We investigate the influence of the boundary conditions and the box size on the existence and stability of various types of localized solutions (particles and holes) of the cubic-quintic complex Ginzburg-Landau equation as it arises as a prototype envelope equation near the weakly hysteretic onset of traveling waves. Two types of boundary conditions are considered for one spatial dimension, both of which can be realized experimentally: periodic boundary conditions, which can be achieved for an annulus and Neumann boundary conditions, which correspond to zero flux, for example in hydrodynamics. We find that qualitative differences between the two types of boundary conditions arise in particular for propagating and breathing localized solutions. While an asymmetry in the localized state is always connected to motion for periodic boundary conditions, this no longer applies for Neumann boundary conditions. In the case of Neumann boundary conditions we observe that breathing localized states can no longer exist below a certain box size, which is comparable to the ‘width’of the localized state.
- Boundary conditions
- cubic-quintic Ginzburg-Landau equation