Dimension-breaking Phenomena in Wave Propagation
This talk describes the formation of coherent structures in multidimensional systems, starting from simple, essentially one-dimensional solutions such as planar fronts or line solitons. Varying system parameters or changing the initial conditions may lead to coherent structures that break the translational invariance of the primary solution, a phenomenon which we refer to as dimension breaking.
Mathematically, dimension-breaking phenomena can be understood as bifurcation problems in parameter-dependent nonlinear systems in extended domains. We show how methods from the theory of dynamical systems can be used to investigate such phenomena. We identify a number of bifurcation scenarios which cause a breaking of dimension. Various multidimensional waves are constructed in this way including curved-like interfaces in reaction-diffusion systems and three-dimensional water waves.