Domain Wall Dynamics in Ginzburg-Landau-Type Equations with Conservative Quantities

In the Ginzburg-Landau equation, there are domain walls connecting two metastable states. The dynamics of domain walls has been intensively studied, but there remain still unsolved but crucial problems even for a single domain. We study the domain wall dynamics in three different Ginzburg-Landau-type equations satisfying conservation laws. In a modified $\phi^4$ model satisfying the law of energy conservation and the Lorentz invariance, the motion of a domain wall is accelerated and the velocity approaches its maximum. In a one-dimensional model of eutectic growth, the order parameter is conserved and a domain wall connecting a metastable uniform state and a spatially periodic pattern appears. We try to find a selection rule for the wavelength of a spatially periodic pattern. In a model equation for martensitic transformation, a domain wall connecting a uniform metastable state and a zigzag structure appears which propagates at a high velocity.