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one of the highly active areas of investigation of hamiltonian nonlinear systems over the past decade has been the examination of nonlinear dynamical lattices of the discrete nonlinear schrdinger ( dnls ) type . chiefly , this development has arisen due to the multitude of applications of pertinent models that have emerged in areas such as nonlinear optics and atomic physics . more specifically , in the optical context , the setting of fabricated algaas waveguide arrays @xcite has been one of the most prototypical ones for the application of dnls models . there , the interplay of discreteness and nonlinearity revealed many interesting features including peierls - nabarro potential barriers , diffraction and diffraction management @xcite , and gap solitons @xcite , among others ; see also the reviews @xcite and references therein . another recent development , which also promoted the analysis of discrete systems in connection with nonlinear optics was the proposal @xcite and creation @xcite of optically induced photonic lattices in photorefractive crystals such as sbn . this paved the way for the observation of a large set of exciting nonlinear wave related phenomena in such crystals . as a representative subset , we mention here the formation of patterns such as dipole @xcite , quadrupole @xcite and necklace @xcite solitary waves , impurity modes @xcite , discrete vortices @xcite , rotary waves @xcite , higher order bloch modes @xcite and gap vortices @xcite , two - dimensional ( 2d ) bloch oscillations and landau - zener tunneling @xcite , wave formation in honeycomb @xcite , hexagonal @xcite and quasi - crystalline lattices @xcite , and recently the study of anderson localization in disordered photonic lattices @xcite . although this setting is mostly studied in the continuum context with a periodic potential ( and sometimes in the presence of the inherent crystal anisotropy ) , it has also spurred a number of studies in the dnls context with the saturable photorefractive nonlinearity @xcite . lastly , another physical realization of such nonlinear dynamical lattices arose over the past few years in atomic physics through the examination of bose - einstein condensates ( becs ) trapped in periodic potentials . there , once again , a reduction of the relevant model can be formulated in the tight - binding approximation within the mean - field limit , reducing the so - called gross - pitaevskii equation with a periodic potential to a genuinely discrete nonlinear schrdinger equation @xcite . in both the nonlinear optical and in the atomic physics setting discussed above , multi - component systems were also examined in recent investigations . more specifically , the first observations of discrete vector solitons in optical waveguide arrays were reported in @xcite , the emergence of multipole patterns in vector photorefractive crystals was presented in @xcite , while numerous experiments with becs were directed towards studies of mixtures of different spin states of @xmath0rb @xcite or @xmath1na @xcite and even ones of different atomic species such as @xmath2k@xmath0rb @xcite and @xmath3li@xmath4cs @xcite . it should be noted that while the above bec experiments did not include the presence of an optical lattice , the addition of such an external optical potential is certainly feasible within the present experimental capabilities @xcite .    our aim in the present work is to propose and analyze a family of solutions particular to multicomponent ( in particular , binary , although more - component generalizations are certainly possible ) systems of dnls equations . we dub these proposed solutions `` interlaced '' discrete solitons and vortices , a name stemming from the feature that the profiles of the modes in the two interacting components will have a vanishing intersection of excited sites in the extreme discrete limit of zero coupling between adjacent nodes of the lattice . in these structures , the first component will be excited where the second component is not and vice - versa . in the one - dimensional case , we show how to interlace in a stable fashion simple , as well as more elaborate , bound states of the system @xcite . for such solutions , we consider their existence and stability properties also from an analytical point of view , using as a starting point the anti - continuum limit ( of no - coupling between the sites ) . then we generalize our considerations to higher dimensional settings , showcasing the potentially stable interlacing of more elaborate structures , such as discrete vortices @xcite ( but also of vortices with non - vortical structures ) . we present detailed stability diagrams of such interlaced structures , and also examine their dynamics when they are found to be unstable . our presentation is structured as follows . in section ii , we present the model and general mathematical setup . in section iii , we illustrate both analytically and numerically the properties of such structures in 1d settings . in section iv , we generalize these considerations to a numerical investigation of higher dimensional settings . finally , in section v , we summarize our findings and present our conclusions . we consider a set of coupled dnls equations @xmath5 where @xmath6 is a @xmath7-dimensional index and @xmath8 is the discrete laplacian in @xmath7 dimensions . we look for stationary solutions @xmath9 , @xmath10 through the relation @xmath11    the dynamical equations ( [ dyn1 ] ) then transform into @xmath12    the stability is determined in a frame rotating with frequency @xmath13 for @xmath14 and @xmath15 for @xmath16 , i.e. , we suppose that @xmath17,\qquad      v_n(t)=\exp(i\lambda_2t)[v_n+\xi^{(2)}_n(t)].\ ] ]    the small perturbations @xmath18 , with @xmath19 can be expressed as @xmath20 leading to the linear stability equations @xmath21 with @xmath22    soliton and vortex solutions are calculated using methods based on the anti - continuous limit . upon calculating these solutions at @xmath23 , we continue them to finite coupling by varying @xmath24 or other parameters ( such as the interspecies nonlinearity strength @xmath25 ) . we are interested in interlaced solitons ( iss ) in 1d lattices and interlaced vortices ( ivs ) in 2d and 3d lattices . the excited sites at @xmath23 are equal to @xmath26 and @xmath27 , except for a phase factor @xmath28 , while @xmath29 at the corresponding excited site . these values are @xmath30    in what follows , we choose @xmath31 and @xmath32 . we also choose @xmath33 as , for @xmath34 interlaced solitons and vortices are unstable for every value of @xmath24 . we consider interlaced solitons which are labeled by @xmath35 , where @xmath36 . this number indicates the `` order '' of the excited state at the anti - continuous limit , whose phase @xmath37 is chosen so that the isolated solitons ( i.e. when @xmath38 ) are stable for any small @xmath24 . for instance , the ground state @xmath39 means @xmath40 and the first excited state @xmath41 will be taken to mean @xmath42 at the ac limit . thus , the state @xmath43 corresponds to @xmath40 , @xmath44 and @xmath45 to @xmath42 , @xmath46 . we first analyze the @xmath43 state , which is stable for @xmath47 . at @xmath48 the iss become unstable through hopf bifurcations ( the value of @xmath49 differs as a function of the rest of the system parameters such as @xmath25 , however the above scenario is robust ) . cascades of this type of bifurcations arise as @xmath24 increases and , when , @xmath50 , the iss become also exponentially unstable . there is a special region for @xmath51 $ ] where the system experiences an inverse hopf bifurcation recovering the stability in a window . the system becomes unstable again through hopf bifurcations for @xmath52 $ ] and exponential instabilities for @xmath53 $ ] . besides , for @xmath54 $ ] there exist windows with only exponential instabilities . fig . [ fig1 ] illustrates all of the above features , by showcasing a typical example of the @xmath43 state , a typical continuation of its principal linearization eigenfrequencies @xmath55 , and a full two - parameter diagram of the stability of this state in the two - parameter plane @xmath56 .    for @xmath45 states , the scenario is essentially similar to the @xmath43 case , although , in essence , it is considerably simpler due to the absence of any inverse hopf bifurcations and restabilization windows . [ fig2 ] shows the corresponding features for @xmath45 , as fig . [ fig1 ] for the @xmath43 case .    [ cols="^,^,^ " , ] in the present work , we have illustrated the possibility to successfully interlace structures which are stable in each one of the components ( either simple ones , such as single site solitary waves , or more elaborate ones , such as bound states and vortices ) in order to produce stable multi - component interlaced solitons / vortices . we have continued the resulting structures from the anti - continuum limit of no inter - site coupling to finite coupling and illustrated the intervals of stability , as well as the ones of both exponential and oscillatory ( hopf ) instabilities . we have given detailed two - parameter diagrams of the stable ranges of the solutions as a function of the inter - site and inter - component couplings . these revealed that the linear stability of the interlaced structures necessitates sufficiently weak coupling ( typically no larger than 0.4 , with the relevant range decreasing as the inter - component interaction is increased ) and sufficiently weak inter - component interaction ( i.e. , @xmath57 ) . finally , we examined the dynamical evolution of the instability of such interlaced structures , which typically resulted in the destruction of the waveforms , in favor of simpler , more stable dynamical patterns . nevertheless , there is still a number of important open questions for future consideration . for instance , it would be particularly interesting to examine whether it would be possible for the inter - component coupling to actually stabilize structures that are dynamically unstable in the single - component setting . also , it would be useful to possess a systematic classification of the solutions ( interlaced and non - interlaced ones ) available in the multi - component system setting , similarly to the one - component classifications of @xcite . such efforts are currently underway and will be reported in future publications . r. morandotti , u. peschel , j. s. aitchison , h. s. eisenberg and y. silberberg , phys . lett . * 83 * , 2726 - 2729 ( 1999 ) ; h. s. eisenberg , y. silberberg , r. morandotti and j. s. aitchison , phys . lett . * 85 * , 1863 ( 2000 ) . fitrakis , p.g . kevrekidis , h. susanto and d.j . frantzeskakis , phys . e * 75 * , 066608 ( 2007 ) ; v.m . rothos , h.e . nistazakis , p.g . kevrekidis and d.j . frantzeskakis , j. phys . a : math . theor . * 42 * , 025207 ( 2009 ) .
in the present work , we propose a new set of coherent structures that arise in nonlinear dynamical lattices with more than one components , namely interlaced solitons . 
 these are waveforms in which in the relevant anti - continuum limit , i.e.  when the sites are uncoupled , one component has support where the other component does not . 
 we illustrate systematically how one can combine dynamically stable unary patterns to create ones such for the binary case of two - components . in the one - dimensional setting , 
 we provide also a detailed theoretical analysis of the existence and stability of these waveforms , while in higher dimensions , where such analytical computations are far more involved , we resort to corresponding numerical computations . 
 lastly , we perform direct numerical simulations to showcase how these structures break up , when exponentially or oscillatorily unstable , to structures with a smaller number of participating sites .