Nonlinear perturbation of a high-order exceptional point:Skin discrete breathers and the hierarchical power-law scaling

We study the nonlinear perturbation of a high-order exceptional point(EP)of the order equal to the system site number L in a Hatano-Nelson model with unidirectional hopping and Kerr nonlinearity.Notably,we find a class of discrete breathers that aggregate to one boundary,here named as skin discrete breathers(SDBs).The nonlinear spectrum of these SDBs shows a hierarchical power-law scaling near the EP.Specifically,the response of nonlinear energy to the perturbation is given by E_(m)∝Γ~(α_(m)),whereα_(m)=3^(m-1)is the power with m=1,...,L labeling the nonlinear energy bands.This is in sharp contrast to the L-th root of a linear perturbation in general.These SDBs decay in a double-exponential manner,unlike the edge states or skin modes in linear systems,which decay exponentially.Furthermore,these SDBs can survive over the full range of nonlinearity strength and are continuously connected to the self-trapped states in the limit of large nonlinearity.They are also stable,as confirmed by a defined nonlinear fidelity of an adiabatic evolution from the stability analysis.As nonreciprocal nonlinear models may be experimentally realized in various platforms,such as the classical platform of optical waveguides,where Kerr nonlinearity is naturally present,and the quantum platform of optical lattices with Bose-Einstein condensates,our analytical results may inspire further exploration of the interplay between nonlinearity and non-Hermiticity,particularly on high-order EPs,and benchmark the relevant simulations.

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.

The effect of cubic potentials on discrete breathers in a mixed Klein-Gordon/Fermi-Pasta-Ulam chain

Nonlinearity has a crucial impact on the symmetry properties of dynamical systems. This paper studies a one-dimensional mixed Klein-Gordon/Fermi Pasta-Ulam diatomic chain using the expanded rotating plane-wave approximation and numerical calculations to determine the effect of cubic potentials on the symmetry properties of discrete breathers in this system. The results will be very useful to researchers in the field of numerical calculations on discrete breathers.

Periodic,quasiperiodic and chaotic discrete breathers in a parametrical driven two-dimensional discrete diatomic Klein-Gordon lattice

We study a two-dimensional （2D） diatomic lattice of anhaxmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers （DBs） can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the 2D discrete diatomic Klein-Gordon lattice with hard and soft on-site potentials. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers （QDBs） and chaotic discrete breathers （CDBs） by changing the amplitude of the driver. But the DBs and QDBs with symmetric and anti-symmetric profiles that are centered at a heavy atom are more stable than at a light atom, because the frequencies of the DBs and QDBs centered at a heavy atom are lower than those centered at a light atom.

Two-Dimensional Breather Lattice Solutions and Compact-Like Discrete Breathers and Their Stability in Discrete Two-Dimensional Monatomic β-FPU Lattice

We restrict our attention to the discrete two-dimensional monatomic β-FPU lattice. We look for two- dimensional breather lattice solutions and two-dimensional compact-like discrete breathers by using trying method and analyze their stability by using Aubry＇s linearly stable theory. We obtain the conditions of existence and stability of two-dimensional breather lattice solutions and two-dimensional compact-like discrete breathers in the discrete two- dimensional monatomic β-FPU lattice.

Unusual slow energy relaxation induced by mobile discrete breathers in onedimensional lattices with next-nearestneighbor coupling

We study the energy relaxation process in one-dimensional(1D) lattices with next-nearestneighbor(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.

Discrete gap breathers in a two-dimensional diatomic face-centered square lattice

In this paper we study the existence and stability of two-dimensional discrete gap breathers in a two-dimensional diatomic face-centered square lattice consisting of alternating light and heavy atoms, with on-site potential and coupling potential. This study is focused on two-dimensional breathers with their frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of the existence of two-dimensional gap breathers by using a numerical method. Six types of two-dimensional gap breathers are obtained, i.e., symmetric, mirror-symmetric and asymmetric, whether the center of the breather is on a light or a heavy atom. The difference between one-dimensional discrete gap breathers and two-dimensional discrete gap breathers is also discussed. We use Aubry＇s theory to analyze the stability of discrete gap breathers in the two-dimensional diatomic face-centered square lattice.

Existences and stabilities of bright and dark breathers in a general one-dimensional discrete monatomic Chain

A general one-dimensional discrete monatomic model is investigated by using the multiple-method. It is proven that the discrete bright breathers （DBBs） and discrete dark breathers （DDBs） exist in this model at the anti-continuous limit, and then the concrete models of the DBBs and DDBs are also presented by the multiple-scale approach （MSA） and the quasi-discreteness approach （QDA）. When the results are applied to some particular models, the same conclusions as those presented in corresponding references are achieved. In addition, we use the method of the linearization analysis to investigate this system without the high order terms of ε. It is found that the DBBs and DDBs are linearly stable only when coupling parameter χ is small, of which the limited value is obtained by using an analytical method.

Discrete gap breathers in a diatomic K2-K3-K4 chain with cubic nonlinearity

The discrete gap breathers （DGBs） in a one-dimensional diatomic chain with K2-K3-K4 potential are analysed. Using the local anharmonicity approximation, the analytical investigation has been implemented. The dependence of the central amplitude of the discrete gap breathers on the breather frequency and the localization parameter are calculated. With increasing breather frequency, the DGB amplitudes decrease. As a function of the localization parameter, the central amplitude exhibits bistability, corresponding to the two branches of the curve ω = ω（ζ）. With a nonzero cubic term, the HS mode of DGB profiles becomes weaker. With increasing K3, the HS mode of DGB profiles becomes weaker and a bit narrower. For the LS mode, with increasing K3, the central particle amplitude becomes larger, and the DGB profile becomes much sharper. But, as k3 increases further, the central particle amplitude of the LS mode becomes smaller.

On some classes of two-dimensional local models in discrete two-dimensional monatomic FPU lattice with cubic and quartic potential

This paper discusses the two-dimensional discrete monatomic Fermi- Pasta-Ulam lattice, by using the method of multiple-scale and the quasi-discreteness approach. By taking into account the interaction between the atoms in the lattice and their nearest neighbours, it obtains some classes of two-dimensional local models as follows： two-dimensional bright and dark discrete soliton trains, two-dimensional bright and dark line discrete breathers, and two-dimensional bright and dark discrete breather.

Compact-like discrete breather and its stability in a discrete monatomic Klein-Gordon chain

This paper studies a discrete one-dimensional monatomic Klein Gordon chain with only quartic nearest-neighbour interactions, in which the compact-like discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. Introducing the trying method, it proves that compact-like discrete breathers exist in this nonlinear system. It also discusses the linear stability of the compact-like discrete breathers, when the coefficient （β） of quartic on-site potential and the coupling constant （K4） of quartic interactive potential satisfy the given conditions, they are linearly stable.

Two-vibron bound states in the β-Fermi-Pasta-Ulam model

This paper studies the two-vibron bound states in the β- Fermi Pasta-Ulam model by means of the number conserving approximation combined with the number state method. The results indicate that on-site, adjacent-site and mixed two-vibron bound states may exist in the model. Specially, wave number has a significant effect on such bound states, which may be considered as the quantum effects of the localized states in quantum systems.