We discuss several ways of how one could classify the various types of soliton solutions related to NLEE that are solvable with the generalized $n \times n$ Zakharov–Shabat system. In doing so we make use of the fundamental analytic solutions, the dressing procedure and other tools characteristic for the inverse scattering method. We propose to relate to each subalgebra $\mathfrak{sl}(p), 2 \leq p \leq n$ of $\mathfrak{sl}(n)$, a type of one-soliton solutions which have $p − 1$ internal degrees of freedom.

I describe work in progress with Baryshnikov and Zharnitsky on periodic billiard orbits that leads one to an exterior differential system (EDS). I then give a brief introduction to EDS illustrated by several examples.

The aim of this paper is to show that stochastic models provide a very good playground to enhance the utility of quantum groups. Quantum groups arise naturally and the deformation parameter has a direct physical meaning for diffusion systems where it is just the ratio of left/right probability rate. In the matrix product state approach to diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra which defines a noncommutative space with a quantum group action as its symmetry. Boundary processes amount for the appearance of parameter-dependent linear terms in the algebra which leads to a reduction of the bulk symmetry.

We propose to extend the concept of geometric phase to quantum dissipative systems, in the case of meta-stable spin states in magnetic resonance. We use the generalized version of Lewis–Riensenfeld invariant theory to study the dissipative systems described by non-hermitian time-dependent Hamiltonian.

Let $(M, \omega)$ be a closed symplectic $2n$-dimensional manifold. According to the well-known result by Donaldson [5] there exist $2m$-dimensional symplectic submanifolds $(V^{2m}, \omega)$ of $(M, \omega), 1 \leq m \leq n - 1$, with $(m − 1)$-equivalent inclusions. In this paper, we have found a relation between the flux group and the kernel of the Lefschetz map. We have present also some properties of the flux groups for all symplectic $2m$-submanifolds $(V^{2m}, \omega)$ where $2 \leq m \leq n - 1$.

Here we derive a new formula for the exponent of an arbitrary matrix in the Lie algebra of the Lorentz group. Our considerations are based on the fact that for each constant electromagnetic field there exist an inertial system in which one can easily solve the respective Lorentz equation and therefore to find the explicit formulas describing the trajectories of the charged particles in this field.

We consider Hamiltonian systems in first-order multisymplectic field theories. In particular, we introduce Hamiltonian systems in the extended multimomentum bundle. The resulting extended Hamiltonian formalism is the generalization to field theories of the extended (symplectic) formalism for non-autonomous mechanical systems. In order to derive the corresponding field equations, a variational principle is stated for these extended Hamiltonian systems and, after studying the geometric properties of these systems, we establish the relation between this extended formalism and the standard one.

In this paper by means of similarity transformation we find some one-dimensional quasi-exactly solvable differential equations and their related Hamiltonians which appear in physical problems. We have provided also two examples with application of these differential equations.

Using the Cartan form of first order constrained variational problems introduced earlier we define the second variation. This definition coincides in the unconstrained case with the usual one in terms of the double Lie derivative of the Lagrangian density, an expression, that in the constrained case does not work. The Hessian metric and other associated concepts introduced in this way are compared with those obtained through the Lagrange multiplier rule. The theory is illustrated with an example of isoperimetric problem.

The reductions of the multi-component nonlinear Schrödinger models related to C.I and D.III type symmetric spaces are studied. We pay special attention to the MNLS related to the $\mathfrak{sp}(4)$, $\mathfrak{so}(10)$ and $\mathfrak{so}(12)$ Lie algebras. The MNLS related to $\mathfrak{sp}(4)$ is a three-component MNLS which finds applications to Bose–Einstein condensates. The MNLS related to $\mathfrak{so}(12)$ and $\mathfrak{so}(10)$ Lie algebras after convenient $\mathfrak{Z}_6$ or $\mathfrak{Z}_4$ reductions reduce to three and four-component MNLS showing new types of $\chi(3)$-interactions that are integrable. We briefly explain how these new types of MNLS can be integrated by the inverse scattering method. The spectral properties of the Lax operators $L$ and the corresponding recursion operator $\Lambda$ are outlined. Applications to spinor model of Bose–Einstein condensates are discussed.

The well known Laplace–Young equation asserts that the pressure difference across the film or a membrane in a equilibrium is proportional to the mean curvature with a proportionality constant the surface tension of the interface. Here we present two variants of this equation leading to the surfaces of Delaunay and the mylar balloon and in this way provide their nonvariational characterization.

In this talk, I will discuss the vortex filament flow, and its connection with the nonlinear Schrödinger equation (NLS), the geometry of solutions to the filament flow that correspond to finite-gap solutions of NLS, and in particular, the relationship between geometric properties of the filament to features of the Floquet spectrum of a periodic NLS potential. These will be illustrated by the example of elastic rod centerlines. I will also discuss the geometric implications of symmetric Floquet spectra, and the topological implications of certain isoperiodic deformations of the spectrum that open up real double points.

The Manev model is known to possess Ermanno–Bernoulli type invariants similar to the Laplace–Runge–Lenz vector of the ordinary Kepler model. If the orbits are bounded these invariants exist only when a certain rationality condition is met and consequently we have superintegrability only on a subset of initial values. On the contrary, real form dynamics of the Manev model is superintegrable for all initial values. Using these additional invariants, we demonstrate here that both Manev model and its real Hamiltonian form have $\mathfrak{su}(2) \simeq \mathfrak{so}(3)$ (or $\mathfrak{so}(2,1)$ depending on the value of a parameter in the potential) symmetry algebra in addition to the angular momentum algebra. Thus Kepler and Manev models are shown to have identical symmetry algebras.

In the present paper we investigate the completeness, minimality and basic properties of the eigenfunctions of one discontinuous Sturm–Liouville problem with a spectral parameter in boundary conditions and transmission conditions.

In the present work the inverse problem of the scattering theory for Sturm-Liouville differential equation with a spectral parameter in the boundary condition is investigated. The Gelfand–Marchenko–Levitan fundamental equation is obtained, the uniqueness of the solution of the inverse problem is proved and some properties of the scattering data are given.

After a brief introduction to several variational problems in the study of shapes of thin structures, we deal with variational problems on two-dimensional surface in three-dimensional Euclidian space by using exterior differential forms and the moving frame method. The morphological problems of lipid bilayers and stabilities of cell membranes are also discussed. The key point is that the first and the second order variations of the free energy determine equilibrium shapes and mechanical stabilities of structures

A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the “gyrolanguage” of this paper one attaches the prefix “gyro” to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this paper share. The aim of this article is to employ recent developments in analytic hyperbolic geometry for the presentation of the relativistic hyperbolic parallelogram law, and the relativistic particle aberration.

The six-parameter group of three dimensional Euclidean motions is recognized as the largest group of point transformations admitted by the membrane shape equation in Mongé representation. This equation describes the equilibrium shapes of biomembranes being the Euler-Lagrange equation associated with the Helfrich curvature energy functional under the constraints of fixed enclosed volume and membrane area. The conserved currents of six linearly independent conservation laws that correspond to the variational symmetries of the membrane shape equation and hold on its smooth solutions are obtained. All types of non-equivalent group-invariant solutions of the membrane shape equation are identified via an optimal system of one-dimensional subalgebras of the symmetry algebra. The reduced equations determining these group-invariant solutions are derived. Special attention is paid to the translationally-invariant solutions of the membrane shape equation.

Here we consider the cubic complex Ginzburg–Landau equation. Applying the Hone’s method, based on the use of the Laurent-series solutions and the residue theorem, we have proved that this equation has no elliptic standing wave solutions. This result supplements Hone’s result, that this equation has no elliptic travelling wave solutions. It has been shown that the Hone’s method can be applied to a system of polynomial differential equations more effectively than to an equivalent differential equation.

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasakian metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient condition to generate a totally geodesic submanifold in $T_{1}S^{n}$.

We consider some recent results about the Poisson structures, arising on the co-algebra of a given Lie algebra when we have on it a structure of a bundle of Lie algebras. These tensors have applications in the study of the Hamiltonian structures of various integrable nonlinear models, among them the O(3)-chiral fields system and Landau–Lifshitz equation

Living human fibroblasts were attached on fibronectin coated surfaces and stained with FITC labeled anti-$\beta_{1}$ integrin monoclonal antibody. The dynamic behaviour of these integrin–antibody complexes were then observed within 2.5 hours by periodic scans using confocal laser scanning microscope. Obtained data were used for analyzing the initial $\beta_{1}$ integrin reorganizations during fibroblasts spreading on fibronectin. Pursuing this aim, a specific physical model and mathematical algorithm was created that permit the corrections of the noise and the fluorescence photobleaching during the scanning. Using specific image analyzing software were defined three “regions of interest” (ROI) and the kinetic changes of integrin densities, as well as, the individual velocity of receptor clusters movement were quantified. Calculated velocities provide novel quantitative information about the centripetal movement of $\beta_{1}$ integrins on the dorsal cell surface of fibroblasts upon ligand binding.