Topological filters and high-pass/low-pass devices for solitons in inhomogeneous networks R. Burioni, 1 D. Cassi, 1 P. Sodano, 2,3 A. Trombettoni, 2 and A. Vezzani 4 1 Dipartimento di Fisica and I.N.F.N., Università di Parma, parco Area delle Scienze 7A, Parma I-43100, Italy 2 Dipartimento di Fisica and I.N.F.N., Università di Perugia, Via A. Pascoli, Perugia I-06123, Italy 3 Progetto Lagrange, Fondazione C.R.T. e Fondazione I.S.I. c/o Dipartimento di Fisica-Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, I-10124, Italy 4 I.N.F.M./C.N.R. and Dipartimento di Fisica, Università di Parma, parco Area delle Scienze 7A, Parma I-43100, Italy Received 20 October 2004; revised manuscript received 24 April 2006; published 27 June 2006 We show that, by inserting suitable finite networks at a site of a chain, it is possible to realize filters and high-pass/low-pass devices for solitons propagating along the chain. The results are presented in the framework of coupled optical waveguides; possible applications to different contexts, such as photonic lattices and Bose- Einstein condensates in optical networks are also discussed. Our results provide a first step in the control of the soliton dynamics through the network topology. DOI: 10.1103/PhysRevE.73.066624 PACS numbers: 42.65.Tg, 02.10.Ox, 05.45.Yv, 42.81.Dp I. INTRODUCTION Soliton dynamics in discrete structures features remark- able effects, presenting a very high—theoretical and experimental—interest 1–3. In particular, a very active and well-established field of research is the study of wave trans- mission in presence of nonlinearity on translationally invari- ant chains, where solitonic and breather solutions have been extensively studied 2,4. For solitons propagating on more general discrete structures, the role played by the topology of the network i.e., how the sites of the network are connected between them and the interplay and competition of topology and nonlinearity in the soliton dynamics are still an open problem, and one expects that new interesting phenomena should arise. The interest on this topic is also motivated by the fact that in several systems, like networks of nonlinear waveguide arrays 3, arrays of super conducting networks 5, Bose-Einstein condensates in optical lattices 6, and silicon-based photonic crystals 7, one can, at some extent, engineer the shape i.e., the topology of the network. On this respect, the effects of inhomogeneity on soliton propa- gation and on localized modes have been investigated in Y junctions 8,9, junctions made of two infinite waveguides and waveguide couplers 10, lattices featuring topological dislocations created by the interference between plane waves and waves with nested vortices 11, and scattering through a topological perturbation 12. In this paper we consider solitons propagating on inhomo- geneous discrete structures, and we show that it is possible to use the inhomogeneity—i.e., the shape of the network—to realize filters for the soliton momentum allowing for the propagation of solitons with a given velocity and high-pass/ low-pass devices allowing for the propagation of solitons with high/low velocity. In particular, we will focus on the discrete nonlinear Schrödinger equation DNLSE on inho- mogeneous networks: the DNLSE is a paradigmatic example of nonlinear model which has been extensively studied on regular lattices. Indeed it describes the properties of several real systems, like coupled waveguides arrays 3, nonlinear discrete electrical networks 13, and Bose-Einstein conden- sates in optical lattices 14we also refer to the reviews 2,15 for more references on applications of the DNLSE. In particular, soliton propagation has been experimentally ob- served in coupled optical waveguides 16 and these systems represent a promising physical setup for the study of the effects of topology on soliton propagation. A topological en- gineering of waveguides appears to be a realizable task, as illustrated by Fig. 1. Other experimental systems could be used to engineer and build inhomogeneous networks. Two- dimensional optically induced nonlinear photonic lattices have been realized and discrete solitons observed 17: using a suitable interference of two or more plane waves in a pho- tosensitive material one could in principle create a non- translationally invariant photonic lattice. For Bose-Einstein condensates in optical lattices, discrete gap solitons and self- trapped states have been recently observed in linear arrays 18: the control of the lattice shape in this system is realized properly superimposing the laser beams creating the optical lattices, as discussed in Ref. 19. Here we address the issue of filtering the soliton propaga- tion in the DNLSE, focusing on a particular class of inho- mogeneous networks, built by adding a finite discrete net- work G 0 to a single site of a linear chain see Fig. 1. We FIG. 1. An inhomogeneous system of coupled nonlinear waveguides extending in the z direction, obtained arranging the waveguides in a chain and attaching another finite chain in the waveguide 0. In the inset we plot the corresponding graph: each waveguide is a site of the graph, and two coupled waveguides are connected by a link. The attached graph G 0 is here a finite chain of length 3; G r is obtained subtracting  from G 0 . PHYSICAL REVIEW E 73, 066624 2006 1539-3755/2006/736/0666246 ©2006 The American Physical Society 066624-1