The inverse scattering transform for Manakov's system is formulated in a covariant form. This leads immediately to a description of the polarization using Fedorov's beam tensor. New polarizational conservation laws are formulated with the help of this tensor. Such laws may be written for other integrable vector evolutional equations.

535.012.22Self-focusing of an optical paraxial beam in a photorefractive cubic crystal exposed to a strong external electric field E 0 of arbitrary orientation has been considered with regard to the Pockels effect. The best localization of radiation is shown to be attained when the vector E 0 is oriented along diagonals of the cubic cell. Numerical modeling revealed that the beam width increased significantly for non-optimal electric field orientations.Introduction. Self-focusing of optical beams in photorefractive crystals has been well studied experimentally and theoretically [1][2][3][4][5][6][7][8]. Thus, the dielectric permeability tensor was expanded into a Taylor series [9] around an external electric field. The linear term corresponded to the Pockels effect; the quadratic, to the Kerr effect. These effects make the formation of solitons dependent on the orientation of the external electric field E 0 and the wave normal n. A cubic crystal becomes biaxial under the influence of a strong field. If the direction of vector E 0 is changed, both induced optical axes change their orientation over broad limits. The dynamics of the optical axes and refractive indices of a perturbed cubic crystal with various orientations of the external field E 0 and wave normal have been investigated [10].Applications require light radiation to be localized in small areas. The localization of a beam can be improved by selecting the direction of the wave normal and the external electric field. Vectors E 0 and n in theoretical studies [2][3][4] were directed along the crystallographic axes. The properties of screened solitons with the external field E 0 oriented in certain fixed planes have also been studied [5, 6]. Self-focusing with E 0 oriented in the (1 1 2 _ ) plane was examined theoretically [5]. It has been reported that the electric field vector belongs to the (1 _ 1 _ 0) plane [6]. It was shown [5, 6] that the best conditions for self-focusing of a light beam are attained for E 0 directed along [111] and for the beam polarized parallel to this direction. The same orientation of the vectors is frequently selected for experimental studies of self-focusing in cubic photorefractive crystals [7,8].Herein we determine the optimum orientations of the vectors of the external electric field, wave normal, and polarization. All possible orientations of E 0 in space are examined. This vector is not required to reside in any fixed plane.The properties of the solitons depend on the direction because of the anisotropy of the linear and nonlinear responses of the crystal that are described by tensors of different ranks. It has been shown [11][12][13][14][15] that the coordinate-free (covariant) Fedorov method is effective for studying anisotropy in optics and acoustics. In the present work this method is used to study the properties of photorefractive solitons in cubic nongyrotropic crystals.Nonlinear Schro .. dinger Equation for Cubic Crystals. Maxwell's equation for the electric field E frequency

535.012.22The 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.

517.958:535The 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.Introduction. Photorefractive media show promise in nonlinear optical applications for a number of reasons [1][2][3][4]. Self channeling beams are formed in these media at optical radiation powers of a few microwatts, while soliton effects are observed in fibers at powers on the order of a watt. Simpler experimental apparatus is used for obtaining photorefractive solitons than for obtaining Kerr solitons.Three major types of spatially photorefractive solitons are known [5]: screening, photogalvanic, and quasistationary. Each of these can be realized in the form of bright and dark solitons. Bright solitons have a bell-shaped intensity profile that approaches zero at infinity. Dark solitons have a profile with an intensity dip relative to its constant value [6].A high, constant electric field is applied to a photorefractive crystal in order to observe screening solitons. As optical radiation propagates in the crystal, the photorefractive effect causes ionization of donors, so that mobile electrons show up in the conduction band. The electrons are displaced by the high external field. Thus, a space charge develops which reduces the external field. Hence, the electric field in the illuminated portion of the crystal falls below the electric field in the dark region. The electro-optical effect changes the refractive index in the illuminated region. With a suitable choice of the external field orientation and of the crystal parameters, it is possible to create a nonlinear waveguide that compensates the diffraction of the beam.When their divergence is disrupted, light beams in a nonlinear medium have a particle-like behavior, as happens with waves in many other natural nonlinear systems [7,8]. Originally the term "soliton" denoted a solution of the wave equation in the form of a solitary stationary wave which asymptotically retained its shape and velocity in collisions with other similar waves. However, this terms is now used in a wider sense, but without a formal definition. In many cases, this definition is too narrow. For example, pulses that fully merit designation as solitons, may interact in a more complicated manner because of their "particle-like nature" [9].We wish to emphasize that the subject of solitons involves a synthesis of various branches of mathematics: geometry, algebra, analysis, algebraic and differential geometry, differential equations, the theory of functions of a real variable, group-theoretical topology, and the theory of operators [10][11][12]. Mathematical models have been constructed for other fields besides electromagnetic fields (Higgs bosons, maximons, Polyakov extremons, etc.) [12]. The major conclusion derived from interference experiments with material waves is that there is no fixed boundary between the classical and quantum worlds [13]. This ...