Recently – via a simple trick, amounting essentially to a change of independent, and possibly as well of dependent, variables – the possibility has been noted to modify a quite general evolution system so that the modified system possess a lot of completely periodic, indeed isochronous, solutions. Generally these isochronous solutions emerge out of an open domain of initial data having full dimensionality in the space of initial data. And many of the isochronous systems obtained in this manner seem rather interesting. In this paper these developments are reviewed, mainly in the context of dynamical systems (systems of ODEs – in particular, systems interpretable as many-body problems), and some specific examples are discussed in detail, including an analysis of the transition (to motions with higher periods, or aperiodic, or perhaps chaotic) occurring when the initial data get outside of the region producing isochronous motions. The applicability of this approach in the context of nonlinear evolution PDEs is also outlined.

The study of systems of differential equations admitting a super-position function allowing us to write the general solution in terms of a set of arbitrary, but independent, particular solutions, and some constants determining each solution, can be reduced to that of an equation on a Lie group. It will be shown that all these systems of differential equations can be seen as the systems determining the horizontal curves on an appropriate connection and we will show how the theory of reduction can be used to simplify the problem of finding the general solution of such Lie systems. The theory will be illustrated with several physical applications

This is a review of the main ideas of the inverse scattering method (ISM) for solving nonlinear evolution equations (NLEE), known as soliton equations. As a basic tool we use the fundamental analytic solutions $\chi^{\pm}(x,\lambda)$ of the Lax operator $L(\lambda)$. Then the inverse scattering problem for $L(\lambda)$ reduces to a Riemann–Hilbert problem. Such construction has been applied to wide class of Lax operators, related to the simple Lie algebras. We construct the kernel of the resolvent of $L(\lambda)$ in terms of $\chi^{\pm}(x,\lambda)$ and derive the spectral decompositions of $L(\lambda)$. Thus we can solve the relevant classes of NLEE which include the NLS equation and its multi-component generalizations, the $N$-wave equations, etc. Applying the dressing method of Zakharov and Shabat we derive the $N$-soliton solutions of these equations.

Next we explain that the ISM is a natural generalization of the Fourier transform method. As appropriate generalizations of the usual exponential function we use the so-called “squared solutions” which are constructed again in terms of $\chi^{\pm}(x, \lambda)$ and the Cartan–Weyl basis of the relevant Lie algebra. One can prove the completeness relations for the “squared solutions” which in fact provide the spectral decompositions of the recursion operator $\Lambda$.

These decompositions can be used to derive all fundamental properties of the corresponding NLEE in terms of $\Lambda$: i) the explicit form of the class of integrable NLEE; ii) the generating functionals of integrals of motion; iii) the hierarchies of Hamiltonian structures. We outline the importance of the classical $R$-matrices for extracting the involutive integrals of motion.

We are dealing below with the tetrad models of gravitation and with certain modifications of this relatively old and partially forgotten subject. The problem is studied in a general $n$-dimensional manifold, not necessarily in the usual four-dimensional space-time. So strictly speaking, we are dealing with fields of frames ($n$-legs) rather than with literally understood tetrads (four-legs, Vierbeine). The main novelty of our models in comparison with traditional ones is that it is the total globally acting $\mathrm{GL}(n,\mathbb{R})$ that is used as the group of internal symmetries. In traditional approaches one uses the globally or locally acting Lorenz group $\mathrm{SO}(1,n − 1)$. And obviously, our models, according to the Einstein–Hilbert paradigm are generally-covariant. We have shown that group spaces of Lie groups or certain their deformations are special vacuum-type homogeneous solutions of the model. Interaction with the bispinor field is briefly discussed and certain cosmological aspects of our models (cosmological expansions, escaping of galaxies) are mentioned. The model sheds new light onto this important problem. It may also have to do with the important problem of dark matter. In a sense, our model may be also interpreted in mechanical terms, as a theory of relativistic continuum with microstructure (relativistic micromorphic continuum), so in a sense, the traditional gap between field theory and mechanics becomes diffused. There are also certain features of our model which seem to be interesting from the point of view of multidimensional Kaluza–Klein universes (models generally covariant in a multidimensional sense, with fibration and fundamental group structures appearing as features of some special solutions). Spherically symmetric solutions of our models are in a sense formally similar to the t’Hooft–Polyakof monopole.

By using the Grassmannian invariant-angle coherents states approach, the classical analogue of the Aharonov-Anandan nonadiabatic geometrical phase is found for a spin one-half in Nuclear Magnetic Resonance (NMR). In the adiabatic limit, the semi-classical relation between the adiabatic Berry’s phase and Hannay’s angle gives exactly the experimental result observed by Suter et al [12].

The present study is concerned with thin isotropic shallow shells interacting with inviscid fluid flow of constant velocity. It is assumed that the dynamic behaviour of the shells is governed by the Marguerre–von Kármán equations. The influence of the fluid flow is taken into account by introducing additional differential and external load terms in the shell equations. It is shown that the system of equations governing such a fluid-structure interaction is equivalent to the von Kármán equations. Thus, the symmetries and conservation laws of the considered fluid-structure system are established.

The application of symmetry analysis in hydrodynamics is illustrated by two examples. First is a description of all irrotational barochronous motions of ideal gas. The second is an exact solution of magnetohydrodynamics equations for infinitely conducting media, which describes the flow of so called “special vortex” type.

The fundamental properties of the multi-component nonlinear Schrödinger (MNLS) type models related to symmetric spaces are analyzed. New types of reductions of these systems are constructed. The Lax operators $L$ and the corresponding recursion operators $\Lambda$ are used to formulate some of the fundamental properties of the MNLS-type equations. The results are illustrated by specific examples of MNLS-type systems related to the D.III symmetric space for the $\mathfrak{so}(8)$-algebra. The effect of the reductions on their soliton solutions is outlined.

This paper applies differential geometry to multi-dimensional affine space. The three-component distributions of affine space are discussed. Some connections of three-component distributions, which allow to generalize theory of regular and vanishing hyper-zones, zones, hyper-zone distributions, surfaces of full and non-full rank, and tangent equipped surfaces in multidimensional affine spaces are constructed.

The geometry of a few simple models closely related to the Cole’s and Yoneda’s experiments are studied in some detail. Their principal differences are through the boundary conditions at the points where the membranes detach themselves from the compressing planes. Mathematically this affect the equilibrium configurations as in all these cases the mean curvature suffers a discontinuity while this is not so for the tangent vector. A new parametrization of the Delaunay’s nodoids is derived.

There are infinitely many hyperbolic transforms of complex abelian surfaces. The corresponding universal covers change from the complex plane to the unit ball, from flat to hyperbolic metrics. Looking back to Jacobi’s periodic functions we were able to construct 2-dimensional abelian functions transformable to automorphic forms on the ball. In this article we prove explicit dimension formulas for all forms of this coabelian types for fixed weights.

Let $T$ be an arithmetic proportional elliptic configuration on a bi-elliptic surface $A \sqrt{-d}$ with complex multiplication by an imaginary quadratic number field $\mathbb{Q}(\sqrt{-d})$. The present note establishes that if $T$ has $s$ singular points and \[ 4s − 5 \leq h \leq 4s \] irreducible smooth elliptic components, then $d = 3$ and $T$ is $\mathrm{Aut}(A \sqrt{-3}−$ equivalent to Hirzebruch’s example $T^{(1,4)}_{\sqrt{-3}}$ with a unique singular point and 4 irreducible components.

We construct Ermanno–Bernoulli type invariants for the Manev model dynamics which may be viewed upon as remnants of the Laplace–Runge–Lenz vector in the Kepler model. If the orbits are bounded these invariants exist only when a certain rationality condition is met and thus we have superintegrability only on a subset of initial values. Manev model’s dynamics is demonstrated to be bi-Hamiltonian and a recursion operator is constructed. We analyze real form dynamics of the Manev model and derive that it is always superintegrable. We also discuss the symmetry algebras of the Manev model and its real Hamiltonian form.

We define a kind of branching process on the loop space by using the branching mechanism of a loop of string theory

We construct $q$-Taylor formula for the functions of several variables and develop some new methods for solving equations and systems of equations. They are much easier for application than well known ones and very useful when the continuous function does not have fine smooth properties. Especially, we will demonstrate their power in solving the equations where the function is defined by some $q$-integral. We will discuss the convergence and accuracy of those methods and compare them with well known methods. The conclusions are illustrated by examples.

The generalized Hénon–Heiles system with an additional non-polynomial term is considered. The standard method for the search of the elliptic solutions is a transformation of an initial nonlinear polynomial differential equation into a nonlinear algebraic system. It has been demonstrated that the use of the Laurent-series solutions allows to simplify the resulting algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial equation in the form of the Laurent or Puiseux series we use algorithm of the Painlevé test.