Confinement of discrete breathers in inhomogeneously profiled nonlinear chains

Department d’Estructura i Constituents de la Mate `ria, Universitat de Barcelona, 08028 Barcelona, Spain Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Physics Department, University of Crete and Foundation for Research and Technology –Hellas, P.O. Box 2208, 71003 Heraklion, Crete, Greece ~Received 4 October 2002; revised manuscript received 11 December 2002; published 31 March 2003 !

Introduction.Moving discrete breathers ͑DBs͒ in homogeneous nonlinear chains have been studied extensively in recent years ͓1-3͔.The scattering of a DB by isolated impurities has also been explored ͓4͔.If the spatial extent of a DB is about ten or more lattice sites, it can readily be described as an envelope soliton ͓5͔.Therefore, many of the results obtained for solitons and, in particular, those obtained using collective-coordinate approaches in continuum ͑or, more refined, quasicontinuum ͓6͔͒ models are directly applicable to DBs' behavior too.
The influence of lattice inhomogeneities of various kinds ͑mass or interaction parameters͒ has been considered previously, e.g., in the context of propagation of energy packets along diatomic Toda chains ͓7͔, soliton interaction in the Ablowitz-Ladik model ͓8͔, soliton propagation in discrete sine-Gordon chain superlattices and aperiodic structures ͓9͔, nonlinear diatomic chains ͓10-12͔, scattering of Toda solitons at a mass interface ͓13͔, etc. Particular attention was paid to the trapping of DBs by an extended impurity region in a nonlinear optical chain with Morse-type on-site potential, intended to model the mechanism of DNA denaturation ͓14͔.However, these latter studies also suggested that many of the features of DBs behavior ͑e.g., DB trapping, DB interactions, phonon production, DB splitting͒ are not properly described ͑qualitatively or quantitatively͒ within the framework of ͑quasi͒continuum approximations.Valuable information on the effect of discreteness can be obtained from the numerical simulations, and this is the approach we adopt here.
The important issue of the scattering of a DB in the presence of ''engineered disorder'' has only recently been addressed in the context of an acoustic Fermi-Pasta-Ulam ͑FPU͒ chain with a junction ͓15͔.We consider here different kinds of ordered inhomogeneities ͑as various mass profiles͒ in FPU chains, obtained by juxtaposing, at a certain distance, two essentially abrupt mass junctions of different types ͑i.e., heavy-light and light-heavy͒ thus producing traps with different profiles ͑insets in Fig. 1͒.We discuss DB reflection, transmission, trapping, splitting, and confinement phenomena, most of them intimately connected with the discreteness of the chain ͑as noted above͒ and the characteristics of the DBs, and therefore difficult to study analytically.Our results have important implications for realistic situations such as junctions between different electron-phonon coupled chains ͓16͔, optical fibers with variable refractive indices ͓17͔, and presumably also for energy transport in biomolecules consisting of various functional moieties ͓18͔.
Model.The FPU model represents a one-dimensional chain of particles with no on-site potential, with the Hamiltonian for an inhomogeneous chain where ␣ and ␤ denote, respectively, the strengths of the linear and nonlinear nearest-neighbor interactions; x n is the elongation at the nth particle ͑with respect to its equilibrium position͒, and m n is the mass of the nth particle.For simplicity, all these quantities are expressed in dimensionless units.The FPU lattice admits DB-like solutions ͓19͔ with periods T DB that are smaller than the minimum period of the phonon spectrum.The details of the model, numerics and how to create various mobile DBs ͑using the algebraic method ͓20͔͒ have been given in Ref. Profiles.In Fig. 1, we show the trapping of a single DB for five different mass profiles in region B. The width of the trap profile is chosen as Lϩ1ϭ41 particle sites.For the rectangular case there is no focusing and the DB keeps bouncing back and forth between the two walls ͑i.e., junc-tions͒ of the trap.The DB slows down after each successive rebound indicating that eventually ͑after a very long time͒ it would get trapped at some lattice site within region B. The second and third panels correspond to a smooth, essentially concave mass profile of the type m i ϭ0.9ϩ␦ tanh 2 (͉L/2 Ϫi͉), iϭ0,1, . . .,L, where i is the label of the ith particle in the trap, and ␦ is chosen, depending on , so that the ͑i͒ Figure 2 depicts the effect of trap width for the rectangular case.In the case of synchronous arrival of the two DBs from the opposite sides, for a large trap comprising 41 lattice sites ͑top panel͒ the DBs continue to collide symmetrically with each other and with the two walls of the trap with a slight reduction in their velocity at each subsequent collision ͑the two DBs lose energy through phonon radiation͒.Note that the ''effective'' breather size for the current choice of parameters is about ten lattice sites.Therefore, if we choose the trap width to be smaller than the breather size then we may expect a qualitatively different collision behavior.
Indeed, the middle panel of Fig. 2 shows that for a trap of five lattice sites the two breathers first coexist in a ͑colliding͒ trapped transient state.However, afterwards one of the DBs is expelled from the trap, while the other one remains trapped with oscillatory collisions with the two walls of the trap ͑akin to the case in the first panel of Fig. 1͒.After the interaction, the two DBs have ͑slightly͒ different frequencies and velocities compared to their initial values: there is an FIG. 1. Trapping of a DB in five different mass profiles ͑insets͒ with a trap width of 41 particle sites.The two horizontal lines at site numbers 200 and 240 delimit the trap region.Notice the more efficient focusing as the trap changes from rectangular-passing through smooth concave-to triangular and then to smooth convex.In these and subsequent figures, the quantities plotted are dimensionless.
exchange of energy between the DBs in the presence of the ''perturbation'' represented by the trap.A similar effect, difficult to account for in a collective coordinate description, was reported by Ting and Peyrard ͓14͔.However, we do not find a general tendency toward energy accumulation in the trap.In some cases the DB that ''escapes'' is more energetic compared to the one arriving at the trap-as in the particular example below.The width of the trap seems to play a role: the larger traps are apparently more efficient in accumulating energy than the smaller ones.For a very narrow trap comprising only two lattice sites ͑bottom panels of Fig. 2͒, the behavior depends on the time of arrival.If the DBs arrive simultaneously at the trap, they collide elastically and are reflected back to their respective A sides ͑no trapping͒.On the contrary, if one DB arrives slightly before the other one ͑asynchronous arrival͒, it gets trapped.When the second DB arrives, it is reflected with some change in frequency and velocity.
͑ii͒ In Fig. 2͑a͒, we showed the case of synchronous arrival of two DBs.The asynchronous arrival in a large rectangular trap is explored in Fig. 3, where we present breather collisions for two different arrival times.If the DBs collide first time at the trap boundary ͑top panel͒ then the delayed DB is reflected ͑usually losing some energy͒.If they collide asymmetrically inside the trap, then, after a few collisions with the trap boundaries and each other, one of the DBs is absorbed.The heavier resulting DB is finally trapped by the discreteness of the lattice at a site inside the trap.Thus, the collision phenomena are sensitively dependent on arrival time.
͑iii͒ In the previous cases ͑Figs. 2 and 3͒, we studied collisions of two DBs in a rectangular trap.To investigate the effect of trap shape on DB collisions, we consider a large triangular trap in Fig. 4 with its width comprising 41 lattice sites, so that the DBs have ''enough space'' to enter and collide inside the trap.However, due to the focusing effect of such a trap once the DBs are inside it and collide, they are forced to be confined in a progressively smaller space.Eventually they ''collapse'' and a new DB results.The resultant DB appears to have a larger velocity ͑the amplitude of oscillation is larger just after the collapse͒ as if with a tendency to escape the trap.Nevertheless, due to the focusing effect of the trap it is soon confined to the center of the trap.Note that there is always a significant amount of ''noise'' ͑phonons and tiny, short lived, breathers resulting from collisions͒ associated with these collisional events, which seems more pronounced in the case of mass profiles ͑as compared to our preliminary results for trap profiles in FPU interaction parameters ␣ and ␤).In Fig. 4, we depicted only the case of synchronous arrival of the two DBs at the triangular trap.If the two DBs arrive asynchronously at the trap ͑not shown͒, they may coexist in a transient ''bound state'' for a short time before collapsing into a single DB.
͑iv͒ We also explored the case of two different DBs ͑i.e., with different frequencies͒ arriving at a large trap ͑41 particle sites͒.The arrival time is usually asynchronous in this case.As a general rule ͑results not shown͒, after entering the trap and colliding ͑with the walls and with the other, heavier DB͒, the lighter of the two DBs is ''absorbed'' by the other one ͑a phenomenon already noticed in different contexts ͓3͔͒, resulting in a single DB ͑presumably heavier, i.e., with higher frequency than the colliding ones͒.The larger the difference in frequency of the two incoming DBs, the quicker the disapperance of the lighter DB.The final, heavier DB is trapped by the discreteness of the lattice ͑combined, eventually, with the focusing effect of a profiled trap͒ at some point of the trap, generally its center.
Conclusion.We presented the results of a systematic numerical study of the trapping and collisions of discrete breathers in an inhomogeneous Fermi-Pasta-Ulam chain consisting of mass profiles with various shapes and widths.We also studied the effect of asynchronous arrival time of two DBs at the trap.In all cases quite different DB focusing, merging, and trapping behavior was found.In general, the emerging DBs are different from the initial ones.Qualita- tively similar behavior is observed if we study profiles with variable interaction parameters (␣,␤) instead of the mass.Some of the observed DB behavior can be rationalized in terms of estimating the Peierls-Nabarro barrier for inhomo-geneous chains ͓15͔.Still, a complete quantitative understanding of the cases considered here remains an open question for further study-the use of collective coordinate approach ͓21͔ that accounts appropriately for discreteness effects ͑e.g., the presence of ''internal modes'' for the FPU breather͒ is a possibility.We believe the phenomena reported here are qualitatively robust in the sense that ͑a͒ the choice of potential ͑other than FPU͒ such as, e.g., in the Frenkel-Kontorova model ͑see also Ref. ͓14͔ for Morse-type poten-tials͒ should give similar results and ͑b͒ these phenomena persist under ͑small͒ perturbations.Finally, we suggest that DBs in electron-phonon coupled chains ͓16͔, pulse propagation in optical fibers ͓17͔, and energy transport in biomolecules ͓18͔ may provide a physical realization of the phenomena found in this study.
͓15͔.All the results we present below are obtained for a ''sandwich'' A -B -A structure, in which we fix the parameters of the A part (␣ A ϭ␤ A ϭm A ϭ1) and the interaction constants of the B part (␣ B ϭ␤ B ϭ1), while choosing different mass profiles in region B. A DB with ͑initial͒ period T DB ϭ2.1 ͑in order to provide a sense to the profile of this DB, we refer to Fig.1in Ref. ͓15͔͒ moves from one of the regions A of the chain towards region B. By ''DB's position'' at a certain moment we mean the site n with maximum absolute value of the relative elongation ͉x n Ϫx nϪ1 ͉.
PHYSICAL REVIEW E 67, 037601 ͑2003͒ 1063-651X/2003/67͑3͒/037601͑4͒/$20.00 ©2003 The American Physical Society masses m 0 ϭm L ϭm A ϭ1.The values ϭ10 and ϭ3, respectively, allow the concave profile to interpolate between the rectangular trap and a triangular trap ͑fourth panel͒.As the profile gets closer to the triangular shape, the DB gets progressively more focused in the center of region B. Finally, the last panel shows an essentially convex profile for ϭ0.5; the focusing is even more rapid, as intuitively expected.Recall that a moving DB ͑that is not an exact solution of the FPU Hamiltonian dynamics͒ continuously emits phonons, at a very slow rate, while moving through the lattice ͑due to the discreteness of the lattice͒.This emission is greatly enhanced inside the profiled trap, leading to DB confinement.Discreteness thus plays an essential role.Breather collisions.Next, we systematically study the collisions of two DBs as a function of ͑i͒ trap width, ͑ii͒ DB arrival time, synchronous versus asynchronous, ͑iii͒ trap shape, and ͑iv͒ whether the two colliding DBs are identical ͑same frequency and velocity͒ or different.

FIG. 2 .
FIG. 2. Scattering of two identical DBs, arriving simultaneously ͓except for the right panel of Fig. 2͑c͔͒ at a rectangular trap from the opposite sides, as a function of trap width.The circle in ͑b͒ indicates the coexistence regime of the two DBs.