Analysis of the three-dimensional time-dependent Landau-Ginzburg equation and its solutions

The time-dependent Landau-Ginzburg equation of the form eta t +D Delta eta =P( eta ) where Delta is the three-dimensional Laplacian operator and P( eta ) is an odd-order polynomial up to fifth order, has been used to describe the kinetics of non-conservative order parameters eta at and near criticality. The symmetry reduction method has been applied to solve this equation when P( eta ) is given by P( eta )=a+b eta +c eta 3 +d eta 5 . This can be used to model both first- and second-order phase transitions which take place with (a not=0) or without (a=0) an external field. When either d not=0 or c not=0, two general cases have been found: (i) a=b=0 or a=b=d=0, where the symmetry group involves translations in (3+1)-dimensional spacetime, rotations in three-dimensional space and a dilation; (ii) otherwise, where the symmetry group involves translations in (3+1)-dimensional spacetime and rotations in three-dimensional space. All the reductions to ODE and the corresponding symmetry variables have been derived. The Painleve-type reduced ODE have been solved exactly while the remaining ones can be analysed numerically or approximately. Physical properties of the obtained solutions have been discussed, including their energies.