Ginzburg-Landau Equation for Fractal Media

Abstract

The Ginzburg-Landau equation (Ginzburg and Landau, 1950; Ginzburg, 2004) is one of the most-studied nonlinear equations in physics (Aranson and Kramer, 2002). It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The Ginzburg-Landau equation can be derived (Lifshitz and Pitaevsky, 1980) from free energy functional. We can define the free energy functional in the form:

$$ F\{ \Psi (x)\} = {F_0} + \frac{1}{2}\int_W {\left( {g{{({D^1}\Psi )}^2} + a{\Psi ^2} + \frac{b}{2}{\Psi ^4}} \right)} d{V_3}, $$

(5.1)

where Ψ= Ψ(x) is a real-valued function. In Eq. (5.1) the integration is over 3-dimensional region W of continuous media. Here F ₀ is a free energy of the normal state, i.e., F{Ψ(x)} for Ψ(x) = 0.