Stability Of Multibreathers Research Articles

Multibreather stability in discrete Klein–Gordon equations: Beyond nearest neighbor interactions

Abstract We present results on multibreather stability in one-dimensional nonlinear Klein–Gordon chains. Our analysis is based on Aubryʼs band theory and perturbation theory. First, we provide an alternative proof of the stability of multibreathers in a chain with nearest neighbor interactions only. Then, we extend our analysis to the case of interactions with up to three neighbors. For next-nearest neighbor and third-nearest neighbor interactions we also extend the theory to study the stability properties of recently found multibreathers that have nonstandard phase shifts (not equal to 0 or π).

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

We study the existence and stability of multibreathers in Klein–Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in- or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard (0 or π) phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either next-nearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined.

Stability of non-time-reversible phonobreathers

Non-time-reversible phonobreathers are nonlinear waves that can transport energy in coupled oscillator chains by means of a phase torsion mechanism. In this paper, the stability properties of these structures have been considered. An analytical study has been performed for low-coupling solutions based upon the so-called multibreather stability theorem previously developed by some of the authors (Archilla et al 2003 Physica D 180 235). A numerical analysis confirms the analytical predictions and gives a detailed picture of the existence and stability properties for arbitrary frequency and coupling.

On the stability of multibreathers in Klein–Gordon chains

In this paper, a theorem, which determines the linear stability of multibreathers excited over adjacent coupled oscillators in Klein–Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive nearest–neighbour inter-site coupling, only structures with adjacent sites excited out of phase may be stable, while only in-phase ones may be stable for negative coupling. The situation is reversed for hard nonlinearities. This method can be applied to n-site breathers, where n is any finite number and provides a detailed count of the number of real and imaginary characteristic exponents of the breather, based on its configuration. In addition, an estimation of these exponents can be extracted through this procedure. To complement the analysis, we perform numerical simulations and establish that the results are in excellent agreement with the theoretical predictions, at least for small values of the coupling constant ε.

Effect of the introduction of impurities on the stability properties of multibreathers at low coupling

Using a theorem dubbed the multibreather stability theorem (Archilla J F R et al 2003 Physica D 180 235) we have obtained the stability properties of multibreathers in systems of coupled oscillators with on-site potentials, and an inhomogeneity. Analytical results are obtained for two-site and three-site breathers, multibreathers, phonobreathers and dark breathers. The inhomogeneity is considered both at the on-site potential and at the coupling terms. All the results have been checked numerically with excellent agreement. The main conclusion is that the introduction of an impurity does not alter the stability properties.

Demonstration of the stability or instability of multibreathers at low coupling

Whereas there exists a mathematical proof for one-site breathers stability, and an unpublished one for two-site breathers, the methods for determining the stability properties of multibreathers rely on numerical computation of the Floquet multipliers or on the weak nonlinearity approximation leading to discrete nonlinear Schrödinger equations. Here we present a set of multibreather stability theorems (MST) that provides a simple method to determine multibreathers stability in Klein–Gordon systems. These theorems are based in the application of degenerate perturbation theory to Aubry’s band theory. We illustrate them with several examples.