Dark Breathers Research Articles

Asymptotic analysis of combined breather–kink modes in a Fermi–Pasta–Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi–Pasta–Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [G. James, Existence of breathers on FPU lattices, C. R. Acad. Sci. Paris 332 (2001) 581–586]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schrödinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to the case for traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.

Dynamics of dark breathers in lattices with saturable nonlinearity

The problems of the existence, stability, and transversal motion of the discrete dark localized modes in the lattices with saturable nonlinearity are investigated analytically and numerically. The stability analysis shows existence of regions of the parametric space with eigenvalue spectrum branches with non-zeroth real part, which indicates possibility for the propagation of stable on-site and inter-site dark localized modes. The analysis based on the conserved system quantities reveals the existence of regions with a vanishing Peierls-Nabarro barrier which allows transverse motion of the dark breathers. Propagation of the stable on-site and inter-site dark breathers and their free transversal motion are observed numerically.

Existence of localized solutions in the parametrically driven and damped DNLS equation in high-dimensional lattices

Instead of using the homoclinic orbit approach, which was commonly taken when studying the localized solutions of the discrete non-linear Schrödinger (DNLS) equation in one-dimensional lattices, we apply the continuation theorem to investigate the existence, stability, and spatial complexity of the localized solutions, including bright breathers, dark breathers, and anti-phase breathers, of the parametrically driven and damped DNLS equation in high-dimensional lattices. In particular, we prove that the sufficient condition that the driving strength exceeds the damping constant is necessary for the system with weak coupling to possess localized solutions.

Generation of solitons by a boxlike pulse in the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

The creation of solitons of the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions from a boxlike initial pulse is considered. The inverse scattering transform is used. An equation for soliton eigenvalues is obtained. It is shown that simultaneous generation of breathers (solitons with internal oscillations) and one-parametric (nonoscillating) bright or dark solitons is possible. For some sets of initial parameters only breathers or only one-parametric solitons emerge. No more than three bright solitons can emerge (possibly, simultaneously with breathers) in any case, while dark solitons or breathers can arise in arbitrary number when the corresponding thresholds are exceeded. Analytical estimates for the number of generated solitons and the corresponding thresholds are given in some particular cases.

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.

Bright and dark breathers in Fermi-Pasta-Ulam lattices

In this paper we study the existence and linear stability of bright and dark breathers in one-dimensional $\mathrm{FPU}$ lattices. On the one hand, we test the range of validity of a recent breathers existence proof [G. James, C. R. Acad. Sci., Ser. I: Math, 332, 581 (2001)] using numerical computations. Approximate analytical expressions for small amplitude bright and dark breathers are found to fit very well exact numerical solutions even far from the top of the phonon band. On the other hand, we study numerically large amplitude breathers nonpredicted in the above cited reference. In particular, for a class of asymmetric $\mathrm{FPU}$ potentials we find an energy threshold for the existence of exact discrete breathers, which is a relatively unexplored phenomenon in one-dimensional lattices. Bright and dark breathers superposed on a uniformly stressed static configuration are also investigated.

Dark breathers in Klein-Gordonlattices. Band analysis of their stability properties

Discrete bright breathers are well known phenomena. They arelocalized excitations that consist of a few excited oscillatorsin a lattice and the rest of them having very small amplitude ornone. In this paper we are interested in the opposite kind oflocalization, or discrete dark breathers, where most of theoscillators are excited and one or a few units of them have verysmall amplitude. We investigate, using band analysis,Klein-Gordon lattices at frequencies not close to the linearones. Dark breathers at low coupling are shown to be stable forKlein-Gordon chains with soft on-site potentials and repulsivedispersive interaction, and with hard on-site potentials andattractive dispersive interactions. At higher coupling darkbreathers lose their stability via subharmonic, harmonic oroscillatory bifurcations, depending on the model. However, mostof these bifurcations are harmless in the sense that theypreserve dark localization. None of these bifurcations disappearwhen the system is infinite. Dark breathers in dissipativesystems are found to be stable for both kinds of dispersiveinteraction.