COLLECTIVE COORDINATES AND LENGTH-SCALE COMPETITION IN SPATIALLY INHOMOGENEOUS SOLITON-BEARING EQUATIONS

Perturbed, one-dimensional, integrable (i.e., soliton-bearing) equatio ns 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 per fect homogeneity is dropped to account for unavoidable impurities or d efects, perturbations depending on the spatial coordinate must be adde d 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 availabl e for the study of this class of problems. To place the work in contex t, we summarize the approaches developed to date and focus on the coll ective coordinate approach as one of the most useful tools. We introdu ce several versions of this perturbative method and relate them to mor e 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 revi sited. In our case study, this approach helps us identify perturbative and nonperturbative regimes, yielding a very simple picture of the fo rmer. Beyond perturbative calculations, the same example of the inhomo geneous 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 integr able systems, as other instances discussed in the review show. Such no nperturbative results are obtained by means of numerical simulations o f the full perturbed problem; numerical linear stability analysis is a lso 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 a re also included. Finally, the paper closes with a discussion of the a bove ideas and a speculative outlook on general questions concerning t he interplay of nonlinearity with disorder.