We have studied the spin-wave dispersion at low temperatures and the transition temperature ${(T}_N)$ of the spin-$\frac{1}{2}$ antiferromagnet and high-$T_C$ parent ${\mathrm{La}}_2{\mathrm{CuO}}_4.$ The values of the in-plane exchange parameters (including first, second, and third nearest neighbors) are determined by an accurate fit to the recently experimentally observed in-plane spin-wave spectrum, obtained by the high-resolution inelastic neutron scattering performed on ${\mathrm{La}}_2{\mathrm{CuO}}_4$ [Phys. Rev. Lett. 86, 5377 (2001)]. The analysis of the Néel temperature shows that the in-plane spin anisotropy $\left(\eta \right)$ is much more significant than the three dimensionality, since $T_N$ of the three-dimensional (3D)-antiferromagnet depends rather weakly on the value of the interlayer coupling $\left({\lambda }_{\perp }\right).\mathrm{}$ We obtain that the Néel temperature of the 3D-antiferromagnet varies only weakly within the very wide interval of ${\lambda }_{\perp }$ and the Néel temperature of the anisotropic 2D $(\eta \ne 0,$ ${\lambda }_{\perp }=0)$ antiferromagnet does not differ from the 3D value for the same $\eta .$ These conclusions are valid for both tetragonal and orthorhombic structures. However, $\eta$ dependence of $T_N$ is essentially different: for $\eta =0,$ $T_N$ of the tetragonal structure becomes 0, while $T_N$ of the orthorhombic structure remains finite. These results are valid within the frame of the Tyablikov approximation.

Size effect of the subharmonic Shapiro steps on the behavior of harmonic ones is studied in the ac driven overdamped Frenkel-Kontorova model with different types of physically important realistic substrate potentials. The properties of harmonic Shapiro steps are directly correlated with the size of halfinteger steps; and according to that in their amplitude dependence, three distinctive types of behavior have been classified. Our results on other realistic systems show that these classifications are not universal, and that not only the size of halfinteger steps but also the type of the substrate potential determines the properties and correlation between the Shapiro steps. While for some potentials such as variable and double-barrier potential, system evolves through different types of amplitude dependence as the size of halfinteger steps changes, in the case of double-well potential, the system maintains the standard behavior (the behavior for small halfinteger steps) even in the presence of large halfinteger steps.