A pr 1 99 6 Analytic Structure of the Landau - Ginzburg Equation in 2 + 1 Dimensions

In this paper, two methods are employed to investigate for which values of the parameters, if any, the two-dimensional real Landau-Ginzburg equation possesses the Painlevé property. For an ordinary differential equation to have the Painlevé property, all of its solutions must be meromorphic but for partial differential equations there are two inequivalent definitions, one a direct investigation of a Laurent series expansion in several complex variables and the other indirect and relying on the symmetry group of the partial differential equation. We check both methods for the Landau-Ginzburg equation in 2 + 1 variables, and each one yields that this equation does not possess the Painlevé property for any values of the parameters. The Landau-Ginzburg equation arises in a wide variety of contexts in physics but the case for which we are interested is that of the order parameter of a binary fluid, φ = c1 − c2 c1 + c2 where ci represents that concentration of the i-th component of the fluid mixture. With the thermodynamic potential