Derivation of the time-dependent Ginzburg-Landau equation on the line
The author proposes a rigorous derivation of the one-dimensional time-dependent Ginzburg-Landau (GL) equation; previously, rigorous derivations were elaborated only for its stationary (time-independent) version. The starting points are nonlinear second-order evolutional PDEs and PDE systems, such the Swift-Hohenberg equation (including not only cubic, but also quadratic nonlinear terms), and reaction-diffusion equations (the so-called Brusselator model). By means of the Lyapunov-Schmidt reduction technique and circumventing a known difficulty related to the contribution from the coninuum spectrum, the author derives time-dependent GL equations that, actually, are not PDEs, but rather partial pseudodifferential equations, i.e., those that contain \(x\)-derivatives of all orders. A formal truncation transforms the pseudodifferential equation into the traditional GL one, but the truncation procedure is not derived rigorously in this work. Some results obtained in the paper admit generalizations for multidimensional GL equations.
scientific article; zbMATH DE number 1129276