Stability Of Breathers Research Articles

Breathers and thermal relaxation in Fermi-Pasta-Ulam arrays.

Breather stability and longevity in thermally relaxing nonlinear arrays depend sensitively on their interactions with other excitations. We review numerical results for the relaxation of breathers in Fermi-Pasta-Ulam arrays, with a specific focus on the different relaxation channels and their dependence on the interparticle interactions, dimensionality, initial condition, and system parameters.

Interplay of nonlinearity and geometry in a DNA-related, Klein-Gordon model with long-range dipole-dipole interaction.

Most of the studies on mathematical models of DNA are limited to next neighbor interaction. However, the coupling between base pairs is thought to be caused by dipole interaction, and, when the DNA strand is bent, the distances between base pairs become shorter, therefore the interactions with distant base pairs have to be taken into account. In this paper we analyze the existence and stability of breathers, i.e., localized oscillations in a simple model of bent DNA with long-range dipole interaction. Breathers have been suggested as precursors of the denaturation bubble.

Discrete breathers in anisotropic ferromagnetic spin chains

We prove the existence of discrete breathers (time-periodic, spatially localized solutions) in weakly coupled ferromagnetic spin chains with easy-axis anisotropy. Using numerical methods we then investigate the continuation of discrete breather solutions as the intersite coupling is increased. We find a band of frequencies for which the one-site breather continues all the way to the soliton solution in the continuum. There is a second band, which abuts the first, in which the one-site breather does not continue to the soliton solution, but a certain multi-site breather does. This banded structure continues, so that in each band there is a particular multi-site breather which continues to the soliton solution. A detailed analysis is presented, including an exposition of how the bifurcation pattern changes as a band is crossed. The linear stability of breathers is analysed. It is proved that one-site breathers are stable at small coupling, provided a non-resonance condition holds, and an extensive numerical stability analysis of one-site and multisite breathers is performed. The results show alternating bands of stability and instability as the coupling increases.

Numerical study of breathers in a bent chain of oscillators with long-range interaction

Most of the studies of breathers in networks of oscillators are limited to nextneighbour interaction. However, long-range interaction becomes critical when the geometry of the chain is taken into account, as the distance between oscillators and, therefore, the coupling, depends on the shape of the system. In this paper we analyse the existence and stability of breathers, i.e. localized oscillations in a simple model for a bent chain of oscillators with long-range interaction.

Existence and stability of quasiperiodic breathers in the discrete nonlinear Schrödinger equation

We show that the discrete nonlinear Schrödinger (DNLS) equation exhibits exact solutions which are quasiperiodic in time and localized in space if the ratio between the nonlinearity and the linear hopping constant is large enough. These quasiperiodic breather solutions, which also exist for a generalized DNLS equation with on-site nonlinearities of arbitrary positive power, can be constructed by continuation from the anticontinuous limit (i.e. the limit of zero hopping) of solutions where two (or more) sites are oscillating with two incommensurate frequencies. By numerical continuation from the anticontinuous limit, some quasiperiodic breathers are explicitly calculated, and their domain of existence is determined. Using Floquet analysis, we also show that the simplest quasiperiodic breathers are linearly stable close to the anticontinuous limit, and we determine numerically the stability boundaries. The nature of the bifurcations occurring at the boundaries of the stability and existence regions, respectively, is investigated by analysing the band structure of the corresponding Newton operator. We find that the way in which the breather stability and existence is lost depends qualitatively on the ratio between its frequencies. In some cases the two-site breather becomes unstable with respect to a pinning mode, so that applying a small perturbation results in a splitting of the breather into one pinned and one moving part. In other cases, the breather develops an extended tail as some harmonic of its frequencies enters the linear phonon band and becomes a `phonobreather', which was found to be linearly stable in some domain of parameters.

Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators

We prove existence and practical stability of breathers in chains of weakly coupled anharmonic oscillators. Precisely, for a large class of chains, we prove that there exist periodic solutions exponentially localized in space, with the property that, given an initial datum (with ) close to the phase space trajectory of the breather, then the corresponding solution remains at a distance from the above trajectory, up to times growing exponentially with the inverse of , being a parameter measuring the size of the interaction among the particles. This result is deduced from a general normal form theorem for abstract Hamiltonian systems in Banach spaces, which we think could be interesting in itself.

Stability of breathers and destructive soliton–soliton collisions in a perturbed nonlinear Schrödinger equation

We consider soliton–soliton interactions in the damped ac-driven nonlinear Schrödinger equation, which is the simplest model of charge-density wave conductor or of a cold plasma driven by an external ac electric field. Analyzing perturbatively the collision of two solitons with zero initial velocity at infinity, we demonstrate that after the collision the solitons separate at a finite velocity, i.e., a two-soliton breather does not exist in this model. We corroborate this analytical prediction by direct numerical simulations. We also demonstrate that soliton–soliton collisions in the model considered are practically always destructive: after collision, the solitons find themselves kicked out from the stable states, phase-locked to the ac drive, and finally they fully decay under the action of dissipation. Using the results obtained, we calculate a contribution of the solitons to the rate of absorption of energy of the ac field, which is the most important experimentally observable characteristic of the above-mentioned ac-driven systems.

Stability of breathers in the ac-driven sine-Gordon system

We study the stability of breathers in the damped sine-Gordon system driven by an external ac force in the case when the amplitude of the driving field is spatially inhomogeneous. Physical applications are e.g. an ac-driven long Josephson junction and ac-driven magnetic solitons. We consider analytically the conditions for the stability of the driven breather against phase and position disturbances. The results are illustrated by numerical simulations.