Numerical Computation of Coherent Structures

Abstract

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.

Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

“…In particular, intersections must be viewed geometrically within a fixed energy level. We refer the reader to [28] for a general discussion of such phenomena and their resolution. Here we take a direct approach, which is suitable because we have chosen a functional analytic setting that is distant from the geometric (phase-space) point of view.…”

To Snake or Not to Snake in the Planar Swift–Hohenberg Equation

“…In order to solve for this additional constraint, we exploit the fact that weak interaction with a boundary turns the formerly stationary solutions into waves that travel with a very small speed c in the y-direction. Thus, introducing the speed c as an extra parameter allows us to solve the phase constraint, and we refer to [11] for more details and a rigorous justification of this procedure. The system consisting of the phase condition and the Swift-Hohenberg equation formulated in a frame that moves with speed c in the y-direction is given by…”

“…In this case, the boundary condition at y = 0 factors out the approximate translation symmetry, and we can solve (3.4) directly using Newton's method, see [11]. Where possible, we will use this approach.…”

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

“…However, this is a much more difficult numerical problem that was solved only recently (Sandstede & Scheel 2000;Bordiougov & Engel 2006;Rademacher et al 2007). Rademacher et al (2007) give a full but rather mathematically oriented account of the method; it involves applying AUTO to a boundary-value problem for the eigenfunctions, and the theory underlying numerical continuation of such problems is reviewed by Champneys & Sandstede (2007). A less technical summary, including computer programs written in a tutorial style, is available at www.ma.hw.ac.uk/wjas/ supplements/ptwreview/index.html.…”