Collective coordinates and length-scale competition in spatially inhomogeneous soliton-bearing equations

Perturbed, one-dimensional, integrable (i.e., soliton-bearing) equations arise in many applied contexts when trying to improve the (usually highly idealized) description of problems of interest in terms of the purely integrable equations. In particular, when the assumption of perfect homogeneity is dropped to account for unavoidable impurities or defects, perturbations depending on the spatial coordinate must be added to the original equation. In this review, we use the one-dimensional sine-Gordon (sG) equation perturbed by a spatially periodic term as a generic paradigm to discuss the main perturbative techniques available for the study of this class of problems. To place the work in context, we summarize the approaches developed to date and focus on the collective coordinate approach as one of the most useful tools. We introduce several versions of this perturbative method and relate them to more involved procedures. We analyze in detail the application to the sG equation, but the procedure is very general. To illustrate this other examples of the application of collective coordinates are briefly revisited. In our case study, this approach helps us identify perturbative and nonperturbative regimes, yielding a very simple picture of the former. Beyond perturbative calculations, the same example of the inhomogeneous sG equation allows us to introduce a phenomenon, termed length scale competition, which we show to be a rather general mechanism for the appearance of complex spatiotemporal behavior in perturbed integrable systems, as other instances discussed in the review show. Such nonperturbative results are obtained by means of numerical simulations of the full perturbed problem; numerical linear stability analysis is also used to clarify the origins of the instability originated by this competition. To complement our description of the techniques employed in these studies, computational details of our numerical simulations are also included. Finally, the paper closes with a discussion of the above ideas and a speculative outlook on general questions concerning the interplay of nonlinearity with disorder.