Related papers: Notes on Riemann geometry and integrable systems. …
The paper proposes a 4-dimensional generalization of the Hamilton equations of motion to the case of the Minkowski space-time. The approach can be applied to quantum as well as to classical, non-relativistic as well as relativistic…
On the base of Lie algebraic and differential geometry methods, a wide class of multidimensional nonlinear systems is obtained, and the integration scheme for such equations is proposed.
A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space-time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized…
This paper constructs a Riemann surface associated to the icosahedron and discusses the geodesics associated to a flat metric on this surface. Because of the icosahedral symmetry, this is a distinguished special case of the example treated…
We theoretically study discrete photonic lattices in more than three dimensions and point out that such systems can exist in continuous three-dimensional (3D) space. We study discrete diffraction in the linear regime, and predict the…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
A new approach to the model of the universe based on work by Rippl, Romero, Tavakol is presented. We have used the scheme for relating the vacuum (D + 1) dimensional theories to D dimensional theories for setting up a correspondence between…
A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One…
Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $\Pi$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or…
In this review we try to capture some of the recent excitement induced by experimental developments, but also by a large volume of theoretical and computational studies addressing multi-component nonlinear Schrodinger models and the…
We apply our recent formalism establishing new connections between the geometry of moving space curves and soliton equations, to the nonlinear Schr\"{o}dinger equation (NLS). We show that any given solution of the NLS gets associated with…
We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.
We introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of ${\bf SO}(3) \times {\bf SO}…
We give an account of some recent development that connects the concept of mass in general relativity to the geometry of large Riemannian polyhedra, in the setting of both asymptotically flat and asymptotically hyperbolic manifolds.
Flows on the moduli space of the algebraic Riemann surfaces, preserving the periods of the corresponding solutions of the soliton equations are studied. We show that these flows are gradient with respect to some indefinite symmetric flat…
There is a hierarchy of commuting soliton equations associated to each symmetric space U/K. When U/K has rank n, the first n flows in the hierarchy give rise to a natural first order non-linear system of partial diffferential equations in n…
The normality equations for the Newtonian dynamical systems on an arbitrary Riemannian manifold of the dimension $n \geq 3$ are considered. Locally the solution of such equations reduces to three possible cases: in two of them the solution…
Effective 4-dimensional theories are investigated which were obtained under dimensional reduction of multidimensional cosmological models with a minimal coupled scalar field as matter source. Conditions for the internal space stabilization…
A solution with the pole configuration in six dimensions is analysed both analytically and numerically. It is a dimensional reduction model of Randall-Sundrum type. The soliton configuration is induced by the bulk Higgs mechanism. The…
The manuscript studies a 3+N+1-dimensional space in which the N extra dimensions are dynamically compact. The 3 large dimensions, behaving as the spacial part of the FRW metric, possess a different scale factor in comparison with the N…