The spontaneous polarization in ferroelectric materials could be expected to be completely destabilized by the backward depolarizing electrostatic field produced by the charge rho=div P of the polarization surface breakdown.  This apparent paradox is avoided by the segregation of the sample into polarized domains with different orientation of polarization, as was initially proposed by Landau and Kittel . This alternates the surface charge and diminishes the bulk depolarizing field.

To model the polarization texture in realistic samples of size of 50-500nm we use an analytical approach solving the nonlinear Ginzburg-Landau equations coupled with the electrostatic equations. By variation of the external field we study the dynamics of polarization switching.  The description of the formation of domains, vortices, skyrmions, etc. recently seen in experiment and in first-principle modeling is possible.  Introducing Landau-Khalatnikov dynamics we study the formation and evolution of such topological textures and their role in polarization switching.