Solutions of the steady-state Landau-Ginzburg equation in external driving fields

Abstract

A steady state equation derived from the variation with respect to the order parameter M(x) of a Landau-Ginzburg free energy density of the form -- [special characters omitted] -- is considered, where h ≠ 0, C ≥ 0, D ≠ 0 is a second rank tensor. This is a generalization of prior work by Winternitz et al. [J.Phys. C 21 4931-4953 (1988)], who studied the case h = 0 and C = 0. Applied to a magnetic system, it describes the behaviour of the magnetization of a critical system in the presence of an external magnetic field h and near a structural phase transition. The Landau coefficients A, B, and C are weakly temperature dependent, but are considered constant near the transition temperature Ttr (the Curie point in magnetic systems), except for A α (T — Ttr). The gradient term allows for spatial inhomogeneities due to nearest neighbour interactions. Two cases are examined: C = 0 (B > 0) and C > 0 (B < 0) which correspond to second and first order phase transitions, respectively. The symmetries of the equation arc exploited by the symmetry reduction method to find exact solutions in terms of varied symmetry variables. These solutions are in the form of kinks, bumps, singular, periodic, and doubly periodic solutions. The physical interpretation of these results and other calculations (e.g. energy, susceptibility) based on these results is discussed.