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The Langevin equation for a two-dimensional (2D) nonlinear guided vortex motion in a tilted cosine pinning potential in the presence of an ac current is exactly solved in terms of a matrix continued fraction at arbitrary value of the Hall effect. The influence of an ac current of arbitrary amplitude and frequency on the dc and ac magnetoresistivity tensors is analyzed. The ac current density and frequency dependence of the overall shape and the number and position of the Shapiro steps on the anisotropic current-voltage characteristics is considered. An influence of a subcritical or overcritical dc current on the time-dependent stationary ac longitudinal and transverse resistive vortex response (on the frequency of an ac-driving Ω) in terms of the nonlinear impedance ten-sorẐ and a nonlinear ac response at Ω-harmonics are studied. New analytical formulas for 2D temperature-dependent linear impedance tensorẐL in the presence of a dc current which depend on the angle α between the current density vector and the guiding direction of the washboard PPP are derived and analyzed. Influence of α-anisotropy and the Hall effect on the nonlinear power absorption by vortices is discussed.

The influence of an ac current of arbitrary amplitude and frequency on the mixed-state dc-voltageac-drive ratchet response of a superconducting film with a dc current-tilted uniaxial cosine pinning potential at finite temperature is theoretically investigated. The results are obtained in the singlevortex approximation, i.e., for non-interacting vortices, within the frame of an exact solution of the appropriate Langevin equation in terms of a matrix continued fraction. Formulas for the dc voltage ratchet response and absorbed power in ac response are discussed as functions of ac current amplitude and frequency as well as dc current induced tilt in a wide range of corresponding dimensionless parameters. Special attention is paid to the physical interpretation of the obtained results in adiabatic and high-frequency ratchet responses taking into account both running and localized states of the (ac+dc)-driven vortex motion in a washboard pinning potential. Our theoretical results are discussed in comparison with recent experimental work on the high-frequency ratchet response in nanostructured superconducting films [B. B. Jin et al., Phys. Rev. B 81, 174505 (2010)].

Explicit current-dependent expressions for anisotropic longitudinal and transverse nonlinear magnetoresistivities are represented and analyzed on the basis of a Fokker-Planck approach for twodimensional single-vortex dynamics in a washboard pinning potential in the presence of point-like disorder. Graphical analysis of the resistive responses is presented both in the current-angle coordinates and in the rotating current scheme. The model describes nonlinear anisotropy effects caused by the competition of point-like (isotropic) and anisotropic pinning. Nonlinear guiding effects are discussed and the critical current anisotropy is analyzed. Gradually increasing the magnitude of isotropic pinning force this theory predicts a gradual decrease of the anisotropy of the magnetoresistivities. The physics of transition from the new scaling relations for anisotropic Hall resistance in the absence of point-like pins to the well-known scaling relations for the point-like disorder is elucidated. This is discussed in terms of a gradual isotropizaton of the guided vortex motion, which is responsible for the existence in a washboard pinning potential of new (with respect to magnetic field reversal) Hall voltages. It is shown that whereas the Hall conductivity is not changed by pinning, the Hall resistivity can change its sign in some current-angle range due to presence of the competition between i -and a-pins.