We study, analytically and numerically, the effect of frequency detunings and relaxation processes in laser media on stability and bifurcations of dissipative optical localized structures (DOLS's) in a transversely one-dimensional laser with a saturable absorber. The approximate envelope equation, with an intensity dependent effective coefficient of the diffusion, is derived. Andronov-Hopf bifurcations resulting from frequency detuning and leading to oscillatory DOLS's are analyzed numerically. A numerical and analytical study of bifurcations of transversely motionless DOLS's in a laser with finite relaxation rates of amplifying and absorbing media is performed. New types of DOLS's are found, including those moving with a large transverse velocity and those moving with a periodically oscillating transverse velocity. Hysteresis between different types of DOLS's is demonstrated.

The main factors affecting the accuracy and computational cost of the calculation of (31)P NMR chemical shifts in the representative series of organophosphorous compounds are examined at the density functional theory (DFT) and second-order Møller-Plesset perturbation theory (MP2) levels. At the DFT level, the best functionals for the calculation of (31)P NMR chemical shifts are those of Keal and Tozer, KT2 and KT3. Both at the DFT and MP2 levels, the most reliable basis sets are those of Jensen, pcS-2 or larger, and those of Pople, 6-311G(d,p) or larger. The reliable basis sets of Dunning's family are those of at least penta-zeta quality that precludes their practical consideration. An encouraging finding is that basically, the locally dense basis set approach resulting in a dramatic decrease in computational cost is justified in the calculation of (31)P NMR chemical shifts within the 1-2-ppm error. Relativistic corrections to (31)P NMR absolute shielding constants are of major importance reaching about 20-30 ppm (ca 7%) improving (not worsening!) the agreement of calculation with experiment. Further better agreement with the experiment by 1-2 ppm can be obtained by taking into account solvent effects within the integral equation formalism polarizable continuum model solvation scheme. We recommend the GIAO-DFT-KT2/pcS-3//pcS-2 scheme with relativistic corrections and solvent effects taken into account as the most versatile computational scheme for the calculation of (31)P NMR chemical shifts characterized by a mean absolute error of ca 9 ppm in the range of 550 ppm.

We study stability and bifurcations of 1D localized structures in a laser with a saturable absorber. Instability leading to oscillating localized structure is described. Our numerical study of light propagation in a dispersive medium with saturable gain and absorption provides evidence for the existence of 3D 'laser bullets'.

We predict a new class of three-dimensional (3D) topological dissipative optical one-component solitons in homogeneous laser media with fast saturable absorption. Their skeletons formed by vortex lines where the field vanishes are tangles, i.e., N_{c} knotted or unknotted, linked or unlinked closed lines and M unclosed lines that thread all the closed lines and end at the infinitely far soliton periphery. They are generated by embedding two-dimensional laser solitons or their complexes in 3D space after their rotation around an unclosed, infinite vortex line with topological charge M_{0} (N_{c}, M, and M_{0} are integers). With such structure propagation, the "hula-hoop" solitons form; their stability is confirmed numerically. For the solitons found, all vortex lines have unit topological charge: the number of closed lines N_{c}=1 and 2 (unknots, trefoils, and Solomon knots links); unclosed vortex lines are unknotted and unlinked, their number M=1, 2, and 3.

We reveal the existence of stable dissipative soliton complexes with curvilinear motion of their center of mass. This type of motion results from the field distribution asymmetry and is well pronounced for asymmetric complexes of laser solitons with strong coupling. We present results of numerical simulations of such complexes in a model of wide-aperture lasers or laser amplifiers with saturable gain and absorption. The complex consists of a pair of strongly coupled vortex solitons weakly coupled with a number of other vortex solitons. Similar complexes are expected to exist in different spatially distributed nonlinear dissipative systems, including schemes with discrete dissipative solitons.