  We explain the phenomena of ferroelectric phase transition temperature $T_c$
enhancement beyond the end members in perovskite solid solutions like
BiMeO$_3$-PbTiO$_3$ (Me=Sc, In, etc.) is related to nonlinear and spatial
correlation effects. The explanation is based on the calculation of $T_c$ in
the framework of our random field theory with additional account for nonlinear
effects in the above substances. We show that the maximum of $T_c$ for certain
PbTiO$_3$ content takes place when coefficient of nonlinearity is positive, the
value of this coefficient is found from best fit between theory and experiment.
This nonlinearity coefficient is the only adjustable parameter of the theory.
We show that enhancement of positive nonlinearity coefficients enhances greatly
the $T_c$ maximum over its value for end members.
  The theory lays the foundation to calculate not only $T_c$ for above solid
solutions but virtually any equilibrium and/or nonequilibrium thermodynamic
characteristics such as static and dynamic dielectric susceptibility, specific
heat etc as a functions of PbTiO$_3$ content, temperature, electric field and
other external parameters.

Materials Science