On Heisenberg models with values in Hermitian symmetric spaces

Heisenberg models with values in Hermitian symmetric spaces are introduced. A matrix form of the corresponding nonlinear Schrödinger equations is also presented. It is shown that a hidden symmetry of the generalized Heisenberg models generates a one-parameter family of solutions.     

Gray solitons in parity-time symmetric potentials

We numerically study the gray solitons in parity-time (PT) symmetric potentials. Simulated results show that there are two kinds of gray solitons, the dip-shaped gray solitons and the hump-shaped solitons, and both of them can be stable. Hump-shaped solitons can always exist, but the grayness of a stable dip-shaped gray soliton should exceed a threshold value. More interesting, it is discovered that when propagating in PT symmetric potentials, the gray solitons have no transverse deviation, and this is a phenomenon different from the usual gray solitons.     

Coherent states and symmetric spaces

Properties of system of the coherent states related to representations of the class I of principal series of the motion groups of symmetric spaces of rank 1 have been studied. It has been proved that such states are given by horospherical kernels and are the generalization of the plane waves for the case of symmetric spaces.     

Extrinsic symplectic symmetric spaces

We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also build a natural ?$?$-quantization on a class of examples.     

Extraordinary solitons in non-centrally symmetric photorefractive crystals

The directions of the phase normals for which extraordinary bright screened solitons can be formed are described analytically. Formulas are derived for the directions in which extraordinary solitons can exist in lithium niobate for arbitrary orientations of the external electric field. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 76, No. 3, pp. 403–410, May–June, 2009.     

On weakly symmetric Finsler spaces

In this paper, we study weakly symmetric Finsler spaces. We first study an existence theorem of weakly symmetric Finsler spaces. Then we study some geometric properties of these spaces and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each weakly symmetric Finsler space is of Berwald type.     

Codimension reduction in symmetric spaces

In this paper we give a short geometric proof of a generalization of a well-known result about reduction of codimension for submanifolds of Riemannian symmetric spaces.     

Bright ordinary solitons in non-centrally symmetric photorefractive crystals

The directions of the phase normal such that bright ordinary screening solitons can exist in non-centrally symmetric crystals are determined analytically. The results are used to calculate the directions of solitons in lithium niobate. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 76, No. 1, pp. 152–160, January–February, 2009.     

Integrability and Huygens' principle on symmetric spaces

The explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of their restricted roots are even. As a corollary in the odd-dimensional case one has that the Huygens' principle in Hadamard's sense for these equations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same symmetric spaces based on the fact that the radial part of the Laplace-Beltrami operator on such a space is related to the algebraically integrable case of the generalised Calogero-Sutherland-Moser quantum system. In the last section of this paper we apply the methods of Heckman and Opdam to extend our results to some other symmetric spaces, in particular to complex and quaternian grassmannians.     

Symmetric spaces and star representations: II. Causal symmetric spaces

We construct and identify star representations canonically associated with holonomy-reducible simple symplectic symmetric spaces. This leads a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley-type and Hermitian symmetric spaces of tube-type.     

Three graded exceptional algebras and symmetric spaces

The exceptional algebras of typeE ₇ are studied from the point of view of their three graded structure. The connection between three-grading and the Jordan Pair structure of such Lie algebras is analyzed. The Jordan Pair content is in turn related to the symmetric spaces and . Coset spaces of this type have been recently suggested as possible scalar manifolds in supergravity. We develop a way of representingE ₇ in a matrix form, which makes the Jordan Pair content of theE ₇ real and complex forms quite transparent and shows whether such forms admit a three graded structure.     

Conformational completion of Lorentz symmetric spaces

Every simply connected Lorentz symmetric space of dimensionn3 has a symmetric quotient which can be conformally imbedded in a quadric of the projective space ⁿ⁺¹ (R).     

Twistor spinors on Lorentzian symmetric spaces

An indecomposable Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature. Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In the present paper we solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicitly. In particular, we show that there are — in contrast to the Riemannian case — indecomposable Lorentzian symmetric spaces with twistor spinors, which have non-constant sectional curvature and non-flat and non-Ricci flat homogeneous Lorentzian manifolds with parallel spinors.     

Defect solitons in optical lattices with parity-time symmetric defect

We report experimental measurements of linear and nonlinear magneto-optical polarization rotation on an intercombination transition of Ba vapor (???=?791.1?nm). We observed a maximum polarization rotation angle in Faraday configuration of 15?mrad, accompanied by reduction of absorption.     

Defect solitons supported by parity-time symmetric defects in superlattices

The existence and stability of defect solitons supported by parity-time (PT) symmetric defects in superlattices are investigated. In the semi-infinite gap, in-phase solitons are found to exist stably for positive defects, zero defects, and negative defects. In the first gap, out-of-phase solitons are stable for positive defects or zero defects, whereas in-phase solitons are stable for negative defects. For both the in-phase and out-of-phase solitons with the positive defect and in-phase solitons with negative defect in the first gap, there exists a cutoff point of the propagation constant below which the defect solitons vanish. The value of the cutoff point depends on the depth of defect and the imaginary parts of the PT symmetric defect potentials. The influence of the imaginary part of the PT symmetric defect potentials on soliton stability is revealed.