Discrete breathers in a two-dimensional Morse lattice with an on-site harmonic potential

Two-dimensional discrete breathers in a two-dimensional Morse lattice with on-site harmonic potentials are investigated. Under the harmonic approximation, the linear dispersion relations for the triangular and the square lattices are discussed. The existence of discrete breathers in a two-dimensional Morse lattice with on-site harmonic potentials is proved by using local inharmonic approximation and the numerical method. The localization and amplitude of two-dimensional discrete breathers correlate closely to the Morse parameter a and the on-site parameter κ.     

Geometric discretization of the Bianchi system

We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion of discrete normal congruences. Finally, considering quadrilateral lattices “with equal tangential invariants” which allow for harmonic normal congruences we obtain, in complete analogy with the continuous case, the integrable discrete analogue of the Bianchi system together with its geometric meaning. To obtain this geometric meaning we also make use of the novel characterization of the circular lattice as a quadrilateral lattice whose coordinate lines intersect orthogonally in the mean.     

Discrete breathers in a model with Morse potentials

Under harmonic approximation, this paper discusses the linear dispersion relation of the one-dimensional chain. The existence and evolution of discrete breathers in a general one-dimensional chain are analysed for two particular examples of soft (Morse) and hard (quartic) on-site potentials. The existence of discrete breathers in one-dimensional and two-dimensional Morse lattices is proved by using rotating wave approximation, local anharmonic approximation and a numerical method. The localization and amplitude of discrete breathers in the two-dimensional Morse lattice with on-site harmonic potentials correlate closely to the Morse parameter a and the on-site parameter к.     

Reconfigurable 3D photonic lattices by optical induction for optical control of beam propagation

We report on our recent theoretical and experimental studies of three-dimensional (3D) photonic lattice structures which are established in a bulk nonlinear crystal by employing different optical induction techniques. These 3D photonic lattices bring about new opportunities for controlling the flow of light via coupling engineering originated from the lattice modulation along the beam propagation direction. By fine tuning the lattice parameters, we observe a host of unusual behaviors of beam propagation in such reconfigurable 3D lattices, including enhanced discrete diffraction, light tunneling inhibition—better known as coherent destruction of tunneling (CDT), anomalous diffraction, negative refraction, as well as CDT-based image transmission. In addition, we propose and demonstrate a new way of creating 3D ionic-type photonic lattices by controlled Talbot effect.     

光诱导光子晶格结构中新型的离散空间光孤子

离散孤子标志着从线性到非线性，从连续到非连续，从相干到非相干，人们对孤子认识的一个飞越．文章简要回顾了近期在二维光致光子晶格结构中有关空间离散光孤子的研究，包括基模离散孤子、类矢量离散孤子、离散偶极孤子、离散涡旋孤子和离散孤子串等．在非线性光折变晶体里用部分相干光诱导的波导阵列中，对每一种离散孤子，都清楚地观测到光从二维的离散衍射状态到自囚禁形成离散孤子的转变过程，获得的结果将对其他离散非线性系统中类似现象的研究有所启发．     

Studies of bosons in optical lattices in a harmonic potential

We present a theoretical study of Bose condensation and specific heat of non-interacting bosons in finite lattices in harmonic potentials in one, two, and three dimensions. We numerically diagonalize the Hamiltonian to obtain the energy levels of the systems. Using the energy levels thus obtained, we investigate the temperature dependence, dimensionality effects, lattice size dependence, and evolution to the bulk limit of the condensate fraction and the specific heat. Some preliminary results on the specific heat of fermions in optical lattices are also presented. The results obtained are contextualized within the current experimental and theoretical scenario.     

Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices

We report the first experimental demonstration of ring-shaped photonic lattices by optical induction and the formation of discrete solitons in such radially symmetric lattices. The transition from discrete diffraction to single-channel guidance or nonlinear self-trapping of a probe beam is achieved by fine-tuning the lattice potential or the focusing nonlinearity. In addition to solitons trapped in the lattice center and in different lattice rings, we demonstrate controlled soliton rotation in the Bessel-like ring lattices.     

Geometry, topology, and physics of non-Abelian lattices

Conclusion  After reviewing in some detail the notion of non-Euclidean lattices, whose domain of physical realization lies mostly in the novel carbon structures of the family offullerenes, we have discussed a number of physical problems denned over such lattices. We have shown that the group-theoretical definition of these lattices leads to “designing” new tubular regular structures, endowed with symmetries unheard of in the frame of customary crystallography, which combine features of extreme complexity and, at the same time, of great regularity. We have compared the role of the non-Abelian symmetries which these super-lattices are characterized by, with that of (discrete) harmonic (Fourier) lattice symmetry typical of customary crystallographic lattices. Many novel features enter into play, due to thenon-flatness of the related lattice geometry, which led us to a novel—sometimes unexpected—insight into the dynamical and/or thermodynamical properties of various physical systems which have these lattices as ambient space. We have analyzed how lattice topology bears on the complex combinatorics (related to loop-counting) of the classical Ising model. These lattices, even though finite, are, of course, much closer to being three-dimensional than regular 2D lattices simply equipped with periodic boundary conditions. We have shown, on the other hand, how the relation between the lattice symmetry (for example, in the case of fullerene, the discrete subgroup ofSU(2) that we have denotedg ₆₀ and the symmetry proper to the Hamiltonian of quantum systems of many itinerant interacting electrons (Hubbard-like models) allows us to reduce the calculation of the system spectral properties to a “size” that can be dealt with numerically with present-day numerical exact diagonalization techniques much more easily than a regular 3D cluster with a quite smaller number of sites.     

Influence of impurities on the dynamics and synchronization of coupled map lattices

The influence of impurities and defects on the dynamics and synchronization of coupled map lattices (CML) is studied. In the context of CML we define impurities as sites in the lattice which have another local dynamics that from the whole lattice and defects as sites in the lattice without any dynamics. We show that synchronization and spatial intermittence are obtained as a function of the number of impurities present on a one-dimensional lattice. We also derive an analytical condition for a signal to “transpose” an impurity. For open flow models, we show that not only the presence of the impurity but also its position along the lattice and its local dynamics can be used to manipulate the lattice in order to obtain a regular or irregular motion. We also show how defects can be used to store information in a lattice.     

Discrete nonlinear Schrödinger equation dynamics in complex networks

We investigate dynamical aspects of the discrete nonlinear Schrödinger equation in finite lattices. Starting from a periodic chain with nearest neighbor interactions, we insert randomly links connecting distant pairs of sites across the lattice. Using localized initial conditions we focus on the time averaged probability of occupation of the initial site as a function of the degree of complexity of the lattice and nonlinearity. We observe that selftrapping occurs at increasingly larger values of the nonlinearity parameter as the lattice connectivity increases, while close to the fully coupled network limit, localization becomes more preferred. For nonlinearity values above a certain threshold we find a reentrant localization transition, viz. localization when the number of long distant bonds is small followed by delocalization and enhanced transport at intermediate bond numbers while close to the fully connected limit localization reappears.     

Unusual slow energy relaxation induced by mobile discrete breathers in one-dimensional lattices with next-nearest-neighbor coupling

We study the energy relaxation process in one-dimensional (1D) lattices with next-nearest-neighbor (NNN) couplings. This relaxation is produced by adding damping (absorbing conditions) to the boundary (free-end) of the lattice. Compared to the 1D lattices with on-site potentials, the properties of discrete breathers (DBs) that are spatially localized intrinsic modes are quite unusual with the NNN couplings included, i.e. these DBs are mobile, and thus they can interact with both the phonons and the boundaries of the lattice. For the interparticle interactions of harmonic and Fermi–Pasta–Ulam–Tsingou-β (FPUT-β) types, we find two crossovers of relaxation in general, i.e. a first crossover from the stretched-exponential to the regular exponential relaxation occurring in a short timescale, and a further crossover from the exponential to the power-law relaxation taking place in a long timescale. The first and second relaxations are universal, but the final power-law relaxation is strongly influenced by the properties of DBs, e.g. the scattering processes of DBs with phonons and boundaries in the FPUT-β type systems make the power-law decay relatively faster than that in the counterparts of the harmonic type systems under the same coupling. Our results present new information and insights for understanding the slow energy relaxation in cooling the lattices.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

Optically induced transition between discrete and gap solitons in a nonconventionally biased photorefractive crystal

We show that optically induced photonic lattices in a nonconventionally biased photorefractive crystal can support the formation of discrete and gap solitons owing to a mechanism that differs from the conventional screening effect. Both the bias direction and the lattice orientation can dramatically influence the nonlinear beam-propagation dynamics. We demonstrate a transition from self-focusing to -defocusing and from discrete to gap solitons solely by adjusting the optical-beam orientation.     

Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit

We demonstrate that certain strictly anharmonic one-dimensional FPU lattices with a suitable quartic site potential appended support almost-compact discrete breathers over a macroscopic localized domain that is essentially fixed independently of the sparseness of the lattice. Beyond that domain the discrete breather tails decay at a double-exponential rate in the lattice-cell index, becoming truly compact in the continuum limit. Furthermore, the discrete breather is stable for amplitudes below a sharp threshold that depends on the sparseness of the lattice. For the two-dimensional version of the problem, the continuum limit of a planar hexagonal lattice with a purely quartic interaction potential begets an isotropic multidimensional nonlinear wave equation. When a quartic site potential of the appropriate sign is appended, the continuum equation has a compactly supported radial breather solution.     

Collective behavior in coupled chaotic map lattices with random perturbations

Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a “thermal bath”, where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x→4x(1−x).     

Linear stability of breathers of the discrete NLS

We consider real breather solutions of the discrete cubic nonlinear Schrödinger equation near the limit of vanishing coupling between the lattice sites and present leading order asymptotics for the eigenvalues of the linearization around the breathers. The expansion is given in fractional powers of the intersite coupling parameter and determines the linear stability of the breathers. The method we use relies on normal form ideas and applies to one and higher-dimensional lattices. We also present some examples.     

Phase diagram of two-component dipolar fermions in one-dimensional optical lattices

We theoretically map out the ground state phase diagram of interacting dipolar fermions in one-dimensional lattice. Using a bosonization theory in the weak coupling limit at half filing, we show that one can construct a rich phase diagram by changing the angle between the lattice orientation and the polarization direction of the dipoles. In the strong coupling limit, at a general filing factor, we employ a variational approach and find that the emergence of a Wigner crystal phases. The structure factor provides clear signatures of the particle ordering in the Wigner crystal phases.     

Observation of discrete solitons in optically induced real time waveguide arrays

We report the first experimental observation of discrete solitons in an array of optically induced waveguides. The waveguide lattice is induced in real time by illuminating a photorefractive crystal with a pair of interfering plane waves. We demonstrate two types of bright discrete solitons: in-phase self-localized states and the staggered (pi out-of-phase) soliton family. This experiment is the first observation of bright staggered solitons in any physical system. Our scheme paves the way for reconfigurable focusing and defocusing photonic lattices where low-power (mW) discrete solitons can be thoroughly investigated.     

Can Disorder Induce a Finite Thermal Conductivity in 1D Lattices?

We study heat conduction in one-dimensional mass-disordered harmonic and anharmonic lattices. It is found that the thermal conductivity kappa of the disordered anharmonic lattice is finite at low temperature, whereas it diverges as kappa approximately N0.43 at high temperature. Moreover, we demonstrate that a unique nonequilibrium stationary state in the disordered harmonic lattice does not exist at all.     

Discrete breathers — Advances in theory and applications

Nonlinear classical Hamiltonian lattices exhibit generic solutions — discrete breathers. They are time-periodic and (typically exponentially) localized in space. The lattices have discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. We will introduce the concept of these localized excitations and review their basic properties including dynamical and structural stability. We then focus on advances in the theory of discrete breathers in three directions — scattering of waves by these excitations, persistence of discrete breathers in long transient processes and thermal equilibrium, and their quantization. The second part of this review is devoted to a detailed discussion of recent experimental observations and studies of discrete breathers, including theoretical modelling of these experimental situations on the basis of the general theory of discrete breathers. In particular we will focus on their detection in Josephson junction networks, arrays of coupled nonlinear optical waveguides, Bose–Einstein condensates loaded on optical lattices, antiferromagnetic layered structures, PtCl based single crystals and driven micromechanical cantilever arrays.