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We investigate the spectral properties of one-dimensional spatially modulated nonlinear phononic lattices, and their evolution as a function of amplitude. In the linear regime, the stiffness modulations define a family of periodic and quasiperiodic lattices whose bandgaps host topological edge states localized at the boundaries of finite domains. With cubic nonlinearities, we show that edge states whose eigenvalue branch remains within the gap as amplitude increases remain localized, and therefore appear to be robust with respect to amplitude. In contrast, edge states whose corresponding branch approaches the bulk bands experience de-localization transitions. These transitions are predicted through continuation studies on the linear eigenmodes as a function of amplitude, and are confirmed by direct time domain simulations on finite lattices. Through our predictions, we also observe a series of amplitude-induced localization transitions as the bulk modes detach from the nonlinear bulk bands and become discrete breathers that are localized in one or more regions of the domain. Remarkably, the predicted transitions are independent of the size of the finite lattice, and exist for both periodic and quasiperiodic lattices. These results highlight the co-existence of topological edge states and discrete breathers in nonlinear modulated lattices. Their interplay may be exploited for amplitude-induced eigenstate transitions, for the assessment of the robustness of localized states, and as a strategy to induce discrete breathers through amplitude tuning.

Condensed matter is thermodynamically unstable in a vacuum. That is what thermodynamics tells us through the relation showing that condensed matter at temperatures above absolute zero always has non-zero vapour pressure. This instability implies that at low temperatures energy must not be distributed equally among atoms in the crystal lattice but must be concentrated. In dynamical systems such concentrations of energy in localized excitations are well known in the form of discrete breathers, solitons and related nonlinear phenomena. It follows that to satisfy thermodynamics such localized excitations must exist in systems of condensed matter at arbitrarily low temperature and as such the nonlinear dynamics of condensed matter is crucial for its thermodynamics. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

We numerically investigate out-of-equilibrium stationary processes emerging in a Discrete Nonlinear Schrödinger chain in contact with a heat reservoir (a bath) at temperature T L and a pure dissipator (a sink) acting on opposite edges. Long-time molecular-dynamics simulations are performed by evolving the equations of motion within a symplectic integration scheme. Mass and energy are steadily transported through the chain from the heat bath to the sink. We observe two different regimes. For small heat-bath temperatures T L and chemical-potentials, temperature profiles across the chain display a non-monotonous shape, remain remarkably smooth and even enter the region of negative absolute temperatures. For larger temperatures T L , the transport of energy is strongly inhibited by the spontaneous emergence of discrete breathers, which act as a thermal wall. A strongly intermittent energy flux is also observed, due to the irregular birth and death of breathers. The corresponding statistics exhibit the typical signature of rare events of processes with large deviations. In particular, the breather lifetime is found to be ruled by a stretched-exponential law.

We study a one-dimensional Sine-Gordon lattice of anharmonic oscillators with cubic and quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the one-dimensional Sine-Gordon lattice no matter whether the nonlinear interaction is cubic or quartic. 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 and chaotic discrete breathers by changing the amplitude of the driver.

Clusters of discrete breathers in graphene and graphane are studied by means of molecular dynamics simulations. For both structures, two-breather and three-breather clusters are considered. Energy exchange between discrete breathers in clusters is strongly dependent on the initial conditions such as initial amplitude and phase. Even small changes in these initial parameters can lead to the considerable changes in the behavior or breather clusters.

The modulational instability associated with discrete breathers in 2D quantum ultracold atoms is studied by using the Glauber’s coherent state combined with a semi-discrete approximation and multiple-scale methods. The linear stability analysis exhibits an intriguing threshold amplitude and instability regions associated with modulational growth rate. In addition, we demonstrate a coexistence of two bright intrinsic localized modes namely, the radial symmetric and bilateral symmetric modes, at the center and at the edges of the Brillouin zone, respectively, by alternating the on-site parameter interaction. Numerical investigations reveal a good agreement with the theoretical analysis.

A nonlinear chain with sixth-order polynomial on-site potential is used to analyze the evolution of the total-to-kinetic-energy ratio during development of modulational instability of extended nonlinear vibrational modes. For the on-site potential of hard-type (soft-type) anharmonicity, the instability of q = π mode ( q = 0 mode) results in the appearance of long-living discrete breathers (DBs) that gradually radiate their energy and eventually the system approaches thermal equilibrium with spatially uniform and temporally constant temperature. In the hard-type (soft-type) anharmonicity case, the total-to-kinetic-energy ratio is minimal (maximal) in the regime of maximal energy localization by DBs. It is concluded that DBs affect specific heat of the nonlinear chain, and for the case of hard-type (soft-type) anharmonicity, they reduce (increase) the specific heat.

The work presents a comparison of characteristics of discrete breathers (DB) excited in the close-packed two-dimensional Morse crystal with two different values of the potential stiffness parameter. The dependence of DB frequency on its amplitude for both crystals is calculated.

Discrete breathers in graphene are studied by means of ab initio calculations using methods of density functional theory. It is shown that in the graphene under uniaxial strain applied in “zigzag” direction discrete breathers exist with frequencies inside the gap of phonon spectrum of the system. Breathers have been observed polarized along the “armchair” direction of graphene. The frequency on the amplitude dependency of studied dynamical objects corresponds to the soft type of nonlinearity.

By method of molecular dynamics discrete breathers with soft and hard types of non-linearity in the Pt3Al crystal are studied. Obtained graphs characterize the spatial localization of the discrete breathers, the dependence of the frequency on the oscillation amplitude and the amount of energy localized on the discrete breathers. It is shown that the collision of moving discrete breathers may lead to the energy exchange between the breathers with some portion of the energy given to the Pt3Al lattice. The probability of excitation of discrete breathers in thermodynamic equilibrium is estimated.