Numerical computation of coherent structures

In many applications one is interested in finding solutions to nonlinear evolution equations with a particular spatial and temporal structure. For instance, solitons in optical fibers and wave guides or buckling modes of long structures can be interpreted as localised travelling or standing waves of an appropriate underlying partial differential equation (PDE) posed on an unbounded domain. Spiral waves or other defects in oscillatory media are time-periodic waves with an asymptotic spatially periodic structure. All of these examples are referred to as coherent structure. They represent relative equilibria, that is, their temporal evolution is determined by a symmetry of the underlying PDE: namely, translational symmetry for travelling waves, rotational symmetry for spiral waves, and phase symmetry for oscillatory structures. Given the complexity of typical PDE models, these nonlinear waves are in general accessible only through numerical computations. One possible approach is via direct simulation which is, however, expensive and fails to capture solutions that are either unstable or may have a small basin of stability. Simulation also fails to provide valuable information on how branches of solutions are organised in parameter space. In this chapter, we give an overview of boundary-value problem formulations for coherent structures which provide a robust and less expensive alternative to simulation. Moreover, setting up well-posed boundary-value problems allows us to continue solutions in parameters space, investigate their spectral stability directly, and continue branches of solutions efficiently as parameters vary. In the next section we outline how PDEs can be supplemented by phase conditions that allow us to compute nonlinear waves as regular zeros of the resulting nonlinear system. In the remaining sections, we treat different kinds of coherent structures, namely travelling waves, time-periodic structures, and planar localised patterns. In each case we explain how to set up a well-posed boundary-value problem and illustrate the theory with the results of an example computation.