| Abstract |
Non-linear waves where observed by J. Scott Russell in 1845 and discussed it in the ``Report of the British Association for the Advancement of Science''. What Russell observed was a single solitary wave which was propagating without dispersion along a water channel. As he mention in his report the lonely wave could travel along the channel for miles without dissipation. The mathematical proof for the existence of such solitary waves came in 1895 by D.J. Korteweger and G. de Vries. They proposed a non-linear wave equation for the description of wave propagation in shallow water. As they discovered, the equation they proposed could sustain isolated localized traveling solution, the same type like the solitary wave observed by Russell. The lonely localized waves observed by Russell where called ``Solitons''. In 1955 E. Fermi, J. R. Pasta and S. M. Ulam in their attempt to study the equipartition of energy between 64 particles, weakly coupled by a non-linear interaction, they discovered that starting with one particle excited, the energy distributed itself over the whole system modes but returned almost completely to the initial state. Thermodynamical equilibrium was not reached and the excitation seamed to be stable in that sense. The discovery of Fermi, Pasta and Ulam for the importance of localized solution into the thermodynamical properties of a non-linear system, initiate lot of interest in the study of solitons. Soliton type propagating waves have been found theoretically and experimentally in a large variety of systems in several fields of science like hydrodynamics, optics, elementary particle physics, electromagnetism, condensed matter physics, and biology. Several soliton models have been proposed for the explanation of strange behavior in every system where the nonlinearity is present. An other breakthrough in the non-linear science came with the work of Henry Poincare. In his attempt to investigate the stability of the solar system, he discovered lots of mathematical as well as geometrical and topological tools for the study of non-linear systems. The work of Poincare was continued by Lorenz, Feigenbaum, Kolmogorov, Arnold, Moser, Lyapunov and many others and lead into the discovery of Chaos and the development of the theory of the non-linear dynamics. A relatively new development in the theory of non-linear dynamical systems is the discovery of discrete breathers. They were initially observed in 1988 by A.J. Sievers and S. Takeno and they where described as spatially localized and time periodic solutions in systems of weakly coupled and non-linear oscillators. Discrete breathers where proven later (1994) by R.S. MacKay and S. Aubry, that they are exact solutions of a large variety of systems. Together with the existence proof the MacKay-Aubry theorem gave a numerical technique for the construction and study of these solutions. Since then, lot of interest has been initiated into further study. Discrete breathers have been proven numerically to be long live and linearly stable solutions in many systems and they can affect their thermodynamical and other properties. It is also found that they can be mobile under certain conditions and therefore can play the role of energy carriers. In the last two years there are also experimental verifications for their existence in system like Josephson junctions and coupled wave-guides. It seems therefore that discrete breathers are present in every system as long as it is discrete and non-linear. It has been shown that in systems of coupled non-linear oscillators, discrete breathers exist, are exponentially localized in space and periodic in time solutions. Since the oscillators are non-linear, apart of the main frequency $omega _b$, there are in the Fourier spectrum of the oscillation all the harmonics $nomega _b$. Since they are periodic in time, their stability can be investigated using the Floquet stability analysis for periodic orbits. It has been found that they are linearly stable is most of the cases, depending on the frequency and the other system parameters. The linear stability means that once they are created they remain in the system for a long time and they are not destroyed when they are perturbed by a small perturbation. If the frequency of the breather lye inside the linear phonon spectrum, then the resonance with the phonons leads to the distraction of the breather and the excitation of small amplitude and extended phonons. Not only the basic frequency but also its harmonics when they are in resonance with the phonons lead to the same effect. For investigating the localization and its relation with the frequency of the breathers we use the rotating wave approximation and the approximation that only one oscillator is non-linear while the others are linear. The last approximation is valid because due to the exponential localization of the breather, only the central particles are oscillating with large amplitude while the others are oscillating in the linear part of the potential. The comparison of the approximate results with the numerically obtained solutions shows that they are in good agreement when the breather frequency harmonics are not close to the phonon band. In terms of these two approximations, we compare the discrete breathers with the well known linear impurity modes studied earlier and we find that the localization in both cases are related. Although discrete breathers exist in many systems, properties such as the stability and the mobility depends on the geometry of the system as well as the details of the interactions. Even the aloud breather frequencies vary, depending on the details of the model. For example the stability and the mobility of a breather is altered when impurities are injected into the system. Therefore for the better understanding of the influence they can have in the physical properties of the system it is necessary to investigate several models with different interactions and (or) different lattice geometries. The experimental observation of discrete breathers in Josephson Junction ladder and the possibility of the existence of discrete breathers in macromolecules which cannot be consider as one-dimensional, lead as to study quasi one-dimensional lattice geometries and how they affect the stability and the mobility of DB. As it is found the lattice geometry affects their stability as well as the mobility. As soon as two one dimensional chains are linked with some inter-chain coupling the stability of the breathers is changed. Except of the stability, the mobility properties are different. The single breather becomes unstable and a double breather (which involves large amplitude oscillations in both chains) becomes stable through a pitchfork bifurcation. After the bifurcation, the double breather it is possible to become mobile while the single breather vanishes. The possibility of the existence of discrete breathers in surfaces or in thin films, leads as to study their existence and their mobility in two dimensional lattices. Similar bifurcations, like in the quasi-one-dimensional chain, occur when we introduce impurities into the system. In this case the single breather bifurcates with the multibreathers which have the impurity site excited. After each bifurcation, the single breather disappear together with one of the multibreathers while the other one becomes the dominant solution in the system. The impurities also modify the mobility of the breathers, since a single mobile breather cannot pas through the impurity without loss of energy, some times it is absorbed (depending on the energy of the breather and its velocity as well as the impurity) while other times it is reflected. Another interesting problem where discreteness and nonlinearity is involved is the case of hydrogen bonds and several mathematical models have been proposed for their study. We consider two of them and we show that it they exhibit discrete breather solutions. Since in some cases they are also mobile, they can be associated with the proton transfer in chains of hydrogen bonds. Discrete breathers are oscillations in a nonlinear medium, mobile breathers therefore can be consider as nonlinear waves. We investigate how these nonlinear waves interact with an other type of nonlinear waves that they can exist in the same systems like topological solitons ie. kinks. As we found there is an effective attraction when a breather and a kink are close. The interactions of a kink with a mobile breather exhibit a large variety of different behaviors. Depending on the energy as well as in the model parameters, the kink acts in some cases as a wall reflecting the breather while in other cases it absorbs the breather and radiate its energy into phonons or transform it into kinetic energy. In other cases it has been observed that a bound state between them with the breather oscillating in a small distance from the center of the kink.
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