Soliton-radiation interaction in nonlinear integrable lattices.

The subject of integrable nonlinear lattices is an important branch of the inverse scattering transform (IST) method. ' Many infinite families of these integrable systems can be derived and classified by means of discrete versions of the Lax-pair technique, ' and some of their members are interesting models for vastly different physical situations. As in the continuous context, two types of (nonlinear) normal modes are manifested in the long-time behavior of discrete integrable systems: namely, solitons and radiation. Analysis of their dynamical properties is of relevance in several areas as, for instance, the classical statistical mechanics of soliton-bearing models. In this paper we are concerned with the effect of radiation modes on soliton motion in nonlinear lattices. Our approach rests on the technique proposed in Ref. 5 for describing soliton-radiation interactions in continuous systems. Thus, the main component involved in our strategy is the asymptotic analysis of Gel'fand-Levitan-Marchen ko (GLM) equations of discrete type. As a result the procedure can be applied to any nonlinear lattice integrable through the IST method. The paper is organized as follows. Section II is devoted to a description of some basic aspects of the IST method associated with a linear difference equation. More concretely, we consider the difference-equation eigenvalue problem which gives rise to the Toda and Langmuir hierarchies of nonlinear lattices. Section III deals with the characterization of the asymptotic trajectories of solitons in the presence of radiation and the derivation of the position shifts caused by the soliton-radiation interaction. As relevant applications, the expressions of these shifts for the Toda chain are obtained. Since the main ideas of our analysis can be easily described, to avoid loss of continuity we have relegated the discussion of several technical points to three appendixes at the end of this paper.