijenhuis structure on the manifold of potentials.