Numerical Computation of Coherent Structures

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 parameter 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 traveling waves, time-periodic structures, and planar localized 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.